Download Chemical Dynamics, Thermochemistry, and Quantum Chemistry

Physical Chemistry
Laboratory Experiments
Chemistry 361 & 362
Chemical Dynamics,
Thermochemistry, and
Quantum Chemistry
Jay Baltisberger
Fall 2000 – Spring 2001
Table of Contents
LABORATORY 1 (QUANTITATIVE COMPUTATION)............................................................................................ 4
ERROR BARS AND ERROR ANALYSIS ...................................................................................................................... 4
LABORATORY 2 (LITERATURE WORK) .................................................................................................................. 7
CURRENT ARTICLE REVIEW....................................................................................................................................7
LABORATORY 3 (THERMOCHEMISTRY)................................................................................................................ 8
DETERMINATION OF THE HEAT CONTENT OF COAL................................................................................................ 8
LABORATORY 4 (THERMOCHEMISTRY/SPECTROSCOPY) ............................................................................ 12
THERMODYNAMICS OF RHODAMINE B LACTONE-ZWITTERION EQUILIBRIUM ..................................................... 12
LABORATORY 5 (THERMOCHEMISTRY).............................................................................................................. 15
THE BINARY LIQUID-SOLID PHASE DIAGRAM OF NAPHTHALENE AND P-DICHLOROBENZENE ............................. 15
LABORATORY 6 (THERMOCHEMISTRY).............................................................................................................. 18
A LIQUID BINARY PHASE SYSTEM........................................................................................................................18
LABORATORY 7 (DYNAMICS)................................................................................................................................... 21
A SIMPLE REDUCTION/OXIDATION REACTION AND CHEMICAL KINETICS USING VISIBLE SPECTROSCOPY......... 21
LABORATORY 8 (DYNAMICS/SPECTROSCOPY) ................................................................................................. 24
MEASUREMENT OF LONGITUDINAL RELAXATION TIMES (T1) FOR13C IN ETHYLBENZENE ................................... 24
LABORATORY 9 (THERMOCHEMISTRY).............................................................................................................. 30
MEASUREMENT OF THE NO2 DIMERIZATION EQUILIBRIUM CONSTANT ............................................................... 30
LABORATORY 10 (QUANTUM CHEMISTRY) ........................................................................................................ 33
DETERMINATION OF MOLECULAR STRUCTURE OF HCL / DCL / CH4 ................................................................... 33
LABORATORY 11 (QUANTUM CHEMISTRY) ........................................................................................................ 38
PARTICLE IN A BOX, HÜCKEL MOLECULAR ORBITAL AND GAUSSIAN98 ANALYSIS OF CYANINE DYE
MOLECULE SPECTRA ............................................................................................................................................ 38
LABORATORY 12 (QUANTUM CHEMISTRY) ........................................................................................................ 42
DETERMINATION OF THE POTENTIAL ENERGY SURFACE OF I2 .............................................................................. 42
LABORATORY 13 (STATISTICAL MECHANICS) .................................................................................................. 47
3D ISING SPIN LATTICE MODEL FOR PHASE TRANSITIONS................................................................................... 47
LABORATORY 14 (DYNAMICS/SPECTROSCOPY) ............................................................................................... 56
EXCHANGE RATE MEASUREMENT ON N,N DIMETHYLACETAMIDE USING SPIN SATURATION NMR
SPECTROSCOPY ..................................................................................................................................................... 56
LABORATORY 15 (DYNAMICS/SPECTROSCOPY) ............................................................................................... 61
IMAGING AND DIFFUSION MEASUREMENTS USING PULSED FIELD GRADIENTS WITH NMR SPECTROSCOPY ....... 61
LABORATORY 16 (THERMOCHEMISTRY)............................................................................................................ 64
MEASUREMENT OF THE HEAT CAPACITY RATIO FOR A NON-IDEAL GAS USING THE ADIABATIC EXPANSION
METHOD ............................................................................................................................................................... 64
LABORATORY 17 (QUANTUM CHEMISTRY) ........................................................................................................ 67
A SIMPLE MEASUREMENT OF FLUORESCENCE QUENCHING OF QUININE WITH NACL ......................................... 67
2
LABORATORY 18 (QUANTUM CHEMISTRY) ........................................................................................................ 68
STRUCTURE ANALYSIS OF A STEROID MOLECULE USING MULTI-DIMENSIONAL PFG NMR SPECTROSCOPY ...... 68
LABORATORY 19 (QUANTUM CHEMISTRY) ........................................................................................................ 72
QUADRUPOLAR INTERACTIONS IN NMR SPECTROSCOPY ..................................................................................... 72
LABORATORY 20 (QUANTUM CHEMISTRY) ........................................................................................................ 78
DIPOLAR COUPLINGS MEASURED IN PARTIALLY-ORIENTED SOLUTIONS USING NMR SPECTROSCOPY .............. 78
LABORATORY 21 (QUANTUM CHEMISTRY) ........................................................................................................ 80
STRONGLY J-COUPLED SPIN SYSTEMS IN NMR SPECTROSCOPY..........................................................................80
LABORATORY 22 (THERMOCHEMISTRY)............................................................................................................ 82
MEASUREMENT OF THE JOULE-THOMPSON COEFFICIENT ..................................................................................... 82
REFERENCES ................................................................................................................................................................. 85
3
Laboratory 1 (Quantitative Computation)
Error Bars and Error Analysis
Maple Introduction
For this lab you will use Maple® on the Macintosh PPC to do a complete error propagation
for two different problems. The write-up should go right into your notebook as you would any other
laboratory experiment. The conclusion in each case should be the final answer reported with a 95%
confidence interval. Remember that in general for a function F(x,y,z,...) the error will be expressed:
 F (x, y,z,...) 



x
 y,z ,...
F
=
 F ( x, y,z,... )
+

y

 x, z,...
2
2
2
x
 F ( x, y, z,... )
+


z
 y, x ,...
2
2
y
2
z
+
(1.1)
Notice that partial derivatives hold all variables explicitly constant, thus making life simple. The σx,
σy and σz are the standard deviations for each of the variables x, y and z respectively. Maple commands you may find useful include:
with(stats):
Will load the stats package into memory.
diff(f,x);
Will calculate explicit partial derivative of f
with respect to x.
subs(vars,f);
Will substitute variable expressions var i n t o a
function f where variable expressions are given by
x = ####, y = ####, etc.
sqrt(f);
Calculate square root of a function f.
evalf(f,d);
Will evaluate as a floating point number a function f to d decimal places.
describe[standarddeviation](l);
This function will calculate the standard deviat i o n o f a l i s t o f n u m b e r s l. This requires
with(stats).
describe[mean](l); Will calculate the mean of a list of numbers l.
This requires with(stats).
:=
definition operator (used to assign a value or expression to a variable name).
=
equality expression used when defining an equation
or a substitution.
[##,##,##,...];
Used to define a list of numbers.
solve(eqns,vars);
Used to solve one or more equations given by eqns
for variables given by vars.
simplify(f);
Used to mathematically simplify an expression f.
expand(f);
Used to expand polynomials in expression f.
exp(f);
Take exponential of an expression f.
I
Capital I used to signify imaginary portion of a
number.
4
int(f,x=a..b);
Integrate a function f with respect to variable x
from limits a to b (which may even be Infinity).
Part A
You are given a cylindrical object. You set out to determine it’s mass with high accuracy by
determining its volume and then using its density. Recall the formula for the volume of a cylinder:
V = π r2 h
(1.2)
You make 7 measurements of the diameter using a Vernier caliper and the data is listed below:
122.31, 121.92, 123.01, 122.77, 122.35, 122.88, 121.78 mm
You measure the height of the object with the same calipers:
312.12, 311.26, 311.94, 313.01, 313.31, 312.81, 312.93 mm
You should, find the average value of the height, the radius and calculate an average volume.
Propagate errors through completely. Assuming that the density is 4.6211 ± 0.0033 g/ml, calculate
the mass of your object, again propagating error. Note the ±0.0033 represents a 2 standard deviation
error (about 95% confidence). What factor of improvement would you expect in the overall error in
mass if we had taken twice as many measurements of height, diameter, and then both? (Assuming
that the non-trivial implication of taking the increased number of measurements is that the overall
standard deviation for each measurement went down by a factor of 2.)
Part B
Suppose you are going to measure an equilibrium constant for an acid dissociation (by analyzing a
saturated H2A solution).
H 2 A (aq) 
→ 2 H+ (aq) + A2− ( aq)
(1.3)
Suppose you measured the pH with a meter and got the following values:
pH = 3.21, 3.23, 3.35, 3.11, 3.25, 3.29, 3.18, 3.13, 3.31
Recall that pH = – log [H+]. Now suppose we additionally measure the concentration of the H2A
spectroscopically. The measured absorbances were:
A = 0.313, 0.351, 0.322, 0.327, 0.320, 0.319, 0.329, 0.340, 0.308
Calculate the equilibrium constant for the reaction using A = k [H2A] and k = (3.51±0.01) x 102 L
mol–1. Propagate all errors through final K.
[H ] [ A ]
+ 2
K =
2-
[H 2 A ]
5
(1.4)
As in part A, determine which measurement is giving the greatest contribution to the overall error
observed for the equilibrium constant, K. Describe how the value for K and its error might change
if this measurement were improved by a factor of 3 (reduce the standard deviation of this measurement by a factor of 3.)
Part C
Given the formula below :
A=
b2 e C / d1
/ 2
(1.5)
The value of b was measured to be (cm):
5.231, 5.331, 6.011, 5.112, 5.523, 5.911, 5.783, 5.651
The value of c was measured to be (unitless):
0.5231, 0.4412, 0.4951, 0.3991, 0.5123, 0.5411, 0.4801, 0.4512
The value of d was measured to be (cm2):
102.52, 103.21, 104.51, 100.13, 110.23, 99.23, 95.21, 111.23, 115.23
Using this data calculate the average values for b, c and d as well as standard deviations for these. In
addition calculate the average value for A as well as a 95% confidence interval for this value.
Consider a calibration data set previously generated for A (cm) as a function of t (hrs) given below.
A = 4.634
8.312 11.852 15.152 18.621 22.092 25.907 30.012
t = 1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
Using Statview calculate the time at which the data was collected (estimate error appropriately). As
in the previous part A and B, you should determine which measurement is responsible for the largest portion of the error in the final time measurement. Describe what you might do to try to improve this situation and get a better measurement for the final value of time.
6
Laboratory 2 (Literature Work)
Current Article Review
Self directed literature search.
You will find a paper concerning a topic in Physical Chemistry.
Journals suggested are the Journal of Chemical Education, Journal of the American
Chemical Society, Journal of Physical Chemistry, Science or Nature.
You will read the paper and discuss it amongst yourselves (groups of 3).
You may discuss the paper with me.
You will present (1 week from today) a talk of length 15 minutes on this topic.
Your talk should address the following points
Overall idea of experiment
What sort of apparatus was used and how did it work
What sort of data analysis was conducted
What overall conclusions are reached
What problems exist in the article
You shall copy the article so everyone in the class has a copy of it.
7
Laboratory 3 (Thermochemistry)
Determination of the Heat Content of Coal
Introduction
This experiment is an adaptation of one published by Mueller and McCorkle1. A traditional
physical chemistry experiment is to determine the heat of combustion of a substance using a Parr
oxygen bomb and a technique called bomb calorimetry. It is particularly useful to know the heat of
combustion (∆CH) of a fossil fuel such as coal, especially if one is involved in industry. It is important to be able to predict how effective coal is as an energy source. Coal is used to generate heat to
boil water and ultimately produce electricity (recall Carnot cycles and impact this had on the industrial revolution in England), therefore it is essential to be able to predict how much coal is
needed to produce a certain amount of heat. Bomb calorimetry is a good method to measure the
heat of combustion of coal. Since the technique is essentially the same no matter what the sample,
this experiment will also provide training that will be applicable whenever it is necessary to measure the heat of combustion or formation of another substance.
Experimental Procedure
Sample pellets may be prepared by grinding the naphthalene or benzoic acid standard or
coal unknown into a powder using a grinder or mortar and pestle. Then the powder can be placed
into a gelatin capsule to be used as a sample pellet in the bomb. An alternative method for preparing
the samples is to use a pellet press to compress the sample into a tight tablet shaped pellet which can
be burned in the bomb. It should be noted that the coal pellets have a tendency to fall apart which
would affect your data, thus the capsule should be used. Once these pellets have been prepared the
bomb calorimeter operation procedure below can be followed.
Calorimeter
1.
Make sure both the bomb and bucket are clean and dry. May use acetone to clean the stainless steel bomb, however, all vapor must be allowed to evaporate before tests are conducted.
2.
Get exactly 2.000 L of distilled water in volumetric flask(s) and fill bucket.
3.
Set up digital thermometer for monitoring the temperature of water in the bucket. If stirrer is
available, set this up in bucket as well.
4.
Prepare a standard sample of 0.8±0.1 g benzoic acid (∆CU = – 26.412 kJ / g and ∆CH =
–26.422 kJ / g) or 0.5±1 g naphthalene (∆CU = – 39.581 kJ / g and ∆CH = –40.232 kJ / g).
Normally pellets are prepared, however a gelatin capsule may be used as well. If a gelatin
capsule is to be used, two empty capsule runs must be completed in addition to the two standard runs before attempting any unknown material.
8
5.
Wire must be weighed carefully both before and after each run, to determine the mass which
has been oxidized. The heat of combustion (complete oxidation) of the fuse wire is ∆U =
–5.89 kJ / g.
6.
A 1.000 mL drop of distilled water should be placed in the bottom of the bomb calorimeter
before the start of the experiment, this insures that there will be no additional water vapor
produced in the combustion reaction (all water produced will be in the liquid form, which
has a heat of formation of –243.83 kJ / mol).
7.
Prepare fuse and pellet in holder. Wire should barely touch the top of the pellet or capsule.
Make certain that the fuse holder contains the wire securely. The combustion crucible
should be tilted to point away from the electrodes and the top of the bomb to prevent ignition of these parts.
8.
Seal the bomb itself by screwing the top until hand tight. DO NOT USE A WRENCH FOR
THIS.
9.
Fill the bomb with O 2 to a pressure of 25-35 atm. To do this, first turn the main regulator for
the O2 cylinder on, making sure that the main pressure valve is closed. Make sure that the
exit port on the bomb is closed and the adapter is attached properly. Upon filling once to 25
atm, the bomb should be vented slowly and refilled to 25 atm. This will have the effect of
diluting the N 2 in the bomb by a factor of 625 = 252 rather than just 25. This means that the
N2 percentage (the most abundant gas impurity) should be less than 0.2%.
10.
Carefully place the bomb in the bucket. Attaching the ignition leads to the correct terminals
(note the grounded terminal is the one on the left when reading the number correctly and has
a small “g” symbol near it.) Take care to avoid splashing any water out and around the
bucket as this could change the effective heat capacity.
11.
Record the temperature of the bomb/calorimeter system every 30 seconds for approximately
5 to 10 minutes, allowing the bomb and water to equilibrate. This will allow you to establish
an initial temperature baseline (it is acceptable if this has a slight rise or fall by a few degrees).
12.
Press the ignition switch and hold it down for 2 seconds. The indicator light should come on
briefly and then go back off. It comes on to indicate that the circuit is complete and that current is passing through the ignition wire. It goes out when ignition begins and the wire
burns/breaks.
13.
Continue to record the temperature. Readings should be taken approximately every 30 seconds. The temperature should within a minute or so undergo a 1 to 2 degree jump. Continue
monitoring the temperature for a 5 to 10 minute period following this jump. Do not stop
monitoring temperature until the slope of the temperature versus time curve is reasonably
constant (i.e. each time step the temperature changes by a constant increment).
14.
The bomb should be allowed to cool for about 10 minutes then the pressure should be released VERY slowly. Once released, you may open the bomb and check to see that combustion was complete. Rinse all parts first with deionized water and then acetone to remove any
acid formed from nitrogen or sulfur atoms in the combusted molecule as well as any residual
9
carbon deposits. In theory there should be no char left in the vessel, if there is, this represents a potential source of error from incomplete combustion.
15.
The remaining wire from the fuse should be carefully removed and reweighed. Weigh only
the unoxidized portion of the wire (not the metal oxide balls you may find in the container).
This mass will be used to compute the heat released due to combustion of the wire.
Experimental Procedure
1.
Run a calorimeter standardization using benzoic acid or naphthalene as per the above instructions.
2.
Plot the temperature versus time data on a full scale graph. Draw straight lines (not necessarily flat) through both the initial and final temperature baselines. These need not be parallel to one another or have zero slope.
Figure 1.1 Temperature versus time graph for a bomb combustion run with ignition occurring
between 300 and 400 seconds.
3.
Draw a vertical line which follows the intermediate points and marks out equal area above
and below the curve (as seen in figure 1.1 above). This line intersects the two baselines and
this separation is the ∆T used in the calculations.
4.
Calculate the amount of heat released by both the wire (qwire) and the benzoic
acid/naphthalene (qstd) which were combusted.
5.
Use the equation qstd + qwire = Ccal ∆Tstd to calculate Ccal for the bomb calorimeter.
10
6.
Run the same experiment and measure ∆Tcoal for an unknown coal sample (approximately
0.2 to 0.5 g sample).
7.
All ∆Tcoal measurements should be conducted two or three times for averaging purposes.
You may calculate the qcoal (which is ∆CU) from the equation qwire + qcoal = Ccoal ∆Tcoal using
the unknown ∆Tcoal.
8.
To convert to ∆CH from ∆CU you need to consider the ∆p V portion. First convert from ∆CU
to ∆CUm using the initial mass of the sample. Second, look at the balanced equation and determine the number of gas phase product molecules minus the number of gas phase reactant
molecules. The ideal gas law tells us that ∆p V = ∆(nRT) which means if we assume temperature is constant (roughly true) then ∆p V = RT ∆n. Where ∆n is the change in number of
gas molecules, that is ∆CHm = ∆CUm + RT ∆n. For the coal sample, you will need to make an
assumption about the approximate molecular formula of coal (C200H400) so that you can write
a balanced equation which accounts for the number of carbon dioxide and water product
molecules.
9.
Then error analysis should be completed, propagating all errors completely. The final result
should summarize the ∆CH and ∆fH of coal (using in both kJ / g and kJ / mol) and give 95%
confidence error limits.
11
Laboratory 4 (Thermochemistry/Spectroscopy)
Thermodynamics of Rhodamine B Lactone-Zwitterion Equilibrium
Introduction
This experiment is an adaptation of one published by Hinkley and Seybold2. This experiment uses a commercially available dye, rhodamine B, to study the thermodynamic concepts of
enthalpy change, entropy change, and Gibb’s free energy change. The dye is dissolved in 1propanol. Thus, an equilibrium between lactone (L) and zwitterion (Z) can be established. This
equilibrium is known to vary with temperature (as do most equilibria for which there is a reaction
energy difference.) The fact that the lactone is colorless and the zwitterion is red allows a convenient spectroscopic approach to measuring the equilibrium position as a function of temperature.
Et 2N
O
NEt 2
Et 2N
O
NEt 2
k
O
CO
COO
Lactone
Zwitterion
Knowing the mole fractions of each species, we can determine several important thermodynamic
quantities (∆rxnH°, ∆rxnS°, and ∆rxnG°) for the lactone-zwitterion equilibrium.
Experimental Procedure
There are a variety of solvent molecules which may be used to dissolve the rhodamine B
molecule. The choice of solvent affects intimately the lactone/zwitterion ratio and for this experiment we recommend using one of the solvents listed in the table below. A solution of rhodamine B
(concentration approximately 10 µM) should be prepared by dissolving 0.2 g of rhodamine B in 50
mL of the solvent. A table
Solvent
Mole Fraction Z (at 25˚C) K (at 25˚C)
is given below which lists
Ethanol
0.706
2.40
the mole fraction of zwit1-propanol
0.652
1.88
terion in a variety of sol2-propanol
0.289
0.41
vents at 25˚ C.
1-butanol
0.541
1.18
A UV/Vis spec2-methyl-1-propanol
0.562
1.28
trometer will be used to
12
find the absorbance of the rhodamine B solution over a range of temperatures from 15 °C to 60 °C
and over a range of wavelengths containing the maximum absorbance. An electronically controlled
water bath is used to control the temperature of the solution in the spectrometer sample compartment.
A table should be generated that includes the following data at each temperature: A (absorbance), x Z, x L, K, and ∆rxnG°. Also all calculations should be shown and include graphs used to determine the temperature independent assumed parameters ∆rxnH° and ∆rxnS°. The following equations relate the absorbances measured to the various parameters.
A(100%) =
A(25˚C )
%Z (25˚C )
(4.1)
where A(100%) is the absorbance one would measure if all of the sample were in the colored zwitterion form. The concentration of the zwitterion form (labeled Z), is given by the equation:
[ Z] =
A(T ) ( 25˚C)
[ RhB]
A(100%) (T )
(4.2)
where ρ(25°C) and ρ(T) are solvent densities at 25°C and some temperature T respectively. The difference in densities is assumed to be negligible, that is ρ(25°C) ≈ ρ(T). Assuming that the density is
constant over the temperature range of which the experiment is performed [Z] becomes:
[ Z] =
A(T )
[ RhB]
A(100%)
(4.3)
[L ] = [RhB] − [ Z]
The equilibrium constant, K, can be easily seen from the equilibrium equation to be:
xZ
[Z ] =
[Z ]
=
[L] [ RhB] − [ Z ] 1− xZ
[Z ]
xZ =
[ RhB]
K=
(4.4)
Then ∆rxnG° can be calculated by the following:
∆Go = –RT ln K
The following equation can be compared to the least squared line of the plot ln K vs. 1/T in order to
determine ∆rxnH˚ and ∆rxnS˚ using the equation below:
ln K =
∆So ∆H o
−
R
RT
13
(4.6)
Errors should be calculated as per standard error propagation for K (at each temperature), ∆rxnH°,
∆rxnS°, and ∆rxnG° (at each temperature) of the lactone-zwitterion equilibrium.
14
Laboratory 5 (Thermochemistry)
The Binary Liquid-Solid Phase Diagram of
Naphthalene and p-Dichlorobenzene
Introduction
This experiment is an adaptation of one published by Blanchette3 and another by Calvert et
al.4 This is also described in a standard physical chemistry laboratory manual5. In this experiment,
we are interested in studying the heterogeneous equilibrium between the liquid and the solid phase
in a two component system. The naphthalene and p-dichlorobenzene are chosen because they are
inexpensive, safe chemicals and their melting points are accessible with boiling water bath. This
experiment provides adequate information to determine the enthalpies of fusion (∆fusionH˚), en-
Figure 5.1 Temperature versus mole fraction phase diagram in the melting region for a pdichlorobenzene/naphthalene mixture.
tropies of fusion (∆fusionS˚), Gibb's free energy of fusion (∆fusionG˚), freezing point depression constants (K f) and ultimately produce a binary liquid-solid phase diagram for the mixture of naphthalene and p-dichlorobenzene.
The naphthalene and p-dichlorobenzene mixture will melt for all compositions below the
100˚C temperature produced from a boiling water bath. As the mixture cools, different cooling
curves with specific thermal arrests and breaks are obtained for each different composition. A thermal arrest (a region where the temperature remains constant for a period of time) indicates the so15
lidification temperature of a pure compound. When a liquid of the pure solid A is cooled, the temperature falls until solid A begins to form. The temperature remains constant until solidification is
complete. This curve is called thermal arrest. A break indicates the temperature at which first solid
appears at the given composition. When a mixture of A and B is cooled, the temperature falls until
solid (or solid solution) of A or B begins to form. This curve is called a thermal break. The solidliquid phase diagram is obtained by
making plots of break and arrest temperatures against the mole fraction.
The heat of fusion, entropy of fusion,
and freezing-point-depression constant for each chemical will be calculated from the data obtained by examining how the initial solid fusion
temperature depends upon concentration of solute in the bulk solvent
phase. The eutectic temperature will
be determined by finding the eutectic
freezing region for a variety of compositions. This temperature should be Figure 5.2 Example cooling curve showing various break
the same for each composition stud- points on a liquid-solid binary phase diagram.
ied.
Experimental
The apparatus used in this experiment utilized several components illustrated to the right. A
13 x 100 mm test tube and a two hole stopper to fit
the test tube should be used. The stopper can be
modified to accommodate a copper tubing stirrer
and the digital thermometer probe. An Erlenmeyer
flask, 250 mL, is to be used along with a rubber
collar to hold the test tube after heating, as an air
jacket. The heating should be done by a boiling
water bath (not shown.)
A total of nine cooling curves should be
measured (with both the initial temperature break
point and the eutectic thermal arrest temperature
seen). The table that follows shows the various
mole fractions that should be used for the nine
compositions. It is useful to see that we will use a
pure naphthalene mixture to obtain the high mole
16
Figure 5.3 Experimental setup to measure
cooling curves..
fraction mixtures. For the low mole fraction mixtures a second tube (or the original cleaned) will be
used that starts with a pure p-dichlorobenzene mixture. For each sample, the mixture will be heated
in the test tube immersed in a boiling water bath. This 100˚C temperature will be sufficient to melt
the binary mixture at all compositions. Once heated, remove the test tube from the water bath and
allow it to cool in the Erlenmeyer flask while taking a temperature reading every fifteen seconds.
Temperature readings should be taken until a constant temperature is reached.
Run
1
2
3
4
5
6
7
8
9
Naphthalene (g)
1.000
1.000
1.000
1.000
1.000
0.000
0.200
0.400
0.600
p-Dichlorobenzene (g)
0.000
0.200
0.400
0.600
0.800
1.000
1.000
1.000
1.000
Mole Fraction Napthalene
1.00000
0.85151
0.74142
0.65654
0.58909
0.00000
0.18658
0.31449
0.40763
Table 5.1 Nine sample mixtures to be used for cooling curves. Masses need not be exactly as
indicated in the table but should be close to each point so that all mole fractions are covered
relatively equally.
A separate cooling curve should be drawn for each of these nine (or more) compositions.
This may be done on the same graph paper for ease of locating the eutectic temperature if needed.
Real cooling curves (as opposed to theoretical curve shown in figure 5.2) will not exhibit perfectly
sharp changes in slope. It is important to plot the data accurately and to extrapolate the slope from
the middle of each region to find the exact temperature of each break/arrest point. Once the temperatures for each break/arrest point are determined, this data should be plotted to produce a phase
diagram similar to figure 5.1. Note that you will need to extrapolate the liquid/solid lines to determine the exact eutectic composition. In addition, you should fit these curves (on each side of the
eutectic) using the Gibbs-Helmholtz equation to extract the ∆fusionH and ∆fusionS for each pure substance. Using the ∆fusionH you may then calculate the freezing point depression coefficient for each
substance as well. For all values, appropriate estimates of 95% confidence intervals should be determined.
17
Laboratory 6 (Thermochemistry)
A Liquid Binary Phase System
Introduction
The procedure for the experiment of solubilities of liquids in a binary two-phase system is
taken from an article by Arthur M. Halperm and Saeed Gozashti6 The classic example of a two
phase liquid-liquid system is vinegar
and oil salad dressing. This type of
two phase system is in fact substantially more complicated than simply
a separation of the two chemicals
into pure phases. Instead, both
phases are dilute solutions of each
compound in the other. The composition of each solution is dependent
on temperature and ultimately may
even exhibit an upper or lower critical temperature where the phase
separation disappears. Although
Figure 6.1 Liquid-Liquid Phase Diagram for water/n-butanol.
there are several different techniques available to obtain the information needed to produce a two
phase liquid-liquid diagram (see figure 6.1 above), one of the easiest was described by A. E. Hill in
1923. In this process the extraction of each individual liquid phase is not required, instead they are
left together and the relative volumes are compared. This approach, called the thermostatic method,
is based on volumetric techniques in which only the overall bulk composition of the system as a
whole is needed. Using this technique the mole fractions of water and n-butanol in each phase can
be obtained. The two liquids are weighed and combined in varying ratios for a number of samples.
Subsequently, the relative volumes of each phase are measured for each sample over a range of
temperatures. Given that in a two phase system the molarity of component A in the B phase (and
similarly for the molarity of B in the A phase, the molarity of B in the B phase, and the molarity of
A in the A phase) will be equal for all compositions:
18
N
MCP = 1MAB = 2 MAB = 3M AB = 4M AB = 5M AB = MAB
MBB = 2MBB = 3MBB = 4MBB = 5MBB = MBB
1
MAA = 2MAA = 3MAA = 4M AA = 5M AA = M AA
1
(6.1)
MB = 2MB = 3MB = 4M B = 5M B = M B
1
A
A
A
A
A
A
where N signifies the sample identification, P the phase, and C the component. Thus:
nc = Mcp V p + Mcp V p
(6.2)
na = M AAV A + MAB V B
(6.3)
nb = M BAV A + MBB V B
(6.4)
where nc is the number of moles of component c in a given sample, P defines the phase, Mcp is the
molarity, and VP is the volume. By measuring the molarities of the components and the volumes of
the phases the moles and the mole fractions can be found. Thus, a phase diagram can be produced.
Procedure
A series of five samples should be prepared in which the mole fraction of n-butanol in water
ranges from 0.200 to 0.800. You should not prepare samples with mole fractions outside this range
since at elevated temperatures these may form a single phase which would necessitate not using
data from that sample from that temperature and above. The samples should be carefully weighed
so that you know exactly how many moles of both n-butanol and water are present in each. Each
sample will be placed in a high precision (better than ±0.1 mL precision) 10 mL graduated cylinder
fitted with a glass top. It is recommended that no more than about 7 mL total be filled into each
graduated cylinder to make sure that expansion does not increase the volume above the 10 mL
maximum for each cylinder. These five samples should be sealed with the glass tops and parafilm
to ensure that material enters or leaves the graduated cylinders over the course of the experiment.
For each temperature, the five cylinders will be immersed in a temperature controlled water
bath. It is critical that the cylinders be maintained at a constant temperature and allowed to equilibrate for at least 10 minutes before any measurements are conducted. In addition at the lower temperatures, it is important that the samples be inverted to stir the solutions to guarantee that there are
no concentration gradients present in the sample as the system achieves equilibrium concentrations.
At the higher temperature, thermal convection will help to stir the mixtures and the increased solubilities of the water and n-butanol will help to guarantee equilibrium is reached faster. An alternative approach is to use an ultrasonic stirring bath as a temperature controlled bath if one is available
that can hold all five samples. For increased accuracy, a sixth graduated cylinder should be in19
cluded which contains a known quantity of water so that you may determine the expansion coefficient for the graduated cylinders as you raise the temperature (and account for this in your final
phase diagram.) Once temperature equilibrium is established for each cylinder, you should read the
volumes (to a precision of ±0.01 mL via interpolation if possible) of both phases in the tube. You
should take care to note which is the water rich and which is the n-butanol rich phase (the relative
positions of these phases will not change from tube to tube or temperature to temperature.) This
process should be repeated at temperatures of 20˚C, 30˚C, 40˚C, 50˚C, 60˚C and 80˚C.
Data Analysis
When you review equations 6.3 and 6.4 you may note that these may be transformed in the
following manner:
B
na
A
B V
=
M
+
M
A
A
VA
VA
B
nb
A
B V
A = MB + MB
VA
V
(6.5)
(6.6)
This form suggests that a graph of nB / VA or nA / VA versus the ratio of the volumes, VB / VA. If these
plots are done for the five data points, the slope and intercept will give the concentrations of water
and n-butanol for each of the phases. This should be reported with appropriate error bars using
StatView, Excel or other similar program. These concentrations should then be compiled into a
phase diagram similar to the one in figure 6.1.
20
Laboratory 7 (Dynamics)
A Simple Reduction/Oxidation Reaction and Chemical Kinetics
Using Visible Spectroscopy
Introduction:
This experiment is an adaptation of one published by Elias and Arden7. The experiment
discussed in this paper is a simple oxidation of iodide ions by peroxodisulfate ions. The reaction is
quick and easy to perform; this experiment has advantages in that one can directly observe the increase in absorbance over time due to the formation of the I 3− ion by using a standard visible spectrometer (such as a Spectronics 20.) The reaction between iodide and the peroxodisulfate ion is
S2 O82−
+ 2I − → 2SO 24 − + I2
(7.1)
Subsequently each I2 reacts with the excess I to produce a triiodide ion, I 3− .
–
I2
+ I− 
→ I 3−
(7.2)
This experiment can be performed effectively by obtaining the Spec 20 absorbances of each solution (made with differing concentrations of peroxodisulfate and iodide ions) at 353nm. The build up
curves will appear essentially linear (rather than showing any curvature due to changes in concentrations of the peroxodisulfate and iodide ions) since the molar absorptivity of the triiodide is so
high and the reaction is relatively slow, maintaining [ I3− ] << [ I− ] for the initial moments of the reaction. This initial rate will relate to a rate law of the form given below:
d [I2 ]
dt
= k [S2 O82- ] [I− ]
n
m
(7.3)
where the rate constant, k, as well as the rate law exponents, n and m, will be determined from this
experiment.
Procedure:
You should first prepare solutions of 0.1M KNO3 (to be used as a
diluting agent to maintain constant
ionic strength in all solutions,) 0.02M
K2S2O8 in 0.1M KNO3, and 0.04M KI
in 0.1M KNO3. The wavelength on
Sample
A
B
C
D
E
F
G
0.02 M K2S2O8
4 mL
4 mL
4 mL
4 mL
1 mL
2 mL
3 mL
0.04 M KI
1 mL
2 mL
3 mL
4 mL
4 mL
4 mL
4 mL
0.1 M KNO3
3 mL
2 mL
1 mL
0 mL
3 mL
2 mL
1 mL
Table 7.1 Sample mixtures suggested for rate law determination of
K2S2O8 / KI reaction (note all have identical total volume and identical ionic strengths.
21
the Spec 20 should be adjusted to 353nm and the instrument should be set to 100% transmission
using 0.1 M KNO3 as the blank. Seven or more experiments should be preformed (as suggested in
table 7.1) In experiments A-D the peroxodisulfate concentration is held constant and the iodide ion
concentration is varied. In experiments D-G, the iodide ion concentrations are held constant and the
peroxodisulfate ion concentration is varied. Note that the sample labeled D is used in both sets. In
each experiment, the 0.1 M KNO 3 was first added to the cuvet, the 0.02 M K2S2O8 was added next,
and the 0.04 M KI was added last. The KI should always be added last since once the solution is
mixed it begins to react immediately to produce triiodide ions. After the addition of the KI, the cuvet was inverted once and then quickly placed in the Spec 20 and the reaction's absorbance was
measured every thirty seconds for five minutes.
Data and Analysis:
The time resolved absorbances should be plotted using a program such as Statview or Excel.
These graphs should be roughly linear (with slight curvature at the higher concentrations). The initial slope of this line (which has units of absorbance per unit time) should be extrapolated back to
zero time and determined along with appropriate error bars. The rate equation 7.3 can be manipulated below:
 d [ I 2 ]
 = lnk + n ln[S 2O82- ] + m ln [I − ]
ln 
 dt 
(7.4)
Modifying this by using the knowledge that A = [ I 3− ] b where e is the molar absorptivity of I 3− and
b is the pathlength for the cell. This allows us to modify equation 7.4 below:
 d [ I 2 ]
 d A b )
dA
 = ln (
ln 
= ln  − ln( b)
 dt 
 dt 
 dt 
 dA
ln   = ln( k b) + nln[S2 O82− ] + mln[I − ]
dt
(7.5)
This suggests that by plotting the logarithm of the rate versus either the logarithm of the peroxodisulfate ion (samples D-G) or the iodide ion (A-D). The slope of the resulting line will be either the
n or m respectively. Ideally the error bars on these parameters should allow you to identify the
closest integer value for both n and m. If these parameters are well established, the multi-step rate
law may or may not be identified. Once the n and m values are determined the plot of rates versus
the concentration product [S2 O82− ] [I − ] . The slope of this line will correspond to the product k b.
n
m
The product of b may be measured by mixing 0.100 mL of the 0.02 M K2S2O8 solution with 5.0
mL of the 0.04 M KI solution and dilute the solution to a total volume of 8.0 mL with the 0.1 M
22
KNO3 solution (allow this to stand for at least 15 minutes for the reaction to proceed to completion.) The final triiodide concentration will then be 2.5x10–4 M and the produce b may be determined by taking the observed absorbance of this solution and dividing by the final triiodide concentration. Your conclusion should report the complete rate law, including the rate constant along
with error bars on the appropriate parameters.
23
Laboratory 8 (Dynamics/Spectroscopy)
Measurement of Longitudinal Relaxation Times (T1)
For13C in Ethylbenzene
Introduction:
This experiment is an adaptation of one published by Fuson and others8,9. The measurement of relaxation times in NMR spectroscopy is an important experiment to learn a number of
things about a molecule. First off, when running NMR spectra the time that must be waited for relaxation to occur limits the ultimate rate of data acquisition and ultimately the signal-to-noise ratio
which may be achieved. Secondly, the relaxation times give insight into the molecular motions present in a molecule as well as the bulk reorientation of a molecule in solution. For 1H NMR spectra,
the T1 relaxation times are normally in the 2-10 s time range whereas the 13C NMR T1 relaxation
times are much longer (in the 10-60 s range).
It is surprising to think that such a low energy
event as a nuclear spin flip requires such a
long time to reach equilibrium after a perturbation. The answer to this problem stems
from the difference in rates of spontaneous
versus stimulated emission. As demonstrated
by the Einstein coefficients, the rate of spontaneous emission increases greatly with energy of that emission. In a very low energy
situation (flipping nuclear spins) the rate of
spontaneous emission is essentially zero. For
nuclear spins to reestablish equilibrium
populations, there must be photons present at
exactly the Larmor frequency which can
Figure 8.1 Relaxation time as a function of temperature. The minimum in the T1 curve arises when
the spectral density cuts out at the Larmor frequency.
cause stimulated emission. This radiation frequency arising from black body radiation is very low
in intensity (black body radiation has maximum intensity in the infrared region at room temperature.) The source of radiation to cause relaxation must therefore arise from non-black body mechanisms (i.e. specific molecular motions.)
Ethylbenzene provides an excellent test case to gain insight into relaxation processes. This
molecule has a very well established and easily interpreted NMR spectrum and possesses both
24
internal (methyl group rotation, ethyl group libration) and external (bulk rotation) molecular
motions. The key to relaxation of an NMR transition is the spectral density (the number of
photons per unit time) at the Larmor frequency.
If this spectral density if high then the nuclear
spins will relax more quickly (see temperature
Tb in figure 8.3) while if the spectral density is
low then the nuclear spins will relax slowly.
Figure 8.2 Structure of Ethylbenzene. Note that
both single bonds are free to rotate and will tend
to reduce the overall correlation time for both
the methyl and ethylene protons.
Note that below the Tb temperature, the relaxation time will increase very quickly whereas
above the temperature Tb the relaxation time also
increases but less quickly.
These features are seen in
figure 8.1 where the graph
has a steep slope on the left
while a less steep slope on
the right at higher temperature. Essentially the
spectral density is a function which measures the
correlation time for a molecule (the mean time for
structural information to be
lost due to molecular motions.) The formula which
describes the spectral density in terms of the correlation time of a specific dipolar interaction is shown
Figure 8.3 Dependence of spectral density on frequency, parameterized by
temperature. Temperature Ta represents a temperature where the molecular
motions are faster than the Larmor frequency (high temperature limit), Tb
represents the T1 minimum temperature and Tc a low temperature distribution. Note that the frequency scale is logarithmic
in equation 8.1 below.
J( ) =
2
1+
c
2
2
c
25
(8.1)
This expression is essentially related to a black box model in which oscillators from zero frequency
up to some frequency approximately 1/τc are roughly equally populated. In systems which have
different molecular subunits (such as methyl groups) which have a second correlation time different
than the overall isotropic correlation time will have a modified formula for the overall spectral density at a given nuclear position. This spectral density relates to the overall longitudinal relaxation
time for a given site, T1,i, due to random fluctuations of like spins (protons relaxing other protons)
by formula 8.2 below, where
0
1 3
=
T1,i 2
is the Larmor frequency.
4
I ( I +1) ∑ J ik (
2
0
) + Jik (2 0 )
(8.2)
k
If the spins are heteronuclear (relaxation of a carbon site by a proton) then the equation for the relaxation time is modified.
3
1
=
T1, I 2
2 2
I S
2
[
S( S +1) ∑
s
1
12
J IS (
I,0
−
S ,0
)+
3
2
J IS (
I,
)+
0
3
4
J IS (
I,0
+
S ,0
)]
(8.3)
For each nuclear site in the molecule the T1 will be strongly affected by the spectral density at that
site. For a molecule like ethylbenzene, the spectral density will be different for the methyl group
Figure 8.4 Saturation Recovery (top) and Inversion Recovery (bottom) pulse sequences. To the left are
sequences and the right are the theoretical data buildup curves. In the case of the inversion recovery the
standard inversion pulse has been replaced with a more efficient composite pulse.
than the aromatic portions. In particular, it is expected that the effective correlation time for a
methyl group will be much shorter than other regions and this short correlation time will result in a
26
wider frequency range for the spectral density and an overall reduced spectral density. This will
reduce the transition rate for the methyl group relative to the aromatic region.
Experimental:
The experimental NMR methods used to study relaxation times are either an inversion recovery sequence or a saturation recovery sequence, shown in figure 8.4. Note that the inversion recovery sequence is actually a modified version in which the initial 180˚ pulse has been replaced by
a composite pulse. The saturation recovery method is the easiest to understand. The series of
pulses applied at the beginning of the sequence are used to saturate the spin transitions of interest
(i.e. the protons on ethylbenzene). By applying a series of pulses the spins are continually disturbed. Note that for a stimulate absorption/emission event, there must be a population difference
between the two energy levels. After a large number of pulses, the rate of emission and absorption
will be effectively averaged as the populations of the energy levels are equalized. The delay that
follows is referred to as the recovery delay. During this period of time, the spins return (to some
degree) to their equilibrium populations. By observing the signal recovered as a function of this
delay we may map out a saturation recovery curve (shown to the right of the pulse sequence in figure 8.4) Since we are merely destroying the population difference the equilibrium buildup curve
will follow a simple exponential growth model leading to the equation 8.4 which describes this
buildup curve.
S(
rec
) = S (∞ )[1 − e −
rec
T1
]
(8.4)
In the case of the inversion recovery sequence, the theory is somewhat more complicated. First, the
inversion recovery sequence is coherent, meaning we actually track the magnetization and not just
the population difference. The initial pulse inverts the magnetization from the +z to the –z axis.
While the magnetization is initially aligned in the –z direction, it immediately begins to return to its
equilibrium position in the +z direction. This recovery follows an exponential buildup in much the
same way as the saturation recovery method. The equation 8.5 is modified to account for the initial
inversion as opposed to the saturation.
S(
rec
) = S (∞ )[1 − 2e −
rec
T1
]
(8.5)
The sequence shown in figure 8.4 is one that includes a composite pulse to replace the standard
180˚ inversion pulse. You may notice that this composite pulse is actually three pulses applied in
rapid succession with phase shifts (90x – 180 y – 90 x). The effect of this pulse is identical for an on
resonance spin (the magnetization is rotated from the +z to –z axis). The benefit of the composite
27
pulse is seen when comparing the overall magnetization rotation for an off-resonance spin versus a
non-composite pulse. For the straight 180X pulse, the off resonance magnetization is rotated about
an axis slighly tilted towards the Z axis from the X axis. The effect is that the magnetization appears to over-rotate beyond 180 degrees and misses the bottom of the sphere (the –z direction) and
instead actually passes the bottom and starts to go up giving an inversion population of this spin of
less than 100% (in some bad case is may be much less than 100%). This will seriously affect the
measured T1 values since the initial starting signal will not be equal and opposite to the infinite time
one. By using a composite pulse sequence made up of a 90X then 180Y then 90X pulse, you may
actually achieve a nearly 100% inversion for even very large offsets. In
the figure 8.5 you will notice that the
magnetization of the off resonance
spin moves away from the Y axes towards the X axis initially. This would
be bad if we completed the 180 degrees of rotation but instead, we stop
at 90. The second pulse rotates this
from the +x side to the –x side along
the surface of the sphere. From this
final point, the last 90X pulse can rotate the magnetization towards the –z
a x i s . T h e 1 8 0Y pulse added just
enough compensation so that the second 90X pushed the magnetization
Figure 8.5 Composite pulse inversion using a 90X–180Y–90X
sequence. Note that on resonance this is identical to a 180X, but
for an off resonance spin, the magnetization follows a somewhat
more complicated trajectory. This trajectory still leads to nearly
perfect inversion which an off-resonance 180 would fail to do.
nearly perfectly onto the –z axis.
The overall use of composite pulses will improve the fitting of the experimental inversion
recovery data substantially. If the inversion were less than 100% the equation 8.5 would need to be
modified with the addition of a third adjustable parameter (inversion efficiency) which ranges from
0 to 2 and precedes the exponential term. For the experiment you will conduct, the 13C and 1H relaxation times of ethylbenzene shall be measured using the composite pulse inversion recovery sequence described. The data will be analyzed on the NMR spectrometer using the software available
on the instrument. Once you have the T1 values for each spin site in the molecule, you can then
construct the effective spectral densities for each site. As a simplification, we can assume for a
small molecule such as this that we are in the fast motion limit (from equation 8.1, ω τc << 1) and
28
therefore the spectral density for a given site is effectively constant over the range of frequencies
used in NMR (i.e. J(ω 0 ) = J(2 ω0) for both the 300 MHz 1H and 75MHz 13C sties). Once the various spectral densities are tabulated for each site, a mean correlation time may be determined for
each portion of the molecule. Note than in the case of specific sites in the molecules you will need
to sum up a number of spectral density contributions from various spin pairs within the molecule.
29
Laboratory 9 (Thermochemistry)
Measurement of the NO2 Dimerization Equilibrium Constant
Introduction:
This experiment is an adaptation of one published by Wettack and modified by others10.
The reaction of nitrogen monoxide with oxygen gas produces nitrogen dioxide that then dimerizes
to form the brown dinitrogen tetroxide. This is a major source of visual irritation in locations of
heavy pollution as well as reacting with water to produce acid rain.
2 N Og
( ) + O2 (g) → 2NO 2 (g)
(9.1)
2NO
→ N 2O 4 ( g)
2 ( g) ←
(9.2)
In a constant volume container, the relative quantities of nitrogen dioxide and dinitrogen tetroxide
will shift as the temperature is changed due to the change in the ∆rxnG for the reaction 9.2 following
the well known equation ∆rxnG = –RT ln K.
Experimental:
To generate a reaction vessel with known quantities of both NO2 (g) and N 2O4 (g), we must
be careful to use volumetric gas handling and not allow the NO (g) to contact O2 (g) until the proper
time in the experiment. The reaction we will use to generate the NO (g) is between copper metal
and concentrated nitric acid.
6H + (aq ) + 3Cu (s) + 2 H N O
→ 2 N Og
( ) + 3Cu 2 + (aq ) + 4H 2O( l)
3 ( g) 
(9.3)
This reaction will produce pure NO (g) but this gas will react with any O2 (g) present in the reaction
vessel immediately to produce the NO2 (g). Essentially we need to keep producing NO (g) in excess until all available O2 (g) is reacted. In this manner we can effectively dilute out any produced
NO2 (g) or N2O4 (g) and thus collect nearly pure NO (g). An alternative approach is to bubble the
NO (g) through concentrated base (NaOH or KOH) and thus eliminate any of the NO2 (g) or N2O4
(g). This NO (g) will be collected in a gas tight syringe (approximately 4 mL) and will be temperature regulated to 25˚ C. A second gas tight syringe will be filled with O2 (g) directly from a
99.99% pure gas cylinder; this syringe should the same volume of O2 (g) as is in the NO (g) syringe. From reaction 9.1, the stoichiometry of these two is exactly 1:1 and when equal volumes of
these are mixed, the product will be exactly the same volume of NO2 (g). This product will almost
30
immediately begin to dimerize to form N2O4 (g) and a faint brown color should appear. By maintaining the product in an airtight syringe and monitoring the volume change as a function of temperature, we may extract the equilibrium constant for the reaction 9.2.
Kp =
pN2 O4
p
2
NO2
=
x N 2 O4 ptotal
2
NO2
x
p
2
total
=
x
x
N 2 O4
2
NO2 total
p
(
)
nNO 2 ,0 − nNO2 Vtotal
n nN2 O4 nN 2O 4 Vtotal
= total
=
=
2
2
n NO
nNO
ptotal
n2NO2 RT
2RT
2
2
(9.4)
The total number of moles of gas will change as the temperature changes and the equilibrium shifts
and this will produce a change in the volume (since the overall pressure is maintained throughout
the experiment.)
nNO2 ,0 =
ntotal
n
ptotal VNO
= nNO 2 + N2O 4
RT
2
p V
= total total = nNO 2 + nN2 O4
RT
(9.5)
Combining these equations, we may solve for the number of moles of NO 2 (g) in terms of the initial
volume of NO (g), VNO, and the equilibrated volume of the mixture, Vtotal.
nNO2 =
ptotal (2VNO − Vtotal )
RT
(9.6)
This may be then substituted back into equation 9.4 to give an expression for the equilibrium constant in terms of the two volumes, VNO and Vtotal.
 ptotal VNO ptotal (2VNO − Vtotal ) 


−
 RT
 Vtotal
RT
(Vtotal − VNO )Vtotal
Kp =
=
2
2
 ptotal ( 2VNO − Vtotal ) 
2RT 2 ptotal ( 2VNO − Vtotal )


RT


(9.7)
This expression is now suitable for calculating the equilibrium constant Kp for each temperature.
The only caveat is that the volumes must be all standardized to a single initial temperature. Therefore for reaction mixtures studied at higher or lower temperature must have the total volumes modified using the ideal gas law per equation 9.8.
adj
Vtotal
=
ntotal RT adj
T adj
= Vtotal
ptotal
T
31
(9.8)
The adjusted volumes may then be used for all calculations of equilibrium constants per equation
9.7. It is clear that the adjusted volumes should never exceed the original volume of NO (g) nor
should it be less than one-half the original volume. If either of these data events were to occur, it
would more than likely indicate that your initial reaction mixture contained some impurities (normally O 2 or H 2O). The second of these (H2O) is particularly insideous as it will react with NO2 (g)
to produce nitric (HNO3) and nitrous acid (HNO2) which is soluble in water and therefore will effectively cause a gas phase mass imbalance. In either case, we may account for these impurities as
long as they do not change over time (i.e. as long as there is no leak into or out of the reaction syringe.)
Data Analysis:
Once the equilibrium constants, Kp, have been determined at each temperature, these may be
converted into ∆rxnG at each temperature. By making a graph of the ∆rxnG versus T, we expect to
see a straight line with a slope of –∆rxnS and an intercept of ∆rxnH (assuming both of these are constant over the temperature range being studied.) Alternatively, a plot of ln Kp versus 1 / T will give
a slope of –∆rxnH / R and an intercept of ∆rxnS / R. In either case, the relative errors for each of
these thermodynamic state functions may be extracted from Statview or Excel as appropriate. This
procedure is very similar to that used in laboratory 4, the study of the rhodamine B equilibrium.
32
Laboratory 10 (Quantum Chemistry)
Determination of Molecular Structure of HCl / DCl / CH4
Introduction:
This experiment is an adaptation of one published in a standard physical chemistry laboratory manual 5,11-14. The determination of bond distances and strengths dates back to the beginning
of chemistry. In early chemistry courses students are given data describing atomic and ionic radii.
These data were collected in a variety of ways, such as crystallographic studies with both X-rays
and neutrons and spectroscopic studies. In this laboratory you will study the molecular structure of
CH4 and/or 1H35Cl, 1H37Cl, 2H35Cl and 2H37Cl with infra-red and near infra-red spectroscopy techniques. This laboratory has been extensively used in many physical chemistry courses by many different instructors. 1-6 The data collected will allow the calculation of the bond length and force constant for each of these molecules.
Figure 10.1 Potential energy surface for rotation and vibrational energy levels. The picture on the right is
zoomed onto the bottom of the surface showing the actual energy levels involved in this experiment.
It is important to discuss the actual energy levels which will be probed with the IR spectroscopic technique. Figure 10.1 shows the potential energy well for a diatomic molecule. This well is
normally modeled with a harmonic potential (shown in the dashed line in the figure.) As such the
vibrational energy levels will be nearly equally spaced (shown on the left.) The rotational energy
levels are shown in the right hand picture of figure 10.1 where the bottom of the potential well is
shown. The rotational levels are additive to the vibrational energy levels and are not spaced evenly
(as expected from equation 10.1.) Remember also that at room temperature, most molecules will
exist in the ground state electronic energy level and to a large extent the ground state vibrational
energy level. Many more of the rotational states will be populated due to the small energy differ33
ences between levels. In the case of a diatomic molecule, there is only one vibrational mode or degree of freedom and two rotational degrees of freedom. These rotational degrees of freedom are degenerate and therefore only one type of rotational energy level structures will appear. The basic
equations describing the energy of a diatomic and spherical molecule are given below.
Etotal = Ee + Evib + Erot
Evib = hc ( +
1
2
)˜
E rot = hcJ ( J + 1) B
(10.1)
,J
The variables which appear in these equations are the
and J quantum numbers which refer to the
quantum vibrational and rotational state of the system. The h is the standard Planck's constant while
the c refers to the speed of light. The variable B
,J
and ˜
represent the rotational and vibrational
interaction constants. These constants however are not unique for all quantum numbers and in fact
may be represented below.
˜ = ˜ e − xe ˜ e ( +
B
,J
= Be −
e
(
1
2
)
(10.2)
+ 12 ) − DJ ( J + 1)
This shows that as the vibrational quantum number gets larger the vibrational interaction constants
get smaller, indicating a potential energy surface which has a different shape than the pure parabolic potential used to approximate it (see figure 10.1.) This comes as a result of the slightly longer
average bond length you encounter as the molecule is found with increasingly larger vibrational
quantum numbers. This feature is referred to as anharmonicity. Secondly, the rotational interaction
constant is related to both the vibrational and rotational quantum numbers. As the vibrational
quantum number gets larger the effective bond distance gets larger as well (as mentioned before),
this leads to an increase in the effective moment of inertia and a reduction in the rotational constant,
where the equilibrium constant is defined below (for a diatomic molecule).
˜
I
,J
=
k
1
2 c
= r
2
,J
B
,J
=
4 cI
,J
(10.3)
m1m2
=
m1 + m2
8
2
The moment of inertia equation 10.3 for CH4 is similar ( ICH4 = mH rCH
) and the expression for the
3
reduced mass used in the vibration is no longer applicable. The equilibrium moment of inertia is
easily related to the equilibrium bond distance in equation 10.3. Additionally, equation 10.2 shows
that the rotational constant is reduced as the rotational quantum number increases. This reduction is
34
called the centrifugal distortion and is due to the added force that pulls the atoms apart as the angular momentum is increased. This is much like the merry-go-round effect where as the velocity is
increased the children are increasingly prone to fly off the wheel. This larger effective intermolecular radius again causes the effective inertia to increase and reduces the rotational constant. In
general the centrifugal distortion term, D, in equation 2 will be dependent on which vibrational
state the molecule is in as well, since higher vibrational states are more susceptible to centrifugal
distortion. This term may be shown (see Herzberg7) to be approximately equal to 4 Be3 ˜ e2 . Additional terms may be included, however for the accuracy and precision of the data collected in this
experiment, these formulae are sufficient to completely describe the system.
In this experiment, since most of the molecules occur in the ground state vibrational energy
level ( = 0) the only transition we can observe is the
= +1. The selection rules for the rotational
levels are more complex, since J = ±1, ±2 or 0 may all be observed in one way or another. The
J = ±2 refer to the Raman transitions and will not be discussed here but the other two refer to the
P, Q and R branches observed in a rotationally resolved IR spectrum. The selection rules are related
to the usual transition dipole moments calculated from the integral below.
fi
= −
e∫
∗
(
f
, J f )r
(
i
, Ji )d
(10.4)
These dipole moments squared lead to the intensity of a given transition. When each of the transitions moments are evaluated in the case of HCl it is found that only the
= +1, J = ±1 are ob-
served. The J = 0 transition is forbidden because the initial and final wavefunctions will have
identical rotational components which will cancel in the integral in equation 4 while the vibrational
wavefunctions will be orthogonal and lead to a zero integral.
With the energy equation 1 we may calculate the IR single quantum energy splitting for the
P branch (
= +1, J = -1) and the R branch (
= +1, J = +1). These IR energy splittings are
given below.
For P Branch
E ( =0→1,J → J −1)
hc
3 ˜1 ˜ 0
−
+ J ( J − 1) B1,J − J ( J + 1) B0,J
2
2
= ˜ e − 2x e ˜ e + ( 2 e − 2 Be ) J − e J 2 − 4 DJ 3
=
35
(10.5)
For R Branch
E ( =0→1,J → J +1)
hc
3 ˜1 ˜ 0
−
+ J ( J − 1) B1,J − J ( J + 1) B0, J
2
2
= ˜ e − 2x e ˜ e + ( 2Be − 3 e + 4D) +
=
( 2Be − 4
e
+ 12 D) J + (12D −
e
(10.6)
) J 2 + 4 DJ 3
These equations may be used to generate the positions of the IR transitions which will be observed.
The overall intensity of each transition will be proportional to the Boltzmann population of the
ground state times the population of the excited state. In the case of the rotational transitions involving J ≠ 0, the degeneracy of the two levels must also be considered. For actually fitting the
experimental data, equations 5 and 6 may be used or alternatively, we may renumber the J quantum
number in the P branch so that m = J and in the R branch so that m = –J – 1. This yields the following expression for the R branch energy splittings as a function of m.
E ( =0→1,−m −1→− m )
hc
3 ˜1 ˜ 0
−
+ m( m − 1) B1,− m − m( m + 1) B0, −m−1
2
2
(10.7)
2
3
= ˜ e (1 − 2x e ) + m (2 e − 2Be ) − m e − 4 Dm
=
This expression may immediately be seen to be equivalent to the expression for the P branch with
these redefined quantum numbers. This allows both branches of data to be fit simultaneously. Note
that there will be no line at m = 0, which corresponds to the location of the Q branch lines.
The actual experiment is conducted by placing HCl and DCl gas in an IR gas cell which has
NaCl plates at both ends which are IR invisible. The gases are placed into the cells either by using
gas cylinders or a chemical reaction to generate the HCl and DCl. In our experiment, we will generate the HCl with the following simple reaction.
CaCl2 (s) + H2 SO 4 (conc.) → CaSO4 (s) + 2 H C lg)
(
(10.8)
Likewise, the DCl is generated with a similar reaction in which D2O is added with the H2SO4. Any
excess water in the concentrated acid solution will be absorbed by the solid precipitate. For our experiment, we will need approximately 2 liters of HCl (DCl) gas. At room temperature this represents approximately 0.1 mol of gas, requiring 0.05 mol of concentrated acid. The CaCl2 should be
supplied in about a 2 or 3 fold excess. The reaction may be performed in a 250 ml Erlenmeyer flask
with a one-hole rubber stopper with a glass tube inserted into it. A rubber hose may be attached to
this tube to supply the gas to the IR gas cell. Be careful not to expose the gas cell to water at any
point in these experiments as this can etch the NaCl plates. The gas cell should initially be filled to
about 1 atm by allowing one stopcock to remain open while blowing gas into the other. If the over36
all pressure in the cell is too great and no isotopic substructure may be seen in the IR spectrum, it
may be necessary to use vacuum rack techniques to fill the cell with less than 1 atm of HCl gas.
The IR spectrum may be taken in either absorption or transmission mode in the region of
interest (about 2800 cm-1 for HCl and about 2100 cm -1 for DCl). It should be possible to observe at
least ten distinct lines (perhaps twenty if the resolution of the spectrophotometer is capable of distinguishing the 35Cl from 37Cl isotopes) in both the P and R branches. The spectrum should be
taken with maximum resolution and expansion for accurate measurement of the peak positions.
Once the peaks have been measured, the spectrum should be assigned and then fit to the energy
function given in equation 7. The four basic spectroscopic constants ( e, Be, e, D) may then be
used to calculate the interatomic distances and bond strengths for each vibrational state as well as
the shape of the potential energy surface near the minimum. Errors should be fully propagated to
determine the overall accuracy of this method. These numbers may then be compared to those
given by Herzberg.7
37
Laboratory 11 (Quantum Chemistry)
Particle in a Box, Hückel Molecular Orbital
and GAUSSIAN98 Analysis of Cyanine Dye Molecule Spectra
Introduction:
This experiment is an adaptation of one published in various forms 15-17 as well as being presented in the standard physical chemistry laboratory manual5. The purpose of this lab is to acquire
the UV/Vis spectra for a series of cyanine dye molecules and then attempt to interpret these spectra
in terms of various levels of theory ranging from the particle in a box model, the Hückel approximations for molecular orbits and finally full blown ab initio calculations with Gaussian94. The basic theory for each of these methods of analysis will not be detailed here, but rather only a rough
outline of the overall procedure will be given.
The series of molecules to be studied come from the cyanine family of dye molecules and
have the general structure shown below (1,1’-diethyl-2,2’-dicarbocyanine Iodide):
The individual molecules to be studied are listed in the table below, along with CAS registration
numbers for each compound.
Name
CAS Registry Number
1,1’-diethyl-2,2’-cyanine iodide
32,376-4
1,1’-diethyl-2,4’-cyanine iodide
28,437-8
1,1’-diethyl-2,2’-carbocyanine iodide
16,651-0
1,1’-diethyl-4,4’-carbocyanine iodide
D9,153-5
1,1’-diethyl-2,2’-dicarbocyanine iodide
39,219-7
1,1’-diethyl-4,4’-dicarbocyanine iodide
39,220-0
The fundamental difference between these molecules is the orientation of the two aromatic end
groups (which are rotated approximately 90˚ in the 4,4’ dyes) and the length of the chain connecting these end groups (either 1, 3 or 5 carbon atoms). These dyes are all extremely strong absorbers
in the visible region and to collect the UV/Vis spectrum you will need to prepare dilute solutions of
38
each. Concentration is not critical, as we will not be determining any molar absorptivity values. All
spectra should be acquired at 25˚ C for standardization purposes.
Once the spectra have been collected, you should identify the λmax in each. This is assumed
to be the highest wavelength peak and hopefully will correspond to the transition from the HOMO
(highest occupied molecular orbit) to the LUMO (lowest unoccupied molecular orbit). To calculate
the energy splitting between these energy levels we shall use three methods, first will be the particle
in a box model. In this model we assume that the electrons (π only) are resonant in a box which
extends linearly over the length of a molecule. The length of this one-dimensional box is determined by assuming an average bond length and calculating the number of bonds over which the
electrons are delocalized. The standard energy expression for the one-dimensional particle in a box
is given by
2
En =
2
hn
8ml 2
(11.1)
where n is the quantum number (1, 2, 3, ...), m is the mass of the electron and l is the length of the
box. The observed transition will be calculated by looking at the number of π electrons and placing
two in each of the levels until you reach the HOMO. For example, if you have 20 π electrons then
the first 10 orbits will be filled and the HOMO will be the n = 10 level. The LUMO will be the
nHOMO + 1 level, giving a transition energy of
h 2 (n + 1)
h2 n2 h 2(2n + 1)
∆En→n +1 =
−
=
8ml2
8ml 2
8ml 2
2
(11.2)
where n is the HOMO level. Calculate the wavelength of the HOMO-LUMO transition for each of
the six dye molecules (making appropriate approximations on the length of the “box” parameter, l).
Compare these calculations to the observed λmax for each molecule.
The second method of data analysis involves using the Hückel approximation method for
calculating the energies of conjugated π orbital systems. In a separate handout, the theory behind
the Hückel approach is described. For each of your dye molecules, you should construct the Hückel
matrix and numerically calculate the eigenvalues (energy levels). These energy levels may be filled
with the same electron numbers as in the previous method and the transition energy will be computed from the HOMO-LUMO energy difference. This is best done in Maple and a sample spread
sheet is included showing the calculation of the eigenvalues for the molecule pyrolle (C4H5N, see
appendix A). Note that in pyrolle, the N atom is included in the calculation via the non-bonding
electrons which would participate in π bonding in this system. There are a total of 6 π electrons
present and thus the first 3 energy levels will be filled. The value chosen for b in this calculation is
39
one which represents an average exchange integral value of 77.5 kcal/mol. The matrices which are
required to calculate the energy of a cyanine dye molecule are substantially larger (20+ individual
eigenvalues) and will require a great deal of care to make sure that the elements are correct. In this
problem, the π bonding difference between an sp2 carbon and an sp2 pyrollic nitrogen are included
by changing the diagonal and off-diagonal elements for integrals involving the nitrogen atoms. The
calculated wavelengths for each dye molecule should again be compared to the experimental values
as well as the values calculated with the particle in a box model.
The third model will be to do a full ab initio calculation on the dye molecule of your choice
using GAUSSIAN98. This is a computer program on the Pentium II LINUX workstation which can
calculate wavefunctions and energy levels using the full Hamiltonian of the molecule within the
Born-Oppenheimer approximation using a density functional theory (DFT) approach. The computation time will take multiple hours at a reasonable level of theory and thus you will only study one
molecule with this program. The results will be given in a lengthy output file which contains more
information then you will need. Dr. Baltisberger will aid in extracting the energy levels of the
HOMO and LUMO and will help you convert the values into a wavelength which may be compared to the other two methods.
40
Sample Maple output for Hückel Theory applied to Pyrolle
with(linalg):
H
e := n*b + a:
k := 1.5:
N
H
l := 0.8:
H
C
size := 5:
huckmat := matrix(size,size, [
a+(k*b)-e, l*b,
0,
0, l*b,
l*b, a-e,
b,
0,
0,
0,
b, a-e,
b,
0,
0,
0,
b, a-e,
b,
l*b,
0,
0,
C
C
C
H
H
b, a-e ]):
y := det(huckmat/b);
y := 1.5 n4 - 4.5 n2 + 2.78 - n5 + 4.28 n3 - 3.56 n
ans:=sort([fsolve(y,n,maxsols=size,-3..3)]);
ans := [ -1.618033989, -1.008258220, .6180339887, 1.188674649, 2.319583570 ]
an := p -> -ans[p]*b + a:
an(1+(size+1)/2)-an((size+1)/2):
DE := subs(b=(-77550*4.184),an(1+(size+1)/2)-an((size+1)/2));
DE := 185155.3185
lam := 6.626e-34*3.00e8*1.00e9*6.022e23/DE;
lam := 646.5129761
41
Laboratory 12 (Quantum Chemistry)
Determination of the Potential Energy Surface of I2
Introduction:
This experiment is an adaptation of one published by McNaught18. Additional approaches
to studying this topic are also proposed in the literature19-21. The determination of potential energy
surface for both ground and excited state molecules is of fundamental importance to understand
elementary reactions. In this experiment,
the potential energy surface of the lowest
excited electronic state (shown in figure
12.1) of Iodine gas will be studied using
UV/Vis spectroscopy. Specifically, the
resolution of the vibrational (and in high
resolution emission spectroscopy the rotational) structure in an electronic transition
will allow us to determine quantitatively
the shape of the excited state electronic
potential energy surface. For additional
information the reader is directed to a variety of references given in the bibliogra-
Figure 12.1 Potential energy surface showing vibrational energy levels for the ground and first excited
states of a diatomic molecule.
phy below14,22-24.
Just as in the HCl laboratory experiment, it is important to discuss the actual energy levels
which will be probed with the UV/Vis spectroscopic technique. Remember that at room temperature, virtually all of the molecules will exist in the ground state electronic energy level and to a
large extent the ground (and a small fraction in the first excited) state vibrational energy level. The
basic equation describing the energy of a diatomic molecule is given below.
Etotal = Ee + Evib + Erot
Evib = hc ( +
1
2
)
el,
E rot = hcJ ( J + 1) Bel,v, J
The variables which appear in these equations are the
(12.1)
and J quantum numbers which refer to the
quantum vibrational and rotational state of the system. The h is the standard Planck's constant while
the c refers to the speed of light. The variable Bel, ,J and el represent the rotational and vibra42
tional interaction constants. Remember that these constants are not unique for all quantum numbers
and in fact the vibrational frequency may be represented below.
el,
=
el
− xel
el
(
+
1
2
) − yel el (
+
)
1 2
2
−
(12.2)
Notice that the vibrational constants in this equation actually refer to equilibrium for each electronic
state (there are separate el, etc. for each different electronic energy level). Also, as in the first
laboratory experiment, these equations show that as the vibrational quantum number gets larger the
vibrational interaction constants get smaller, indicating a potential energy surface which has a different shape than the pure parabolic potential used to approximate it. This feature is referred to as
anharmonicity, which may be seen in figure 12.1. As the interatomic radius is reduced, the potential
grows rapidly as the inner electrons begin to interact strongly (and ultimately as the nuclei begin to
interact). As the radius is increased, the potential levels off at a constant value. This is the dissociation energy above which the molecule dissociates into atomic species. It may be seen that the curvature of the right hand side of the potential energy surface goes from concave up to concave down,
indicating a change in sign of the second derivative of the potential at some point. This may be approximated using the higher order terms in equation 12.2. Additional terms may be included, however for the accuracy and precision of the data collected in this experiment, these formulae are sufficient to completely describe the system.
In this experiment, since most of the molecules exist in the ground state vibrational level of
the ground electronic state, X 1
g,
the primary transition we observe is from the
= 0 vibrational
level of the X state to any of the higher vibrational levels of the excited electronic state, B 3
may be shown in the usual manner that there will be no restriction on
+
0u .
It
. The selection rules for the
rotational states are the same as in the first laboratory ( J = ±1). These energy levels will have the
effect of broadening the observed vibronic transition because of the range of energies which are observed for the P and R branches of a rotation/vibration spectrum. In addition, at higher temperatures
the higher energy
= 1 and
= 2 vibrational levels of the X state will be increasingly populated and
will lead to two additional bands of transitions to vibrational states of the B state. The usual transition dipole moment integral may be used to calculate the intensity for each of the observed vibrational bands.
fi
= −e ∫
*
B
(
f
)
,Jf r
X
(
i , Ji
)d
(12.4)
The X and B subscripts on the wavefunctions refer to the respective electronic states. The intensity
of lines within each of the three observed bands will be determined from equation 12.4. These inte43
grals are often referred to as Franck-Condon overlap integrals and are primarily related to the overall overlap between the two wavefunctions. The intensity of each transition manifold relative to another will be determined only by the population of the X state vibrational levels. The intensity of
transitions within a given manifold should remain constant while the relative intensity of the vibronic manifolds should change as the temperature is changed in this experiment.
With the energy equation 1 we may calculate the UV/Vis energy splitting observed for each
of the three transitions.
E
(
E
E
=0→
hc
(
(
=1→
hc
=2→
hc
)


f )

= E B − E X + B,v f 

f )

= EB − E X + B,v f 

f
= EB − EX +
B,v f
f
f
f
1
+ −
2
1
+ −
2
1
+ −
2
1
2
3
X,1
2
5
X, 2
2
(12.5)
X, 0
(12.6)
(12.7)
These equations may be used to generate the positions of the UV/Vis transitions which will be observed. These may not easily be simplified, however there will some common terms to all three
bands.
E
E
(
(
=0→
hc
f
)=E
(
=1→
hc
)=E
f
(
∆E
(
= 2→
hc
f
)
B
B
− EX +
B
− xB
B)
− EX +
B − xB
B)
= EB − EX +
(
B
− xB
B
B
f
B
)
f
B
−
2
f
− xB
−3
2
xX
X
− xB
4
B
+
(12.8)
2
Bv f
+
X
− xB
−5
2
− xB
+
X
9x X
X
− xB
4
B
+
(12.9)
2
B f
+
X
B
25x X
− xB
4
X
B
+
2
f
(12.10)
It is immediately apparent that all three splittings depend quadratically on the final vibrational
quantum number with the same coefficient, xB B. Second, all three splittings depend linearly on
the final vibrational quantum number with a second coefficient, B −xB B. The three expressions
differ most in the first term which is independent of the final vibrational quantum number. This
means that the spacing between successive vibrational lines arising from the same ground state in
the spectrum should be given by equation 12.11. (Notice that all of these energy differences are independent of which initial vibrational state we start from and therefore all three initial states may be
used to help graph the energy splitting differences versus final vibrational state.)
44
E
(
i→
f +1
)− E (
i→
hc
f
)=
(
=(
B
− xB
B − 2x B
B
) − xB B ( 2
B ) − 2x B B
f
)
+1
(12.11)
f
Additionally, the difference between energy splittings arising from the same final state but different
starting states may be shown below. (Again, notice that this is purely a constant and therefore as
many final vibrational state pairs should be averaged together as is possible to give good values for
both equation 12.12 and 12.13.)
E
E
(
(
i = 0→
i =1→
f
f
)− E (
hc
− E
hc
)
(
i =1→
i = 2→
f
f
)
)
=
X
− 2 xX
X
=
X
− 4 xX
X
(12.12)
(12.13)
Therefore using equations 12.11, 12.12 and 12.13, the molecular constants X, B, xX X, xB B and
+
EB−EX may be extracted. These may then be used to generate both the X 1 g and B 3 0u
potential
energy surfaces.
The actual experiment is conducted by placing I2 in a 10.0 mm UV/Vis silica cell with a
Teflon stopper. This cell should be both dry and clean. No more than a few small crystals are necessary to produce enough I2 vapor in this cell.
The UV/Vis spectrum may be taken in absorption mode in the region of interest (about 400
to 700 nm). It should be possible to observe at least ten to thirty distinct lines from each of the three
bands. The spectrum should be recorded at 40˚C and at 90˚C so that the lines coming from the
and 2 vibrational states may be differentiated from the
=1
= 0 lines. The spectrum should be taken
with maximum resolution and expansion for accurate measurement of the peak positions. Once the
peaks have been measured, the spectrum should be assigned (for the
= 0 band, the f = 30 line oc-
curs at 537.87 nm, see attached sheet for sample data and assignments). The five basic spectroscopic constants may then be computed. Errors should be fully propagated to determine the overall
accuracy of this method. These numbers may then be compared to those given by Herzberg.1 (Potential minimum splitting, EB − EX, is 15641.6 cm−1, B is 128.0 cm-1, xB B is 0.834 cm-1, X is
214.6 cm−1 and xX X is 0.6127 cm−1.)
For further data analysis, we would like to calculate DX and DB, the dissociation energies
for the ground and first excited electronic states. The second may be determined from a BirgeSponer plot. In this type of graph, the vibrational energy level splittings are plotted versus vibrational quantum number. In this fashion as the potential energy surface becomes wider and the energy spacing between vibrational levels becomes smaller, the point of dissociation may be seen
when the vibrational spacing becomes nearly zero. The total dissociation energy, Del,0, will then
45
represent the sum of all of these splittings, or the area under the curve. To a good approximation
this curve will be nearly linear and may be approximated as such. Note that this dissociation energy
is from the
= 0 level, which is not the bottom of the potential energy well and therefore this disso-
ciation energy is labeled DB,0. To get DB from this, we need to add on the zero point energy B / 2.
To calculate the energy from the bottom of the ground electronic state to the dissociation point of
the first excited electronic state, E*, we need to add together the electronic splitting EB − EX and
DB,0. The difference between the ground state dissociation products and the excited state products
in the I2 dissociation will be the final state of one of the I product atoms. This atom will be in an
excited electronic state which is 7598 cm-1 higher in energy than a ground state iodine atom. The
dissociation energy, DX,0, for the ground electronic state from the
= 0 vibrational level will be
given by the difference between E* and the atomic iodine excitation energy, 7598 cm-1. Again, this
dissociation energy must be corrected by the zero point vibrational energy, X / 2, to yield the correct value of DX.
46
Laboratory 13 (Statistical Mechanics)
3D Ising Spin Lattice Model for Phase Transitions
Introduction:
This experiment is partially based on work by Steckline25 and by Chandler26. Phase transitions have always been of interest to both physicists and chemists due to the difficulty in modeling
a transition from one phase to another. In a pure single phase system, the equations of state are
usually rather simple and small changes in temperature and pressure do not change the state to a
great degree. At a phase transition, however, small changes in state variables can lead to dramatic
changes in the appearance of the system. In particular at the critical point where the phase transitions approach second-order the changes a system may show may provide completely new properties previously unknown. An example of the lattice gas model is shown in figure 13.1 at three different temperatures. What can clearly be seen in these figures is the transition from a gas-like low
Figure 13.1 Example of lattice gas Ising model using a 40x40 lattice at three different temperatures
(high, near Tc and low.) Note that at the high temperature state the molecules are spread evenly
throughout the lattice. As the temperature is lowered, the molecules clump together and then ultimately these clumps condense to form a solid phase.
density state to a condensed phase (liquid or solid like depending on the mobility and order in the
condensed lattice). The intermediate temperature might represent a critical state if the overall density is correct (in this case however the density is too low). Recall that a critical point occurs when
the two phases become indistinguishable from a statistical mechanics point of view and the energy
of phase transition becomes zero (by definition.)
The purpose of this lab is to use a three dimensional lattice gas model to construct a density
versus temperature phase diagram. An example of this type of phase diagram is shown in figure
13.2; notice that this diagram resembles a more conventional pressure versus temperature phase
diagram. The fundamental difference between pressure and density is that a high density is akin to
a high pressure while a low density behaves like a low pressure. In both cases, raising the density
47
(pressure) will raise the melting point
of the condensed phase. At each density the temperature of the phase transition is determined by plotting the
configuration energy of the system at
each temperature. The slope of this
line will represent the heat capacity of
the system at that temperature/density
point. As the system undergoes the
phase transition you will observe a
discontinuous change in the heat capacity (as well as a discontinuity in
the overall energy of the system.)
Note that since the system being
studied is finite, the discontinuities
Figure 13.2 Density versus Temperature phase diagram for
a lattice gas. Units on the temperature axis are expressed in
terms of the atom-atom coupling constant, CJ, and the
Boltzmann constant, kB.
will be smoothed out to some degree. Figure 13.3 shows an example of the energy versus temperature and heat capacity ( E T ) versus temperature plots which are generated using the LINUX
computer program modeling a 50x50 lattice at a density of 1% filling of the lattice. It is clear that
at a temperature of approximately 0.30 the lattice undergoes a phase transition from an average of
3.5 neighbors to 0.5 neighbors (seen in the energy scale which will effectively count the average
0.00
25.0
-0.50
20.0
Heat Capacity
-1.00
Energy
-1.50
-2.00
-2.50
15.0
10.0
-3.00
5.0
-3.50
-4.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.0
0.00
0.80
Temperature
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
Temperature
Figure 13.3 Energy (left) and Heat Capacity (right) as a function of Temperature. As in figure
13.2 the units of temperature are expressed in K / kB CJ, energy in J / CJ atom, and heat capacity as J
kB / K where kB is the Boltzmann constant and CJ is the atom-atom coupling strength in J.
number of neighbors for an atom.) In addition, since there are no additional degrees of freedom
other than the configuration, the heat capacity is essentially the same for the gas and the condensed
phase. Note that in the real world these heat capacities will normally differ considerably as the gas
48
can store energy in translational degrees of freedom which are absent in the solid phase or reduced
in the liquid. Also in the real world as the temperature is raised the various high energy vibrational
degrees of freedom will be increasingly populated (of course this is not an issue for a vapor phase
made up of isolated atoms, such as He gas.) Notice in figure 13.3 that the spike in the heat capacity
does not go to infinity but is rounded. This is due to the limited size of the lattice used for the
simulation. If the number of atoms were increased, the sharpness of this peak would also increase
and ultimately would approach a true singularity as the lattice size approached the size of
Avogadro’s number.
Experimental:
For your experiment you will be using the C program written by Dr. Jay Baltisberger running on a Pentium II LINUX workstation. This program allows you to run the Ising lattice simulation over a full range of temperatures at a given lattice density. Your goal is to construct a phase
diagram for the lattice gas and determine the critical point (above which there is no phase transition). You should choose the largest lattice possible (within practical limitations of computer time)
for each experimental run. In addition you should consider what interatomic coupling strength to
choose; in the LINUX program you can set this parameter to an appropriate value (ranging from
120 kJ/mol for strongly hydrogen bonded gases to under 100 J/mol for weakly interacting Van der
Waals gases). For whatever coupling constant you choose, compare the observed phase diagram
with that of a gas with a similar coupling constant.
49
The Maple code to conduct a two-dimensional Ising model study is shown below:
restart():
with(plots):
randomize():Size := 40:
Temp := 1.0:
Ising := array(1..Size,1..Size):
showplot := proc()
listdensityplot([seq( [ seq( Ising[i,j],
i=1..Size ) ], j=1..Size )], style=PATCH,
axes=none);
end:
MaxDen := 5:
r1 := rand(MaxDen+1):
r2 := rand(Size)+1:
make_new := proc()
global spincount, Ising, Size, MaxDen,
where, nospins, notwhere, r3, r4;
local i,j,found,notfound;
spincount := 0:
for i to Size do
for j to Size do
Ising[i,j] := `if`(r1()>=MaxDen,0,1);
spincount := spincount + Ising[i,j];
od;
od;
where := array(1..spincount,1..2):
nospins := Size*Size-spincount:
notwhere := array(1..nospins,1..2):
move_spin := proc()
global Ising,where,notwhere,Temp;
local
spin1,spin2,en1,en2,test1,test2,was_i,was_j;
en1 := 4:
while (en1 = 4) do
spin1 := r3():
en1 := count_neighbors( where[spin1,1],
where[spin1,2] ):
od:
spin2 := r4():
en2 := count_neighbors( notwhere[spin2,1],
notwhere[spin2,2] ):
test1 := exp(evalf(en2-en1)/Temp);
test2 := rand();
test2 := evalf( test2 / 999999999999 );
test1 := test1 / test2;
if (test1 > 1.0 ) then
flip_two(spin1,spin2);
fi;
end:
found := 0:
notfound := 0:
for i to Size do
for j to Size do
if (Ising[i,j] = 1) then
found := found + 1;
where[found,1] := i;
where[found,2] := j;
else
notfound := notfound + 1;
notwhere[notfound,1] := i;
notwhere[notfound,2] := j;
fi;
od;
od;
r3 := rand(spincount) + 1:
r4 := rand(nospins) + 1:
end:
flip_two := proc(spin1,spin2)
global Ising,where,notwhere;
local was_i,was_j;
was_i := where[spin1,1]:
was_j := where[spin1,2]:
Ising[was_i,was_j] := 0:
where[spin1,1] := notwhere[spin2,1]:
where[spin1,2] := notwhere[spin2,2]:
Ising[where[spin1,1],where[spin1,2]] := 1:
notwhere[spin2,1] := was_i:
notwhere[spin2,2] := was_j:
end:
make_new():Temp := 10.0;
for i to 5000 do move_spin() od:
InitEnergy := total_energy();showplot();
Temp:=1.0;
for i to 5000 do move_spin() od:
FinalEnergy := total_energy();showplot();
Temp:=0.1;
for i to 5000 do move_spin() od:
FinalEnergy := total_energy();showplot();
spin_before := proc(i)
local i_new;
i_new := i - 1;
if (i_new = 0) then i_new := Size fi;
i_new;
end:
spin_after := proc(i)
local i_new;
i_new := i + 1;
if (i_new = Size+1) then i_new := 1 fi;
i_new;
end:
count_neighbors := proc(i,j)
global Ising;
local neigh;
neigh := Ising[spin_before(i),j];
neigh := neigh + Ising[spin_after(i),j];
neigh := neigh + Ising[i,spin_before(j)];
neigh := neigh + Ising[i,spin_after(j)];
neigh;
end:
total_energy := proc()
global Ising;
local energy,i,j;
energy := 0;
for i to spincount do
energy := energy +
count_neighbors(where[i,1],where[i,2]);
od;
energy;
end:
50
The C program to perform the three-dimensional Ising model study is shown below:
/*
Ising model for 3D lattice gas
*/
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
<time.h>
<math.h>
/* Defines */
/* Globals */
double temp,j_coup,j_coup_kb;
long int ***data_array;
long int **spin_positions;
long int size,energy,spins,idum;
/* structures */
/* Prototypes */
long int count_spins(void);
double ran2(long int idum[]);
long int get_energy(void);
long int **matrix(long int nrh,long int nch);
long int ***lattice(long int nrh,long int nch,long int ndh);
long int count_neighbors(long int i, long int j, long int h);
double *array(long int size);
long int randim(void);
long int ranspin(void);
long int flip_spins(long int flips);
void init_data(void);
/* Main program */
void main(void) {
long int i,iterations,flips,num_total;
long int num_temps,num_tem2;
char ans;
char filename[25],errfilenm[25];
FILE *output;
double *temps,*errs,*energies,temp_min,temp_max,run_energy,temp_spacing;
double total_energy,total_eng2;
printf("What is size of array (NxNxN, with N < 2500)?\n");
scanf("%li",&size);
if (size>2500) size = 2500;
printf("The array will be %li x %li x %li\n",size,size,size);
printf("What is number of gas spins (must be less than %5li)?\n",size*size*size/2);
scanf("%li",&spins);
if (spins>3*size*size*size/4) spins = 3*size*size*size/4;
printf("The array will be filled with %li spins\n",spins);
data_array = lattice(size+1,size+1,size+1);
spin_positions = matrix(spins+2,5);
idum = -time(NULL);
printf("Random seed: %li\n",idum);
if (idum>=0) idum = - idum - 1;
ran2(&idum);
printf("How many spins to flip per energy check?\n");
scanf("%li",&flips);
if (flips<=100) flips = 100;
if (flips>=size*size*size) flips = size*size*size;
printf("%li spins will be flipped each reporting period.\n",flips);
printf("How many iterations of spin flips?\n");
scanf("%li",&iterations);
if (iterations<=10) iterations = 10;
if (iterations>=500) iterations = 500;
printf("%li reporting periods will be averaged over.\n",iterations);
printf("Do you want to choose your temperature or use a range [y/r]?\n");
scanf("%1s",&ans);
if (ans != 'r' && ans != 'R') ans = 'y';
num_temps = 1;
num_tem2 = 0;
if (ans == 'r' || ans == 'R') {
printf("What is the highest temperature you want (K)?\n");
scanf("%lf",&temp_max);
if (temp_max<0) temp_max = 2.0;
printf("What is the lowest temperature you want (K)?\n");
scanf("%lf",&temp_min);
if (temp_min<0) temp_min = 0.1;
printf("How many temperature points do you want?\n");
scanf("%li",&num_temps);
if (num_temps<0) num_temps = 10;
temp = temp_max;
temp_spacing = (temp_max-temp_min)/(double)(num_temps-1);
temps = array(num_temps+1);
energies = array(num_temps+1);
errs = array(num_temps+1);
num_tem2 = num_temps;
printf("What is the filename for the energies [15 character max]?\n");
scanf("%15s",filename);
printf("What is the filename for the errors [15 character max]?\n");
scanf("%15s",errfilenm);
}
j_coup = 0.0;
while (j_coup <= 0.0) {
printf("What is the interatomic coupling (note this is positive\n");
printf("constant for attractive coupling in kJ/mol)?\n");
scanf("%lf",&j_coup);
if (j_coup <= 0.0) {
printf("Interatomic coupling should be positive.\n");
}
}
j_coup_kb = j_coup / 0.00831441;
init_data();
while (num_temps > 0 || (ans == 'y' || ans == 'Y' || ans == 'r' || ans == 'R') ) {
if (ans == 'y' || ans == 'Y') {
printf("What is the temperature (k T / J)?\n");
scanf("%lf",&temp);
if (temp<=0.0001) temp = 0.0001;
printf("The temperature will be %lf\n",temp);
}
printf("Initial Status:");
printf("
Energy (per spin): %lf\n",-(double)get_energy()/(double)spins);
total_energy = 0.0;
total_eng2 = 0.0;
num_total = iterations/5;
if (num_total<7) num_total = 7;
printf("Iteration ");
for (i=1;i<=iterations;i++) {
printf(" #%li ",i);
if (!(i%12)) printf("\n");
flip_spins(flips);
get_energy();
if (iterations-i < num_total) {
total_energy += (double)energy;
total_eng2 += (double)energy*(double)energy;
}
}
printf("\nNumber of Spins:%li",count_spins());
printf("\nFinal Status:");
printf("
Energy (per spin): %lf\n",-(double)get_energy()/(double)spins);
run_energy = total_energy / (double)num_total;
printf("\nAverage Energy per spin (@ temperature %lf): %10.8lf\n",temp,run_energy/(double)spins);
total_eng2 = (double)num_total*total_eng2 - total_energy*total_energy;
printf("Error Energy per spin: %10.8lf\n\n",sqrt(total_eng2)/(double)num_total/(double)spins);
if (ans == 'y' || ans == 'Y') {
printf("\nRun another temperature [y/n]?\n\n");
scanf("%1s",&ans);
} else {
temps[num_temps] = temp;
energies[num_temps] = -run_energy*j_coup/(double)spins;
errs[num_temps] = sqrt(total_eng2)*j_coup/(double)num_total/(double)spins;
temp -= temp_spacing;
num_temps--;
if (num_temps==0) ans = 'n';
}
}
if (num_tem2) {
remove(filename);
output = fopen(filename,"w");
if ( output == NULL )
printf("\nOutput file could not be generated.");
else {
for (i=1;i<=num_tem2;i++) {
fprintf(output,"%lf %lf\n",temps[i],energies[i]);
}
}
fclose(output);
remove(errfilenm);
output = fopen(errfilenm,"w");
52
if ( output == NULL )
printf("\nOutput file could not be generated.");
else {
for (i=1;i<=num_tem2;i++) {
fprintf(output,"%lf %lf %lf\n",temps[i],energies[i],errs[i]);
}
}
fclose(output);
}
}
long int randim(void) {
return( size*ran2(&idum) + 1 );
}
long int ranspin(void) {
return( spins*ran2(&idum) + 1 );
}
void init_data(void) {
long int i,j,k,h;
for (i=1;i<=size;i++) {
for (j=1;j<=size;j++) {
for (h=1;h<=size;h++) {
data_array[i][j][h] = 0;
}
}
}
for (i=1;i<=spins;i++) {
j = randim();
k = randim();
h = randim();
while ( data_array[j][k][h] == 1 ) {
j = randim();
k = randim();
h = randim();
}
data_array[j][k][h] = 1;
spin_positions[i][0] = j;
spin_positions[i][1] = k;
spin_positions[i][2] = h;
}
}
long int flip_spins(long int flips) {
long int total,new,old,h1,i,j,h,k,i1,j1,choosespin;
double test;
total = 0;
for (k=1;k<=flips;k++) {
choosespin = ranspin();
i = spin_positions[choosespin][0];
j = spin_positions[choosespin][1];
h = spin_positions[choosespin][2];
i1 = randim();
j1 = randim();
h1 = randim();
while ( data_array[i1][j1][h1] == 1 ) {
i1 = randim();
j1 = randim();
h1 = randim();
}
if (i==i1 && j==j1 && h==h1) { exit(EXIT_FAILURE); }
old = count_neighbors(i,j,h);
new = count_neighbors(i1,j1,h1);
test = ( (double) (new-old) ) * j_coup_kb / temp;
test = exp( test ) / ran2(&idum);
if (test >= 1.0) {
data_array[i][j][h] = 0;
data_array[i1][j1][h1] = 1;
total++;
spin_positions[choosespin][0] = i1;
spin_positions[choosespin][1] = j1;
spin_positions[choosespin][2] = h1;
}
}
return(total);
}
long int count_neighbors(long int i, long int j, long int h) {
long int neighbors = 0;
if (j==size) {
neighbors = data_array[i][1][h];
}
} else {
neighbors = data_array[i][j+1][h];
53
if (i==size) {
neighbors += data_array[1][j][h];
}
} else {
neighbors += data_array[i+1][j][h];
if (h==size) {
neighbors += data_array[i][j][1];
}
} else {
neighbors += data_array[i][j][h+1];
if (j==1) {
neighbors += data_array[i][size][h];
}
} else {
neighbors += data_array[i][j-1][h];
if (i==1) {
neighbors += data_array[size][j][h];
}
} else {
neighbors += data_array[i-1][j][h];
} else {
neighbors += data_array[i][j][h-1];
if (h==1) {
neighbors += data_array[i][j][size];
}
return(neighbors);
}
long int get_energy(void) {
long int i,j,h,count;
energy = 0;
for (i=1;i<=size;i++) {
for (j=1;j<=size;j++) {
for (h=1;h<=size;h++) {
if (data_array[i][j][h] == 1) {
energy += count_neighbors(i,j,h);
}
}
}
}
return(energy);
}
long int count_spins(void) {
long int i,j,h,count;
count = 0;
for (i=1;i<=size;i++) {
for (j=1;j<=size;j++) {
for (h=1;h<=size;h++) {
if (data_array[i][j][h] == 1) {
count++;
}
}
}
}
return(count);
}
long int **matrix(long int nrh,long int nch) {
long int i,j;
long int **m;
m = (long int **) malloc( (nrh+2)*sizeof(long int*));
if(m ==NULL) {
printf("\nMalloc failed (2nd dim of 2d array)");
exit(EXIT_FAILURE);
}
for(i=0;i<=nrh;i++) {
m[i]=(long int *) malloc( (nch+2)*sizeof(long int));
if(m ==NULL) {
printf("\nMalloc failed (2nd dim of 2d array)");
exit(EXIT_FAILURE);
}
}
for(i=0;i<=nrh;i++) {
for(j=0;j<=nch;j++) m[i][j] = 0;
}
return(m);
}
long int ***lattice(long int nrh,long int nch,long int ndh) {
long int i,j,k;
long int ***m;
m = (long int ***) malloc( (nrh+2)*sizeof(long int**));
if(m ==NULL) {
printf("\nMalloc failed (1st dim of 3d array)");
exit(EXIT_FAILURE);
}
54
for(i=0;i<=nrh;i++) {
m[i] = (long int **) malloc( (nch+2)*sizeof(long int*));
if(m[i] ==NULL) {
printf("\nMalloc failed (2nd dim of 3d array)");
exit(EXIT_FAILURE);
}
for(j=0;j<=ndh;j++) {
m[i][j] = (long int *) malloc( (ndh+2)*sizeof(long int));
if(m[i][j] ==NULL) {
printf("\nMalloc failed (3rd dim of 3d array)");
exit(EXIT_FAILURE);
}
}
}
for(i=0;i<=nrh;i++) {
for(j=0;j<=nch;j++) {
for(k=0;k<=ndh;k++) m[i][j][k] = 0;
}
}
return(m);
}
double *array(long int size) {
double *m;
long int i;
m = (double *) malloc( (size+3)*sizeof(double));
if ( m == NULL ) {
printf("\nMalloc failed (1d Array)");
exit(EXIT_FAILURE);
}
for(i=0;i<=size;i++) m[i] = 0.0;
return(m);
}
55
Laboratory 14 (Dynamics/Spectroscopy)
Exchange Rate Measurement on N,N Dimethylacetamide
using Spin Saturation NMR Spectroscopy
Introduction:
This experiment is an adaptation of one published by Bell27 and Jarek28. NMR spectroscopy is a powerful technique which may be used to elucidate the structure of both small and very
large molecules as well as provide medical images
using field gradients. NMR spectroscopy is also a
potent tool in the determination of dynamics of
molecular processes ranging from less than one
microsecond to longer than many hours. Few
other chemical tools can be used to measure dynamics over so wide a range without substantial
changes in instrumentation design. In this experiment we will examine the effects of chemical
exchange on an NMR spectrum and learn to inter-
Figure 14.1 Structure of N,N-dimethylacetamide
pret these changes in terms of a rate measurement. The structure of N,N-dimethylacetamide is
shown in figure 14.1; it is interesting to note that all of the carbon, nitrogen and oxygen atoms are
coplanar. Of course the three methyl groups may freely rotate about the three-fold symmetry axes
but the nitrogen to carbonyl carbon bond is not as mobile as a normal single bond. This is a result
of the pz orbital on the nitrogen forming a partial bond with pz orbitals of the carbon and oxygen atoms which make the
bond. The effect of this partial double bond between the carbon and the ni-
trogen atoms is to limit rotation about the bond axis. Thus the methyl groups on the amide nitrogen
are actually chemically distinguishable on an NMR timescale (the exchange rate is measured in microseconds.) At very low temperature the NMR spectrum for a system such as N,Ndimethylacetamide will show distinct peaks for each of the methyl groups. In addition there is the
possibility of long range J-coupling if the chemical shifts of the two sites are different enough and
the atoms are close enough together. As the temperature rises, the methyl groups begin to rotate
about the amide bond and the two peaks begin to shift towards one another. At some temperature
(termed the coalescence temperature) the two peaks merge into a single peak. Above this temperature the methyl groups will be indistinguishable, though the line width of this coalesced peak
56
will provide exchange rate information. There are three distinct temperature regions that may be
studied using different NMR techniques (see figure 14.1 for sample two-site spectra.) First, in the
low temperature regime the peak positions/line widths do not provide useful information that may
be used to extract the exchange rate. In this temperature regime, a method called spin-saturation is
used to determine the rate of exchange. In the intermediate temperature regime where the two
4.50
4.70
4.90
5.10
5.30
5.50
5.70
5.90
6.10
Frequency (ppm)
Figure 14.1 Two-site exchange spectra for various exchange rates ranging from very slow (bottom) to very
fast (top).
peaks are moving towards one another and begin to coalesce, the exchange rate may be determined
using spectral simulations of standard NMR spectra. In the high temperature limit the same spectral simulations may also be used, but in addition there are closed form equations relating the observed line width to the exchange rate. In molecular systems where the exchange involves multiple
sites or molecules, the equations are substantially complicated but still modern computers may be
used to simulate all but the most complex systems.
Theory:
Any description of exchanging spins in NMR requires the use of either a full quantum mechanical spin simulation or the modified Bloch equations. The second of these is substantially easier to understand if the number of spins is small and we will use this description for this laboratory.
Note that the Bloch equations are not always appropriate in other situations and should be used with
care. Normally for a single site the Bloch equations (14.1) shown below describe the magnetization
57
vector (MX,A, MY,A, MZ,A) in the Larmor frequency ( ) rotating frame and have two types of terms,
those that relate to the chemical shift of the site A (
) and those that relate to the relaxation (T1,A,
T2,A).
1
M˙ Z ,A =
( M Z, A,0 − M Z, A)
T1, A
) MY , A −
1
M
T2, A X , A
) MX , A −
1
M
T2, A Y ,A
(
0
−
A
M˙ Y, A = (
0
−
A
M˙ X , A =
(14.1)
This is a fairly complicated set of coupled differential equations, but the terms are relatively easy to
understand qualitatively. The T1,A relaxation term essentially restores the original equilibrium magnetization, MZ,A,0 when the magnetization vector has been disturbed from equilibrium. This is a familiar exponential growth equation which is independent of the other two vector components. In
fact we studied T1,a measurements in laboratory 8 and you may return there for additional information. The T2,A terms are line broadening terms that essentially exponentially damp out the magnetization in the x-y plane. The chemical shift terms serve to rotate the magnetization about the x-y
plane at a frequency
. In the presence of chemical exchange, the Bloch equations for site A are
modified as shown below (note that in the case of exchange there are now two spins labeled A and
B). Similar equations can be written for the B site.
1
M˙ Z ,A =
( M Z, A,0 − M Z, A)
T1, A
) MY , A −
1
M + k B MX, B − kA MX , A
T2, A X , A
) MX , A −
1
M + k B MY ,B − k A MY , A
T2, A Y ,A
(
0
−
A
M˙ Y, A = (
0
−
A
M˙ X , A =
(14.2)
The additional terms involve rate constants kA and k B which are the rates for exchange from A to B
and B to A respectively. Note that these are not independent and will relate to one another via the
equilibrium populations of the two sites such that pA kA = p B kB. The effect of this exchange on the
Bloch equations is rather mathematically challenging since we now have six coupled differential
equations to simultaneously solve. The net outcome from these equations is that an equation for the
lineshape may be written below.
58

S ( ) = Im 

A
=
1
kA
B
=
1
−(
T2 B
A
B
+
B
(1 +
=
0
+
A
A A
1
kB
−
B
)(1+
A
B
−
(
=
A
pB )


)
−1

B
pA +
B
1
−(
T2 A
B
0
−
A
−
)i
(14.3)
)i
Note that that the function Im() indicates that the expression is evaluated for a given
and then the
imaginary portion of this resulting value is extracted. Needless to say, this is also not a rather easy
equation to understand without a computer. Fortunately, numerous graphical simulations have been
written based on this formula and we will never be forced to evaluate this by hand! There are a few
features that may be described from this equation. First, if the exchange rate is fast, then the two
peaks will collapse into a single peak. As the rate is reduced the single peak separates into two individual population weighted peaks and ultimately will approach the non-exchanging limit (see figure 14.1 for simulated two-site spectra with such a program.)
As mentioned earlier, there
are three distinct exchange rate regimes that may be studied with
NMR spectroscopy. In the low temperature regime, the two peaks appear clearly separated and very close
to their non-exchanged positions in
the NMR spectrum. Under these
conditions, any lineshape simulation
is doomed to fail since the small rate
constant is indistinguishable from
zero! To solve this problem an experiment called spin-saturation is
Figure 14.2 Effect of Spin Saturation on Energy Level Populations. Note that the site on the left is being saturated and therefore
the populations of the up and down states are equal. By considering all of the various processes (given by arrows), an equilibrium
population description may be reached.
used. The pulse sequence used is a
rather simple one in that the transmitter is placed exactly on resonance with one of the two exchanging chemical shift locations. This transmitter is then turned on at low power which will saturate one transition and leave the other unaffected directly. To test that this condition is properly set,
you should move the transmitter to the other side of the unaffected peak and place it the same distance from that peak as it is on the saturated peak. Since there is no peak at this location, the saturation will have no affect on the observed spectrum. The impact on the spectrum (in the presence
59
of slow exchange) is that the unsaturated exchanging peak will be reduced in amplitude from the
equilibrium intensity. In the case of N,N-dimethylacetamide, the three methyl groups all should
have equal intensity initially (and show no J-coupling). Under the spin saturation experiment, the
non-exchanging methyl group will remain the original intensity while the other peaks are reduced
(one in fact disappears under saturation). The effect of spin saturation is shown in figure 14.2
where the two peaks will develop different population differences than a normal equilibrium population difference in the absence of spin saturation. The mathematics of evaluating the various rate
constants in this figure are somewhat complicated (since it will be a system of coupled first order
differential equations) but straightforward in solution. By comparing the height of the exchange
peak to the non-exchange peak, the rate constant for the exchange process may be extracted via
equation 14.4 (the solution to the coupled equations).
k=
I0 − I
I 1T
(14.4)
Where I is the intensity of the peak in the presence of spin saturation and I0 is the intensity of the
non-exchanging methyl group. The T1 is the spin lattice relaxation time (as measured identically to
laboratory 8) of the exchanging methyl group. For greater accuracy, the T1 for the exchanging site
should also be measured under spin-saturation of the other exchange site before substitution into
equation 14.4. If this is not done, you can expect approximately 5 to 10% errors in the measured
exchange rate constant, k.
Experiment:
For the experiment, you will conduct a series of T1 and spin saturation measurements at a
minimum of four different temperatures. Our NMR spectrometer is capable of covering temperature ranges from -40˚ C up to over 100˚ C. By measuring the rate constants using the procedure
outlined in the theory section, you can examine the rates as a function of temperature. By plotting
the ln k versus temperature, you may determine the activation energy of the rotational process, that
is to say the energy needed to rotate the methyl group by 90˚ (half-way to exchange). This information is useful because it will give you insight into the amount of double bond character present
between the amide nitrogen and the carbonyl carbon atoms.
60
Laboratory 15 (Dynamics/Spectroscopy)
Imaging and Diffusion Measurements
using Pulsed Field Gradients with NMR Spectroscopy
Introduction:
This experiment is an adaptation of one published by Hull29 and another by Quist30. NMR
spectroscopy is a powerful technique which may be used to elucidate the structure of both small
and very large molecules as well as provide medical images using field gradients. NMR spectroscopy is also a potent tool in the determination of dynamics of molecular processes ranging from
less than one microsecond to longer than many hours. Few other chemical tools can be used to
measure dynamics over so wide a range without substantial changes in instrumentation design. In
this experiment we will examine the effects of chemical exchange on an NMR spectrum and learn
to interpret these changes in terms of a rate measurement.
Theory:
Normally when researchers acquire high resolution NMR spectra of molecules with the purpose of structure determination, the instrument must be carefully shimmed. By shimming, I mean
that the magnetic field in the region of the sample must be perfectly (ideally) homogeneous to the
level of parts per billion or better. Superconducting magnets used today are constructed by having
a large superconducting coil which is powered with 30-50 A and maintained at a temperature of 4
K. This coil produces the bulk 7 T field (sometimes higher or lower depending on the specific instrument). Unfortunately, unless the turns
of this coil are perfectly
constructed, there will
be slight field inhomogeneities even in the
center of the coil. Inside the superconducting magnet are three or
Figure 15.1 Effect of a field gradient on observed chemical shifts. On the left
is an NMR tube filled with water showing the field gradient in the z direction.
Notice that different portions of the tube feel different magnetic field
strengths. These manifest themselves as different chemical shifts across the
sample tube leading to spectra seen on the right.
61
more additional gradient
coils which are also superconducting. These
are used to apply gradi-
ents in the x, y and z directions to try to improve the overall homogeneity of the magnetic field.
Using superconducting shims, the field homogeneity may achieve a part per thousand level. To
reach the highest resolution, there is a room temperature set of shims which have higher order coils
that may be used to reach the part per billion homogeneity level.
In the experiments we are going to perform in this laboratory, we will abandon this need for
homogeneity and will actually apply a strong gradient (see figure 15.1) in the z direction. The difference between this gradient and the bad gradients we eliminate with shimming is that this gradient is perfectly linear. This means that the magnetic field will vary exactly linearly from the bottom
of the sample to the top of the sample. As see in figure 15.1, the effect of this gradient is to render
the spins in the NMR tube labeled by position with chemical shift differences. The molecules at the
bottom will have chemical shifts much lower than those at the top of the sample. The net effect is
the acquired one-dimensional spectrum provides a projection of spatial water density within the
sample coil versus z axis position. Additional gradients could be applied in the x or y direction
with appropriate NMR equipment and a full three-dimensional density weighted image could be
acquired. This is the basic technology which is the heart of magnetic resonance imaging (MRI)
used in the medical field today.
For this experiment we will be using
gradients to measure the
rate of diffusion of a liquid molecule in solution.
This may be the solvent or
it may be a solute molecule in practice, but for
today's experiment we
will be studying only neat
solutions. First, the measurement of the gradient
strength, G, is done by
simple gradient imaging
Figure 15.2 Pulsed Field Gradient Echo Diffusion Experiment pulse sequence. The time from the beginning of the first gradient pulse to the second refocusing gradient pulse is the period that spins which diffuse out of a
region are lost from the echo intensity.
of a sample containing
small plastic beads. By applying a set gradient strength (in our case an integer value from -15 to
15) onto this sample a spectrum will appear which has dips and peaks indicating the positions of
each bead. By measuring the number of kHz spacing between bead you may then calculate the
62
gradient strength (after converting from kHz to T using the gyromagnetic ratio of the sample.) The
basic idea which we will use to measure diffusion is to use gradients to encode spins with specific
z-axis position information (in the form of an evolved chemical shift.) By using a 180˚ pulse to refocus the chemical shift interaction and applying a second gradient pulse, only those spins which
have not moved will be effectively refocused. The pulse sequence described is shown in figure
15.2. Note that the gradient pulses are relatively short with a long delay in between them. By
varying this delay (τ), the number of molecules which diffuse away from their starting locations
changes. Alternately, varying the gradient strength may achieve the same effect by providing
changing sensitivity to position. This approach in fact has the advantage of having a constant relaxation contribution to the overall echo intensity. The dependence on the diffusion constant is
shown below:
I 

ln  g  = −

 I0 
2
2
g

2 g 
 D
G 2  + ∆+

3  
(15.1)
Where γ is the gyromagnetic ratio of the sample (γ = 2.675x108 T–1 s–1 for 1H in water,) G is the
gradient strength (measured in T / m,) and Ig and I0 are the echo intensities in the presence and absence of gradients respectively. In fact this equation suggests that the best way to measure the diffusion constant is to plot the ln(Ig/ I0) as a function of G2. This plot would then be linear with a
slope of the form:
−
2
2
g

2 g
 +∆+
 D

3 
(15.2)
By inserting the appropriate values for the delays, this would then give an experimental measure of
the diffusion constant which might be compared against accepted values or theory:
D = kB T / 6 π η r
(15.3)
where kB is the Boltzmann constant, T is the absolute temperature, r is the molecular radius and η is
the viscosity.
63
Laboratory 16 (Thermochemistry)
Measurement of the Heat Capacity Ratio for a Non-Ideal Gas
using the Adiabatic Expansion Method
Introduction:
This experiment is an adaptation of one published in a standard physical chemistry laboratory manual5. The heat capacity ratio, γ = C p,m / C V,m, is a unitless parameter that occurs frequently
in various equations used in thermochemistry. In particular, when an ideal gas undergoes a reversiγ
ble adiabatic expansion the product p V = constant (as opposed to p V = constant for a reversible
isothermal expansion). Recall that in an adiabatic expansion the gas neither gains nor loses heat
upon expansion (though it will change in energy in the form of work performed on surroundings.)
Experimental:
For this experiment the
gas to be studied is placed in a
large carbuoy that is maintained
at a fixed temperature (ideally
using a water bath). This should
be pressurized slightly above atmospheric pressure (approximately 250 mm of oil.) Be aware
that the manometer will measure
pressure in units of mm of oil
rather than mm of Hg (torr). To
convert from mm of oil to torr
you must multiply by the ratio of
the densities of the two fluids.
Figure 16.1 Experimental setup for heat capacity ratio measurement.
The pressure differences measured with this manometer will be much more sensitive than if measured using a pressure gauge or mercury based manometer. For this experiment once the carbuoy is
charged with the gas, it is allowed to equilibrate to room temperature. The pressure inside the carbuoy is measured precisely, then the stopper is removed and reinserted. The delay between remove
and reinsertion should be long enough to allow the pressure to drop to atmospheric level while short
enough to prevent any heat exchange with the atmosphere. This entire process may be viewed in a
two step path:
64
S1( p1, V1,m, T1 ) expands to
S2( p2, V2,m, T2 )
(16.1)
S2( p2, V2,m, T2 ) is heated to
S3( p3, V2,m, T1 )
(16.2)
Note that the external pressure will be p2 and the external temperature will be T1. This path will indeed be adiabatic since during the expansion no heat will enter the gas through the thick carbuoy
walls. Since this is an adiabatic expansion, the final temperature, T2, will be lower than the initial
temperature, T1. Note that the molar volume is constant during the last step since the carbuoy is
plugged during this time of reheating to room temperature. These various state variables do of
course relate to one another. If we assume a perfect gas behaviour, then the first step will be governed by the ideal gas law (p Vm / T = R) as follows:
p1 V1,m / T1 = p2 V2,m / T2
(16.3)
Similarly for the second step we may write:
p2 V2,m / T2 = p3 V2,m / T1
(16.4)
Combining equations 16.3 and 16.4, we may arrive at the relation below:
p1 / p3 = V2,m / V1,m
(16.5)
In addition for the first step (which is adiabatic) the initial and final pressures and volumes must be
γ
related via an adiabat (p V = constant) giving the equation below:
ln (p1 / p2) = γ ln (V2,m / V1,m)
(16.6)
Combining equation 16.5 and 16.6, we can write a closed form for the heat capacity ratio:
γ = ln (p1 / p2) / ln (p1 / p3)
(16.7)
This expression is valid for an ideal gas but will be reasonable for a non-ideal gas for which the interactions between molecules are not huge or where the pressure is not large. Note that p1 > p3 > p 2
and therefore this expression will generate a γ greater than 1 (as is required.) When performing the
actual experiment, be certain to remember to add the atmospheric pressure to the measured ∆p (p1 =
p2 + ∆p1 and p3 = p2 + ∆p2) for each experimental cycle (release pressure/equilibrate). You should
repeat the release/equilibrate cycle multiple times until the differential pressure, ∆p, is too small to
be measured effectively. Calculate the γ for each cycle (note that the end point of the previous cycle acts as the initial point for the subsequent cycle.) An alternative derivation of the heat capacity
65
ratio in which the expansion is not viewed as reversible gives a somewhat different relation shown
below:
γ = (p1 / p2 – 1) / (p1 / p3 – 1)
(16.8)
Notice that this expression is very similar to 16.7 and is in fact related via a power law expansion.
You should calculate the heat capacity ratios with this formula as well and compare all results to
the accepted literature values. In addition, errors should be propagated fully and you should answer
the question whether you can effectively distinguish the 16.7 and 16.8 heat capacity ratios.
66
Laboratory 17 (Quantum Chemistry)
A Simple Measurement of Fluorescence Quenching of Quinine with NaCl
Introduction:
This experiment is an adaptation of one published by Sacksteder et al.31 The purpose of this
is to understand the physical process of fluorescence and to learn how a quenching agent changes
that process.
67
Laboratory 18 (Quantum Chemistry)
Structure Analysis of a Steroid Molecule using
Multi-dimensional PFG NMR Spectroscopy
Introduction:
The use of multi-dimensional NMR experiments to elucidate the structure of molecules of
increasing complexity has been substantially simplified by the introduction of pulsed field gradient
(PFG) probes and techniques in the late 80’s. Since that time NMR acronyms like HMQC32-34,
HSQC, HMBC and DQF-COSY have become second nature in the structural biology laboratories
actively using NMR spectroscopy to determine the three-dimensional solution phase structure of
proteins and nucleic acids. In this laboratory, you will learn how to use these experiments, along
with standard one-dimensional 1H and 13C spectra to determine the complete NMR assignment of a
steroid molecule. This is the first step in a
O
structure determination of any molecule. The
subsequent step of studying NOESY/ROESY
data at various temperature/compositions will
not be attempted at this time. If a student
wishes to pursue this kind of work, additional
experiments can be arranged outside of the
HO
laboratory to learn additional NMR experi-
Figure 18.1 Structure of -Estrone.
ments.
Experimental:
The first step in assigning a large molecule such as β-estrone (or other steroid derivatives) is
to acquire basic 1H and 13C NMR spectra. For molecules with 20 or more proton peaks (as is the
case with most of the steroid family of molecules) the one-dimensional spectrum is generally more
complicated than can be assigned. In particular for a molecule like β-estrone there will be a large
number of peaks in the 1 – 3 ppm region arising from the saturated ring portions of the molecule.
At a high field (more than 7 T) where the J-couplings are reduced in relative size this sort of molecule may be assigned using a basic DQF-COSY experiment. In our 300 MHz instrument, the
overlap is substantial and we need to perform additional experiments. The 13C APT experiment is a
very powerful tool 35-37 to assign the 13C peaks for a molecule such as β-estrone. The APT has two
advantages over a standard single pulse experiment (as used to study the 1H peaks.) First, the APT
has the 1H-13C J-couplings decoupled during acquisition. In the 1H NMR spectrum, the J-couplings
68
can be used to help identify which protons are close to other protons, whereas in a non-decoupled
13
C spectrum, the J-couplings produce excessive cluttering of the spectrum which is virtually im-
possible to decipher. The APT does not ignore the J-couplings completely, but instead uses them to
label peaks as arising from a
—CH3, —CH2—, —CH—, or
tertiary carbon group. In particular, this labeling is accomplished by having an evolution
period (the sections of the pulse
sequence labeled τ) which allows free 1H-13C J-coupling in
the absence of chemical shift
interactions. The net effect will
be that for —CH— and —CH3
carbon sites the polarization
180˚ out of phase from the tertiary and —CH2— sites. Thus
the NMR spectrum will show
peaks pointed in opposite direc-
Figure 18.2 Attached Proton Test (APT) experimental pulse sequence.
tions for carbons with odd versus even numbers of hydrogen atoms directly attached (and thus the APT acronym.) The second
advantage of this sequence is that the 1H—13C J-couplings are decoupled during acquisition which
results in a single line for each carbon atom. The solvent (normally deuterated pyridine for steroids) will appear as 1:1:1 triplets in the aromatic region pointed the same direction as the other even
carbon atoms. When reading the APT pulse sequence in figure 18.2, it is critical to set the pulse
lengths correctly as well as choosing appropriate delays for τ and δ. The τ delay will normally be
set to 1 / 1J(C,H) = 7 ms, assuming 1J(C,H) == 140 Hz; this delay is appropriate for a directly
bonded H atom. If you wish to test for longer ranged attached protons, then this delay might be
modified. The second delay, δ, is the preacquisition delay. This is the time it takes to turn off the
transmitter and turn on the receiver and is normally very short (on the order of 10 - 25 µs).
Once you have acquired both the one-dimensional 1H and 13C APT data, it is time to begin
to look at heteronuclear connectivities. By this we mean looking at which specific 1H sites are directly bonded to which specific 13C sites. The APT experiment already gives you good information
about each 13C site in terms of how many attached hydrogen atoms are present. Using simple
69
Figure 18.3 HMQC pulse sequence, including pulsed field gradients (PFG).
chemical shift correlations, you can make reasonable predictions about which 1H correspond to the
attached protons in the APT experiment. Basically, the chemical shift ranges (200 ppm for 13C versus 12 ppm for 1H) are very different but the order of the peaks will be very similar. To confirm
this assignment strictly, the HMQC experiment is a very powerful tool with the pulse sequence
shown above in figure 18.3 (in the absence of a PFG NMR probe, the HETCOR experiment may be
substituted). The delays are set such that τ is equal to 1 / 2J(C,H) = 3.5 ms and δ will be equal to τ
less the length of the three gradient pulses. This selection of the τ delay functions very similar to
that of the same τ delay in the APT sequence. The fundamental difference is that the J coupling
evolves during both τ delays in the HMQC but only the second τ delay in the APT (and thus the
factor of two time difference). The other difference is that this is a two-dimensional (2D) sequence
(note the t1/2 delays on either side of the 180˚ pulse.) In a 2D experiment, the t1 values are incremented by 1 / SW 1 where SW1 is the 13C sweep width in this case. Thus the data will be an array of
free induction decays (FID's) which are collected for a series of t1 delays. Once the data is Fourier
Transformed (FT) with respect to both time axes, the resulting 2D spectrum will have cross peaks
indicating the two frequencies of the single-bond connected spins. This correlation will very
quickly help you to determine exactly which proton peaks correspond to which carbon peaks. This
is the first step in getting a complete NMR assignment of a complicated molecule. This data should
70
be tabulated in a spread sheet (such as Excel) so that you can clearly see which sites have which
chemical shift values. The HMBC is analagous to the HMQC with the first 13C pulse replaced by a
90-τ2-90 sequence which is tuned to detect two and three bond J couplings (τ2 = 1 / 2 nJ(C,H) = 60
ms where nJ(C,H) is a long range two or three bond J coupling.) The net effect is that the cross
peaks observed in the HMBC will arise from one, two and three bond correlations. The added
benefit is that the 13C is not decoupled at the end and thus the one bond cross peaks will be split in
the proton dimension by 140 Hz. Therefore it is clear whether a given peak is a short or long range
correlation. By using the HMBC data, you may begin to connect the dots so to speak. By choosing
a peak which you clearly know (i.e. a methyl group for instance,) you may then look for HMBC
cross peaks to protons which are near this methyl peak. Once you determine the neighboring protons, you may use the HMQC to determine to which carbons these correspond. This process is repeated as you cycle around the entire molecule. A different experiment (DQF-COSY) may be used
to obtain the same correlation data but the drawback with this experiment is that both dimensions
are 1H. Since the 1H spectrum is rather crowded in the 1 - 3 ppm region, the cross peaks will overlap very strongly and thus the connectivity information will be rather difficult to extract.
71
Laboratory 19 (Quantum Chemistry)
Quadrupolar Interactions in NMR Spectroscopy
Introduction:
A vast majority of the atoms on the periodic table possess nuclear spin greater than 1/2. For
these nuclei, the charge distribution is non-spherical and thus interacts with local field gradients
present at the nucleus. In general these non-spherical nuclei may interact with even higher order
terms in the expansion of the electric potential energy but in general the quadrupolar terms will
dominate all others. The reason the quadrupolar terms are the largest arises since these second order terms do not produce any direct forces on the nucleus (other than some torque on the nucleus).
The first order dipolar term is generally going to be zero since the nuclear location will always be at
a minimum in the potential energy field (i.e. the location where the first derivative is zero.) The zeroeth order term will merely act as an energy offset term on all nuclear energy levels and this constant offset will not appear in any NMR energy expressions. Therefore the quadrupolar interaction
remains as the largest electric field – nuclear interaction.
Theory:
Quadrupolar interactions arise when expanding electromagnetic energies using a Taylor expansion. In general the electric field gradient may be represented as a 3 x 3 tensor of elements.

V =

 V 


Vxx Vyx
Vxy Vyy

Vxz Vyz
,



Vzx 
Vzy 
Vzz 
(19.1)
,
(19.2)
This tensor may always be rewritten in an axis system which renders it diagonal. In this frame the
3 x 3 tensor is reduced to three diagonal components Vxx, V yy, V zz. These three components must
obey the Poisson equation which demands that the field gradient tensor be traceless, i.e. Vxx + Vyy
+ Vzz = 0. Thus the quadrupolar interactions will ultimate depend only on the two parameters, Vzz
and η, where the second term is defined below.
=
Vxx − Vyy
Vzz
72
(19.3)
Normally the three field gradient elements will be arranged such that | Vxx | < | Vyy | < | Vzz | and thus
the value of
will range from 0 to 1.
So far, we have focused on describing the electric field present at the nucleus. This field
gradient will be affected primarily by the population of various local atomic orbitals (which may
produce an asymmetric charge distribution around the nucleus) or by the charged atoms in close
proximity to the nucleus. The degree to which these field gradients affect the nuclear energy levels
is determined by the degree to which the nuclear charge is non-spherically distributed as well as
how effectively the inner core electrons shield the nucleus from the local field gradient. The combination of these two effects will serve to change (and in general reduce) the overall Vzz seen by the
nucleus. In general we shall define a quadrupolar coupling constant, CQ which effectively combines these factors.
CQ =
e 2 Qq eQVzz
=
h
h
(19.4)
This constant can range from the 50-100 kHz range (for small nuclei in symmetric environments) to
the 50-100 MHz range (for halogens in asymmetric environments.) To study a quadrupolar system
with NMR it is essential that the CQ be no more than about 25% of the Larmor frequency. For halogen nuclei (35Cl, 37Cl, 81Br, 79Br, 131I) in asymmetric environments this condition makes NMR experiments difficult if not impossible (though NQR may work in these cases.) For other quadrupolar
nuclei the CQ is small enough to be ignored (51V, 133Cs, and others) and using NMR to extract field
gradient information may be almost impossible. This is true in most situations for nuclei including
27
Al, 23Na, 17O, 87Rb, 85Rb, 11B, 10B, 39K, 2H, and 7Li. A representative example might be the 17O nu-
cleus in SiO2 (quartz) which has a CQ of 5 MHz and a Larmor frequency of 67 MHz at an 11.7 T
field strength. Studies have been conducted on
similar oxygen sites with coupling constants as
large as 10 MHz but rarely above this strength.
Experimental:
For this experiment we will be studying a
sample of RbNO3 which has three crystallographically distinct sites or Rb2SO4 which has
two distinct sites. The overall goal shall be to
determine the quadrupolar coupling constants
for each site and use simple point charge models
to assign each set of constants to specific sites in
73
Figure 19.1 Charge distributions shown for electric
field gradients with (a) = 1 and (b) = 0. In these
pictures the darker spots represent negative charge
and the light spots represent positive charge. The
non-spherical nucleus is shown as an ellipse.
the crystal lattice. In particular, the difference between different values for the
asymmetry parameter are very descriptive
of the local charge distribution. In figure
19.1 you can see how two distinctly different charge distributions give rise to different
asymmetry parameters. By placing point
charges on each atom in the crystal lattice,
you may approximate the charge distributions and determine the appropriate assignments. The approach we will use to determine the quadrupolar coupling constants
Figure 19.2 Energy levels for a spin 3/2 nucleus in the
presence of a strong magnetic field.
will be a mixture of various solid state NMR experiments. The first is a Hahn echo experiment on a
static (non-spinning) sample. Under these conditions, none of the various NMR energy interactions
will be averaged. In particular, we must account for the static quadrupolar, chemical shift and dipolar interactions. With the 25% natural abundance of 87Rb atoms in naturally occurring samples,
the dipolar couplings will be relatively small (no more than about 500
Hz). At the relatively low magnetic
field of our instrument (7 T), the
chemical shift anisotropy will also be
relatively small and we can to first
order consider only the quadrupolar
interaction. By looking at static samples, we can get a good estimate of
the quadrupolar coupling parameters
when the multiple sites are sufficiently different or non-overlapping.
In the case of Rb2SO4, this experiment
will give a very good estimate for
these parameters. For a sample such
as RbNO3, the sites overlap very
strongly and static experiments will
Figure 19.3 Theoretical spectra from static quadrupolar sample.
result in a very broad peak which is
74
difficult to interpret. In almost all cases where the quadrupolar coupling is reasonably large (more
than 1 MHz) NMR is used to examine only the central transition (between the ±1/2 energy levels,
see figure 19.2). The first order correction to the energy levels is exactly the same for both the ±1/2
spin states and therefore the central transition is not changed by this correction. Figure 19.3 shows
the theoretical spectrum obtained from a powder sample with spin 5/2. Since the first order correction can be on the order of 1 to 5 MHz for the outer transitions, it is a good approximation that only
the central transition can be effectively excited. The second order correction unfortunately is not
orientation independent and therefore the central transition will not be sharp but instead will be 1 to
100 kHz wide.
The second portion of this experiment will be to
use a Hahn echo sequence to study the central transition
of these compounds under magic-angle spinning (MAS)
conditions. Historically in liquid state NMR spectroscopy, sample rotation has been used to average inhomogeneities in the magnetic field. By spinning a sample
about an axis either parallel or perpendicular to the magnetic field direction, some portion of the magnetic field
inhomogeneities will be averaged. This is due to the fact
that over a short period of time, a given molecule will
sample a range of magnetic field strengths by moving in
periodic fashion through space. If this time is faster than
Figure 19.4 Magic-Angle Spinning (MAS)
rotor assembly. Vertical line represents the
static magnetic field and θ the angle with
respect to that field. Normally θ will be
54.74˚ (the magic angle).
the NMR sampling time some of the inhomogeneities will be effectively averaged. Note that by
spinning about the z-axis will not average inhomogeneities along the z-axis, but only the x and y
axes. Thus when shimming an NMR instrument, the Z1, Z2, Z 3 and Z 4 shims are generally those of
greatest importance since inhomogeneities along this axis are not averaged by spinning.
For a solid sample (see figure 19.4), the axis of rotation is generally not parallel to the magnetic field (since the inhomogeneities are the least of our problems) but instead is rotated 54.74˚
relative to the static magnetic field. This so called magic-angle (which is the root of the second order Legendre polynomial, 3 cos 2 - 1) is utilized since the chemical shift anisotropy (CSA), dipolar,
and first-order quadrupolar interactions are all averaged to zero if the rotation is rapid enough. In
practice, it is virtually impossible to spin the sample fast enough to average all of these interactions
(the dipolar coupling in 1H-1H systems is often times 80 kHz, the quadrupolar coupling in 17O might
be 5 MHz, the CSA in 209Pb systems can be 100 kHz or more wide) since the fastest spinners today
go about 30-50 kHz and ours at Berea can only go 10-15 kHz. Even though we may not spin out
75
the interaction completely, the resulting spinning sideband patterns may be analyzed. For the quadrupolar central transition, however, there still remains a second order quadrupolar interaction which
is not completely averaged under MAS conditions. Unlike a spin 1/2 nucleus (such as 13C or 31P,) a
quadrupolar nucleus (such as 87Rb) will produce a broad powder pattern under MAS conditions (see
figure 19.5 above). These patterns, just as in the static case, may overlap if multiple sites are present in a sample. In practice, if the patterns have sharp singularities, the quadrupolar coupling pa-
Figure 19.5 MAS Powder patterns produced from quadrupolar samples with different
asymmetry parameters ( ). Overall width of each pattern is determined by the quadrupolar coupling constant (CQ).
rameters may be extracted (even from overlapping patterns). For the RbNO3 it is possible to do
such a three site deconvolution to extract the coupling constants; for the Rb2SO4, one of the sites is
easy to simulate but the second site is complicated due to the large number of spinning sidebands
which change the shape of the observed pattern. Until ten years ago, this would be where this lab
ended and no solution would be possible. With the discovery of double-rotation (DOR)38-40, dynamic-angle spinning (DAS)39,41-48 and the newest multiple-quantum magic-angle spinning
(MQMAS)49-55 experiments the playing field has been leveled and we can take the next step to obtain high resolution spectra from these samples.
In the third experiment of this laboratory you will obtain the MQMAS spectrum of your
sample. The MQMAS experiment is basically a sheared triple-quantum (3Q) / single-quantum
(1Q) correlation spectrum. If the 3Q and 1Q evolution periods are added in the appropriate ratio,
the anisotropic portions of the evolution will cancel each other out and the resulting spectrum will
be isotropic (i.e. sharp single lines for each site in the sample.) This type of correlation is most
easily performed with a simple three pulse sequence shown in figure 19.6. This pulse sequence has
a number of features which make it superior to other MQMAS sequences. First, the phase cycle is
only 24 steps long and selects only two pathways which may contribute signal. One is the desired
pathway and the other is one that goes through a –3 coherence during t1 and then remains –1 after
the second pulse. This type of coherence will not generate an echo and will be add no signal to the
spectrum. If the phase cycle were expanded by a factor of two, this pathway could also be eliminated as well but this is not critical. Second, by acquiring a whole echo, the observed spectrum
may be phased purely absorptive. This will lead to the most narrow lines in both dimensions. This
76
sequence does have the drawback that the echo will shift in the acquisition window with time. This
is accounted for by either doing a shearing transformation after processing both dimensions or alternatively by applying a time dependent first order phase correction following the first dimension
FT. For processing our data, we will use a program written by Philip Grandinetti of Ohio State
University called RMN. The P1 and P2 are high power pulses (usually 180-720˚ of rotation in
length) that must be optimized to achieve maximum signal-to-noise, while the 180˚ pulse will be a
standard central transition selective pulse. To increase the overall signal to noise ratio, the second
pulse (P2) may be replaced with a repeating 90x-90-x (or other appropriate alternating phases)
sequence called fast amplitude modulation (FAM). In this case, the phase cycle would have to be
Figure 19.6 Multiple-Quantum Magic-Angle Spinning pulse sequence with shifted whole
echo detection.
modified since cycling the FAM pulse through four phases would be difficult. In either case, once
the two-dimensional data set is processed, slices through each isotropic peak will be extracted in the
MAS dimension and fit using simulation software.
A fourth approach to studying quadrupolar spectra with substantial sideband overlap is to
use a sequence called QPASS developed by Massiot et al. Using this sequence of 180˚ pulses, the
spinning sideband amplitudes will be modulated in such a way that they may be folded back into
the main peak. The resulting spectrum can be projected sideband free (as if spinning at a rate much
greater than the overall anisotropy) and thus simulated as you would any multiple site overlapping
powder pattern.
77
Laboratory 20 (Quantum Chemistry)
Dipolar Couplings Measured in Partially-Oriented
Solutions Using NMR Spectroscopy
Introduction:
Dipolar coupling, like quadrupolar coupling (see experiment 19) is normally not present in
liquid state experiments (other than providing a relaxation mechanism, see experiment 8.) This is
purely because the random motions of the molecules in solution effectively average the orientation
dependence of these interactions. However, in a solid, orientational dependences will not average
and in fact are what give rise to broad (oftentimes uninterpretable) static spectra which require advanced spinning approaches to achieve high resolution. A similar situation occurs when studying
samples which are prepared in a partially orienting medium (i.e. liquid crystalline solvents, oriented
membranes). In this case, the time dependence will not average completely. It is this dependence
that will be used to measure the dipolar couplings in a small molecule.
Theory:
For homonuclear dipolar couplings, the orientational dependence is given in equation 20.1
below
Dij =
3cos 2 ij − 1
rij6
2
2
where rij is the distance between the two spins, is the gyromagnetic ratio and
(20.1)
ij
is the angle be-
tween the internuclear vector between spins i and j and the static external magnetic field. In a liquid solution the rapid tumbling will average this dipolar coupling.
Dij =
2
6
ij
2r
3cos 2
ij
−1
(20.1)
Notice that this equation is valid when measuring dipolar couplings between spins in the same
molecule (where molecular geometry is fixed) whereas for spins in different molecules the rij term
would also be averaged. For an isotropically tumbling liquid state molecule, this angle dependent
term will average to nothing but in a liquid crystalline system this term will be non-zero with a
value between –2 and 2 (though it is normally quite nearly zero).
78
Figure 21.1 Spectrum and Simulation of Acetone dissolved in a liquid crystalline solvent.
79
Laboratory 21 (Quantum Chemistry)
Strongly J-Coupled Spin Systems in NMR Spectroscopy
Introduction:
One of the most important interactions in liquid state NMR spectroscopy is J-coupling; this
interaction forms the basis of many of the multi-dimensional pulse sequences used in structural
analysis by NMR. For systems where the chemical shift differences are much larger than the Jcoupling, the spectral impact is relatively simple to predict (as you might recall the n + 1 multiplet
rule from organic chemistry for a spin J-coupled to n other magnetically equivalent spins). When
the chemical shift difference is small, however, the impact of J-coupling is substantially more difficult to determine without numerical simulations. For this experiment you will explore how Jcoupling can affect an NMR spectrum and produce different spectral patterns. The ultimate goal is
to learn how to interpret these patterns and recognize them where and when they occur.
Theory:
The two-spin J-coupling is described by a Hamiltonian which is not strictly diagonal in the
eigenbasis of the simple individual spin states.
HJ = 2 J12 I1 • I2 = 2 J12 (I Z 1I Z 2 + IY1 IY 2 + IX 1I X 2 )
(21.1)
In the case of high field experiments where chemical shift differences are large, the J-coupling
Hamiltonian is truncated to only the Iz terms. The standard basis functions given below are modified using second-order perturbation theory or full Hamiltonian matrix diagonalization.
= c1
1 =
2 =
3 =
4 =
+ c2
+ c3
(21.2)
+ c4
(21.3)
These would mix (equation 21.3) to form new eigenfunctions which lead to slight shifts in the energy level diagram. The effect on a two-spin NMR spectrum is seen in figure 21.1 in which the
pair of 1:1 doublets present in the weak coupling case (a) slowly merge to nearly a single peak (c)
in the case where the J-coupling is much larger than the chemical shift difference. Notice that even
in the case of the strongest J-coupling, the four peaks are still present but the intensities and positions shift considerably. Another way to imagine this spectrum is to imagine that the inner two
80
peaks of the pair of doublets cannot cross one another but must always remain separated by the Jcoupling from the outer peaks. In the case of the two spins having identical chemical shifts, the inner doublet peaks indeed collapse to one peak and the intensity of the outer peaks vanishes. This is
why there is no splitting observed between chemically equivalent sites.
Figure 21.1 Simulated spectra for a two spin system with a) ±180 Hz chemical shifts and
5 Hz J-coupling, b) ±50 Hz chemical shifts and 100 Hz J-coupling and c) ±10 Hz chemical shifts and 200 Hz J-coupling.
In situations where the spins are more numerous than the simple two spin case, the simulation becomes more difficult. In general, the problem scales as the diagonalization of a 2n x 2n matrix for
an n-spin system. In practical terms this limits the size of the spin system to no more than about 12
spins (unless there exist degeneracies of chemical shift values).
81
Laboratory 22 (Thermochemistry)
Measurement of the Joule-Thompson Coefficient
Introduction:
The determination of various gas constants has been one of the standard experiments performed in physical chemistry laboratories for most of the history of the subject. We recognize that
these experiments measure constants the form the basis of the rest of physical chemistry (recall that
all chemical potentials are related to one another via the chemical potential in the gas phase). The
Joule-Thompson coefficient is measured normally using an adiabatic expansion through a membrane or porous plug (see figure 22.1). This laboratory has been modified throughout the years, but
fundamentally has remained the
same adiabatic expansion. You
will be using this approach to
determine the Joule-Thompson
coefficient for three gases (CO2,
He and N2) and comparing this
result to both theoretical (via the
Van der Waals equation) and
other experimental values.
Theory:
The basic theory used to
measure the Joule-Thompson
coefficient is to consider an
adiabatic expansion through a
porous plug (see figure 22.2).
Figure 22.1 Joule-Thompson Expansion Experimental Setup.
The basic idea is that the gas
expands (from a high pressure region to a low pressure region) in an adiabatic fashion (meaning no
heat is transferred to the gas from the surroundings). Consider the implications of such an expansion as a two step process. In the first process, the gas is compressed under constant pressure on
the high pressure side of the porous plug and then is expanded under lower pressure on the low
pressure side of the plug. The work done to compress the gas on the high pressure side will be
simply w H = p1 V 1. On the low pressure side the work of expansion will be wL = – p2 V 2. Now if
we consider the system as a whole ∆U = U2 – U1 = p1 V1 – p2 V2. This equation may be rewritten to
82
bring various terms together giving H2 =
U2 + p2 V2 = U 1 + p 1 V1 = H1. The implication is that the adiabatic expansion is a
constant enthalpy process. Recall the
definition of the Joule-Thompson coefficient itself, µ = ( T / ∂p)H. This implies
that if we measure the infinitesimal
change in temperature through an infinitesimal change in pressure under constant enthalpy, we have a direct measure
of the Joule-Thompson coefficient. This
Figure 22.2 Adiabatic Expansion Through a Frit.
is exactly what the experimental apparatus in figure 22.1 (and figure 22.2) are designed to perform.
By graphing the ∆T as a function of ∆p and extrapolating the line back to ∆p = 0 should give a
good estimate of µ. In fact, using a program such as Statview we can fit this data using a first or
second order polynomial and arrive at a good estimate of the slope at ∆p = 0 as well as the error
implicit in this measurement.
The second portion of this experiment is to compare the measured µ and compare it to both
theory and experiment. The experimental values of µ may be found in the literature or your textbook and they have been determined using instrumentation similar to ours. The theoretical values
one arrives at from the Van der Waals equation is a more interesting problem to tackle. First we
need to consider how to perform the derivative in question ( T / ∂p)H under constant enthalpy. Obviously it is rather difficult to perform this analytically since enthalpy does not appear as an explicit
variable in the Van der Waals equation (p = R T / (Vm – b) – a / Vm2). The first step is to try to rewrite the Joule-Thompson coefficient to remove enthalpy from the equations.
µ = ( T / ∂p)H = – ( T / ∂Hm)p ( Hm / ∂p)T = – ( Hm / ∂p)T / Cp,m
(22.1)
Recall that Gm = Hm – TSm and that dGm = –SmdT + Vmdp = (∂Gm/∂T)p dT + (∂Gm/∂p)T dp which
may be used to simplify equation 22.1.
µ=–(
m
/ ∂p)T / Cp,m = – [ Vm + T (
Now by using a Maxwell relation to simplify the (
m
m
/ ∂p)T] / Cp,m
(22.2)
/ ∂p)T term we arrive at the following rela-
tion.
µ = [ – Vm + T (
m
/ ∂T)p] / Cp,m
83
(22.3)
The only difficulty here lies in taking the partial derivative (
m
/ ∂T)p due to the impossibility of
solving the Van der Waals equation explicitly for Vm. To simplify our lives we may rewrite this
derivative below.
(
m
/ ∂T)p = – (
m
/ ∂p)T ( p / ∂T)V = – ( p / ∂T)V / ( p / ∂
)
m T
(22.4)
Of course this leaves us a pair of derivatives of pressure as a function of T and Vm which are
straightforward to perform for the Van der Waals equation of state.
(
m
/ ∂T)p = – (R / (Vm – b) ) / ( – RT / (Vm – b)2 + 2 a / Vm3)
(22.5)
This is then used in equation 22.3 to give the expression below.
µ = { – Vm – [(R T / (Vm – b) ) / ( – RT / (Vm – b)2 + 2 a / Vm3)] } / Cp,m
(22.6)
An ambitious mathematics problem might be to try to rewrite this expression in terms of p and T
instead of V m and T. In any case, to determine the theoretical value of the Joule-Thompson coefficient you need to insert the appropriate values for Vm and T under the conditions of the experiment
(you will need to use a computer to solve for Vm). The only constants you will need to have are the
a, b and Cp,m values for a specific gas under the given conditions.
Experimental:
For this experiment you will simply attach the gas to be studied (CO2, N 2 or He) to the experimental apparatus as shown in figure 22.1. You will measure a series of temperature differences
as a function of pressure drops and should begin with a low pressure drop and slowly increase this
until the ∆p is a maximum value of about 60 psi. You will need to let the system equilibrate at each
pressure drop for about 10 minutes (or longer if the temperature difference is not stable). It is suggested that you do pressure drops of 10, 20, 30, 40, 50 and 60 psi for each gas. Once you have the
data, you should convert to standard units and graph the ∆T as a function of ∆p. From the zero
pressure drop slope you may obtain the values for the Joule-Thompson coefficient (see theory section for additional details). Errors should be propagated for all variables in this experiment to determine if your values agree with theory and other experiments.
84
References
1
M. R. Mueller and K. L. McCorckle, Journal of Chemical Education 71, 169 (1994).
2
D. A. Hinckley and P. G. Seybold, Journal of Chemical Education 64, 362 (1987).
3
P. P. Blanchette, Journal of Chemical Education 64, 267 (1987).
4
D. Calvert, M. J. Smith, and E. Falcào, Journal of Chemical Education 76, 668 (1999).
5
D. P. Shoemaker, C. W. Garland, and J. W. Nibler, Experiments in Physical Chemistry, 6th ed.
(McGraw-Hill, New York, NY, 1996).
6
A. M. Halperm and S. Gozashti, Journal of Chemical Education 65, 500 (1988).
7
H. Elias and A. P. Zipp, Journal of Chemical Education 65, 737 (1988).
8
M. M. Fuson, Journal of Chemical Education 71, 126 (1994).
9
D. J. Wink, Journal of Chemical Education 66, 810 (1989).
10
F. S. Wettack, Journal of Chemical Education 49, 556 (1972).
11
H. W. Thompson, A Course in Chemical Spectroscopy (Oxford University Press, London, England,
1938).
12
F. Daniels and e. al., Experimental Physical Chemistry, 6th ed. (McGraw-Hill, New York, NY,
1962).
13
C. F. Meyer and A. A. Levin, Physical Review 34, 44 (1929).
14
G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed.
(D. Van Nostrand Co., Princeton, NJ, 1950).
15
K. M. Jinks, Journal of Chemical Education 52, 312 (1975).
16
R. S. Moog, Journal of Chemical Education 68, 506 (1991).
17
B. D. Anderson, Journal of Chemical Education 74, 985 (1997).
18
I. J. McNaught, Journal of Chemical Education 57, 101 (1980).
19
R. B. Snadden, Journal of Chemical Education 64, 919 (1987).
20
G. Long, D. Sauder, G. M. Shalhoub, R. Stout, M. H. Towns, and T. J. Zielinski, Journal of Chemical Education 76, 841 (1999).
21
C. J. Pursell and L. Doezema, Journal of Chemical Education 76, 839 (1999).
22
G. Herzberg, Molecular Spectra and Molecular Structure III. Electronic Spectra of Polyatomic
Molecules (D. Van Nostrand Co., Princeton, NJ, 1966).
23
F. E. Stafford, Journal of Chemical Education 39, 626 (1962).
24
R. D'alterio, R. Mattson, and R. Harris, Journal of Chemical Education 51, 283 (1974).
25
V. S. Steckline, Journal of Chemical Education 47, 128 (1970).
26
D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York,
NY, 1987).
27
H. M. Bell, Journal of Chemical Education 53, 665 (1976).
85
28
R. L. Jarek, R. J. Flesher, and S. K. Shin, Journal of Chemical Education 74, 978 (1997).
29
L. A. Hull, Journal of Chemical Education 67, 782 (1990).
30
P.-O. Quist, Journal of Chemical Education 73, 751 (1996).
31
L. Sacksteder, R. M. Ballew, E. A. Brown, J. N. Demas, D. Nesselrodt, and B. A. DeGraff, Journal
of Chemical Education 67, 1065 (1990).
32
J. Ruiz-Cabello, G. W. Vuister, C. T. W. Moonen, P. van Gelderen, J. S. Cohen, and P. C. M. van
Zijl, Journal of Magnetic Resonance 100, 282-303 (1992).
33
R. E. Hurd and B. K. John, Journal of Magnetic Resonance 91, 648-653 (1991).
34
W. Wilker, D. Leibfritz, R. Kerssebaum, and W. Bermel, Magnetic Resonance in Chemistry 31, 287292 (1993).
35
A. M. Torres, T. T. Nakashima, and R. E. D. McClung, Journal of Magnetic Resonance Series A
101, 285-294 (1993).
36
J. C. Madsen, H. Bildsøe, H. Jakobsen, and O. W. Sørensen, Journal of Magnetic Resonance 67,
243-257 (1986).
37
S. Patt and J. N. Shoolery, Journal of Magnetic Resonance 46, 535-539 (1982).
38
B. F. Chmelka, K. T. Mueller, A. Pines, J. Stebbins, Y. Wu, and J. W. Zwanziger, Nature 339, 42-43
(1989).
39
K. T. Mueller, B. Q. Sun, G. C. Chingas, J. W. Zwanziger, T. Terao, and A. Pines, Journal of Magnetic Resonance 86, 470-487 (1990).
40
K. T. Mueller, Ph.D., University of California, Berkeley, 1991.
41
A. Llor and J. Virlet, Chemical Physics Letters 152, 248-253 (1988).
42
K. T. Mueller, E. W. Wooten, and A. Pines, Journal of Magnetic Resonance 92, 620-627 (1991).
43
K. T. Mueller, G. C. Chingas, and A. Pines, Review of Scientific Instruments 62, 1445-1452 (1991).
44
K. T. Mueller, Y. Wu, B. F. Chmelka, J. Stebbins, and A. Pines, Journal of the American Chemical
Society 113, 32-38 (1991).
45
E. W. Wooten, K. T. Mueller, and A. Pines, Accounts of Chemical Research 25, 209-215 (1992).
46
P. J. Grandinetti, Y. K. Lee, J. H. Baltisberger, B. Q. Sun, and A. Pines, Journal of Magnetic Resonance A 102, 195-204 (1993).
47
P. J. Grandinetti, J. H. Baltisberger, A. Llor, Y. K. Lee, U. Werner, M. A. Eastman, and A. Pines,
Journal of Magnetic Resonance A 103, 72-81 (1993).
48
S. L. Gann, J. H. Baltisberger, and A. Pines, Chemical Physics Letters 210, 405-410 (1993).
49
L. Frydman and J. S. Harwood, Journal of the American Chemical Society 117, 5367- (1995).
50
A. Medek, J. S. Harwood, and L. Frydman, Journal of the American Chemical Society 117, 1277912787 (1995).
51
J. H. Baltisberger, Z. Xu, J. F. Stebbins, S. H. Wang, and A. Pines, Journal of the American Chemical Society 118, 7209-7214 (1996).
52
G. Wu, D. Rovnyak, and R. G. Griffin, Journal of the American Chemical Society 118, 9326-9332
(1996).
86
53
A. Samoson, Chemical Physics Letters 247;, 203-206 (1995).
54
G. Wu, D. Rovnyank, B. Sun, and R. G. Griffin, Chemical Physics Letters 249, 210-217 (1995).
55
D. Massiot, B. Touzo, D. Trumeau, J. P. Coutures, J. Virlet, P. Florian, and P. J. Grandinetti, Solid
State Nuclear Magnetic Resonance 6, 73-83 (1996).
87