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Chm 118
Fall 2015, Exercise Set 4
Quantum Levels, Populations,
Entropy, Enthalpy, and Free Energy
Mr. Linck
Version 3.8
November 10, 2015
4.1 Probability
Craps is a game in which rolling a “7” as the sum of the two dice has consequences. What
is the probability that one will roll a “7” with two dice? What is the probability that one
will roll a “2” with two dice?
4.2 Probability
What is most probable in craps, rolling a “7” or rolling a “2” with two dice?
4.3 Definitions
What is the difference between a microstate and a configuration?
4.4 Number of Microstates
NOTA BENE: In this problem and those that directly follow the symbol 9/3/1/0/1 means
nine particles in the lowest level, 3 in the first excited level, 1 in the second excited level,
etc., as illustrated in Figure 1. Energy spacing is always one unit unless stated otherwise.
Find the number of microstates in the configuration 9/0/0/1/0. What is the energy of this
configuration?
4.5 Number of Microstates
Find the number of microstates in the configuration 9/1/1/0/0. What’s its energy?
4.6 Number of Microstates
Are the configurations of the last two problems different? Why?
4.7 Number of Configurations
Is there any other configuration with the same energy as found in the last two problems?
If so, what is it? What is the predominant configuration?
4.8 Number of Microstates, W
Find the number of microstates in the configuration 5/1/1/1/0, a quantity we call W.
4.9 Number of Microstates, W
Find the number of microstates in the configuration 6/3/1/0/0, a quantity we call W.
1.16
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Figure 1: An Example of a Configuration
4.10 W Depends on N
Consider 10 particles with a total energy of 3 units. Find all the configurations and the W
of each. Identify the most probable configuration and note W for it.
4.11 W Depends on N
Consider 20 particles with a total energy of 3 units. Find all the configurations and the W
of each. Identify the most probable configuration and note W for it.
4.12 W Depends on N
From the last two problems, what do you conclude about how W for the most probable
configuration changes as N goes up? What would you do to be sure of your conclusion?
4.13 Most Probable Configuration
Imagine twelve particles in the set of energy levels. Find the configuration that is most
probable for a system with a total of two units of energy.
4.14 Most Probable Configuration
Same number of particles as in problem 13. Find the configuration that is most probable
for a system with a total of three units of energy.
4.15 Most Probable Configuration
Same number of particles as in problem 13. Find the configuration that is most probable
for a system with a total of four units of energy.
Chm 118
Exercise Set 4
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4.16 Changing Configurations
Show that W increases in going from the most probable configuration of problem 13 to that
of 14.
4.17 Changing Configurations
Show that the ratio of W15 /W14 is less than W14 /W13 , where the subscripts refer to problem
numbers, and the W’s are for the most probable configuration, hereafter Wmpc , and then
latter still, just W for reasons that will become clear.
4.18 Changing Configurations
How do you state the conclusion of the last problems in general terms? HINT: There are
two issues concerning the Wmpc , how it changes with an increase in E by a constant amount,
and how it changes at a low E (temperature) versus a high E (temperature).
4.19 Adding Heat Changes W
In the last several problems we have been talking about energy levels that remain equally
spaced. If we consider these levels for translation, then what parameters must be constant
in order to keep the energy levels equally spaced? HINT: Think about the POP problem
which is a reasonable model for translational motion.
4.20 Work and Heat
What is work? What is heat? Be careful to distinguish between work, heat, and internal
energy, the E of our problems above. HINT: Because we are looking for a precise definition,
it might be advisable to look these up.
4.21 Predominant Configuration
Find the predominant configuration for 6 particles with 2 units of energy.
4.22 Predominant Configuration
Find the predominant configuration for 26 particles with 7 units of energy. This takes some
trial and error, but is worth the effort if you examine the nature of your answer.
4.23 Changing Configurations
If one of the particles of the last problem is taken from the v = 0 level and raised to the v
new
= 1 level, does the value of W go up or down? What is the value of W
Wold ? HINT: Since we
are going to use the letter “n” to represent something in the following development, I have
labeled the quantum levels with a different letter, “v”.
4.24 Predominant Configuration
Find the predominant configuration for 26 particles with 2 units of energy. HINT: Relatively
to problem 22 this is easy.
4.25 Changing Configurations
If one of the particles of the last problem is taken from the v = 0 level and raised to the v
new
= 1 level, does the value of W go up or down? What is the value of W
Wold ?
4.26 Changing Configurations
new
How do your results for the ratio of W
Wold from problems 23 and 25? What do you conclude?
HINT: I am trying to get you to restate something you should have already stated because
it is important.
Chm 118
Exercise Set 4
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4.27 Ratio of Particles in the MPC
Does the ratio of particles in the most predominant configuration (MPC) between two
v values depend on the energy difference between those v values? You might look at
problem 22 for instance and compare the ratio nv=1 /nv=0 with nv=2 /nv=0 , where ni is the
number of particles in the ith level.
4.28 Finding the Energy Dependence of nv=1 /nv=0
In the last problem, you determined (presumably) that the ratio nv=1 /nv=0 depended in
some way on the energy. Energy is a quantity for which we can arbitrarily set a zero. Try
the following: If nv=1 /nv=0 depends on the energy of the lower level, and we change the
zero of energy, then the ratio would change. That situation is impossible! So let’s see what
happens if the ratio is dependent on the energy difference:
nv=1
= f (e1 − e0 )
nv=0
Now change the zero of energy. What happens to the ratio? Is that better?
4.29 The Ratio Must Depend on the Exponential of the Energy Difference
Given the result of the last problem and that nnv=2
= f (e2 − e1 ), show that since
v=1
nv=2
nv=2 nv=1
=
nv=0
nv=1 nv=0
if follows that
f (e2 − e0 ) = f (e2 − e1 )f (e1 − e0 )
which in turm requires that
f (ei − ej) = eβ(ei −ej)
where β is some constant (at least at a given temperature); please NOTE there are two
uses of “e” in this equation, distinguished by the presence or lack of subscripts. This leads
us to the Boltzmann equation,
ni = n0 e−(ei −e0 )/(kb T )
(1)
where β has been evaluated as 1/(kb T).
4.30 Boltzmann Distribution
Show that you get the same answer for nv=2 /nv=1 from direct use of equation 1 as you get
if you calculate the ratio in level 2 relative to that in level 0 and the ratio in level 1 relative
to level 0 and use those two numbers to compute nv=2 /nv=1 . HINT: I am simply asking
you to verify that this form for our function in equation 1 is consistent with the equation:
f (e2 − e0 ) = f (e2 − e1 )f (e1 − e0 )
4.31 Boltzmann Distribution
You have a set of equally spaced energy levels, the lowest of which has zero units of energy
and each energy level gap is one unit. If this system is in the predominant configuration
and there are 1920 particles in the e=0 level and 480 in the e=1 level, how many are in the
e=5 level? HINT: Don’t try to calculate this by evaluating factorials!
Chm 118
Exercise Set 4
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4.32 Boltzmann’s Law
If you have 1000 particles in the n = 0 level, the spacing between the levels is 4.14×10−21
J/K, and the temperature is 300 K, what is the number of particles in n = 1? in n = 2?
4.33 Changing Energy Levels and the Predominant Configurations
Imagine a system with energy spacing of two units, comprised of 20/2/0 particles for a total
energy of 4 units. Hold the energy constant, but increase the size of the container such that
the spacing between the levels becomes one unit. What configuration is roughly consistent
with 4 units of energy now?
4.34 Changing Energy Levels and the Predominant Configurations
What is W for each of the configurations of problem 33?
4.35 Changing Energy Levels and the Predominant Configurations
Does W increase in going from the small container of problem 33 to the large one?
4.36 Changing Energy Levels and the Predominant Configurations
Will a gas expand to occupy a larger volume if given the opportunity? In view of the last
problem, do you see why?
4.37 Energy Levels and Entropy
The last few problems illustrate how important energy level spacing is in determining entropy. There are generally four kinds of energy level spacings that we divide molecular
energy into. What are they?
4.38 Energy Levels and Entropy
Which of the four kinds of energy level spacings (see last problem) has the most closely
spaced energy levels? the most widely spaced?
4.39 Macroscopic versus Microscopic Views
Describe a gas in terms of the behavior of individual particles, their motions and their
relative motions. Temperature is roughly a measure of the energy of the particles, but
given populations of various levels yielding an average energy, what does the temperature
of a gas represent in terms of individual particles?
4.40 Macroscopic versus Microscopic Views
Imagine a set of billiard balls rolling around on a table with random speeds distributed
among them. Upon the collision of two balls the speed of each could be changed. However,
these are magic billiard balls, which can take up energy from a collision and store it internally. But they can do this only if the energy of the collision is sufficiently high. When the
billiard balls are moving gently on the table, is it likely that any given collision will be able
to excite the internal energy of the ball? What happens if the average velocity on the table
increases?
Chm 118
Exercise Set 4
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Figure 2: The Folly of Using “Disorder” after Lambert, F. L.; J. Chem. Ed, 2002, 79,
187-192.
4.41 Macroscopic versus Microscopic Views
We have the same situation as in the last problem, but the billiard balls now have two
different (quantum) methods of storing internal energy. One method, call it A, requires a
minimum energy of x; the second, call it B, requires 10x. As temperature is increased (what
does he mean by that?) which internal system, A or B, becomes activated first? If a given
amount of heat is added to the table (what does he mean by that?) does it go entirely to
translational energy? If not, why not? What can you say about the change in translational
energy as heat is added at low temperatures? At high temperatures?
4.42 Entropy
Let’s define entropy, S, as kb ln W, where kb is Boltzmann’s constant, 1.38 x 10−23 J /K,
and W is the number of microstates in the predominant configuration. For real systems, W
is a very large number but S is considerably smaller. Why?
4.43 Energy Levels and Entropy
At room temperature, are vibration energy level spacings (of about 2×10−22 J) important
in determining entropy? Why or why not?
4.44 Don’t Use “Disorder” in Your Arguments
Look at Figure 2 and determine which side is most “disordered”. If you say that the right
hand picture has roughly even spacing of the particles, then it is “ordered” and the left
hand picture is “disordered”. Now consider the real situation where there is a barrier at the
midpoint of the container to keep all the particles on the left hand side in the initial state.
Use the population of energy levels to assess which picture with the highest entropy.
4.45 The Change in Entropy of the System
Give a complete and logical statement about how you would assess the entropy change of
some system.
4.46 The Change in Entropy of the Surroundings
Give a complete and logical statement about how you would assess the entropy change of
the surroundings of some system. HINT: The surroundings are large. To visualize them on
a molecular level is difficult. Also, since the surroundings are large, the temperature of the
same changes little when some heat is taken into it.
4.47 Review
What the relative spacing of energy levels for electronic motion, vibration, rotation, and
translation?
Chm 118
Exercise Set 4
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4.48 Entropy
Determine which system has the highest entropy, Ag(s) at 298K or Ag(s) at 340K? How
did you reach your conclusion?
4.49 Atomic Motion in the Gas Phase
If you have a bottle of Li(g) and a bottle of the same size of Cs(g), both at the same
temperature, determine which is moving more rapidly. HINT: All the energy is kinetic and
total energy is proportional to temperature.
4.50 Another Example Taken from Lambert–see Figure 2
Given your answer to the last problem, which would you imagine is more chaotic, less
ordered, Li(g) or Cs(g). What does “disorder” predict for the highest entropy?
4.51 Another Example Taken from Lambert–see Figure 2
Now consider Li(g) and Cs(g) from the point of view of microstates, not disorder. If you
use a POP model, which has energy levels more closely spaced (in the same size container),
Li(g) or Cs(g)? Which do you predcit has the highest W and the highest entropy?
4.52 Entropy
Look up some data to determine which system has the highest entropy Cs(g) or Li(g) at
298K, both at the same pressure?
4.53 Entropy
Determine which system has the highest entropy Na(s) at 371K or Na(`) at 371K? How did
you reach your conclusion?
4.54 Entropy
Determine which system has the highest entropy NO(g) or NO2 (g) at 298K and the same
pressure? How did you reach your conclusion? HINT: There are a couple of things going
on here.
4.55 Entropy
Determine which system has the highest entropy KClO3 (s) and a liter of water or KClO3 (aq)
(in that liter of water)? HINTS: Ionic solution. Also, maybe ambiguous?
4.56 Extensive and Intensive Properties
An extensive property is one that depends on the amount of material. Give some examples.
4.57 Extensive and Intensive Properties
An intensive property is one that does not depend on the amount of material. Give some
examples.
4.58 Extensive and Intensive Properties
Is energy extensive or intensive? How about volume? Temperature? Entropy?
Chm 118
Exercise Set 4
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4.59 Thermodynamic Properties of Reactions
As chemists we most often want to express the value of a thermodynamic quantity for a
reaction of some sort. Since the quantity of material that we deal with is variable, the
most useful kind of quantity is an intensive one. What kind of property is entropy? What
would happen if we expressed entropy per mole of material transferred from the left of our
equation to the right? What kind of property—intensive or extensive—would that quantity
be?
4.60 Thermodynamic Properties of Reactions
We have a symbol for the change in entropy of a substance, dS or ∆S. Now we need one
for the quantity we just invented in the last problem, one to be used with reactions. There
are several suggestions for such symbols in use, including ∆r S, which is used in advanced
g which was suggested but never adopted,
books, but is often regarded as rather clumsy, ∆S,
and even ∆S, which is often used in introductory texts (and courses) and which leaves it
up to the user to understand the context–probably seldom the case. What would I mean if
I told you the hunk of gold in going from state a to state b has ∆S of -20 J/K?
4.61 Thermodynamic Parameters of Reactions
Referring to the last problem, what would I mean if I told you that ∆r S for
H2 O(s, T=273K) = H2 O(`, T=273K)
was 22 J/(mole K)?
4.62 Thermodynamic Parameters of Reactions
Given the equation in the last problem, how would you interpret the following sentence:
“The entropy change, ∆S, for melting water at 273K is 11 J/K.”
4.63 Thermodynamic Parameters of Reactions
What would you conclude with the information that “∆S for the melting of water at 273K
is 22 J/(mole K)”?
4.64 Reference or Standard State
A second issue we need to address is that changes in chemical reactions depend upon the
conditions of the reagents and products, their pressures, temperatures, etc. Generally, it has
been agreed that in the standard state all materials are pure, gases are at one atmosphere
pressure and all concentration are equal to one molar, and, usually, the temperature is
298.15K If this is so, the thermodynamic parameter is given a “super o”. Would the
entropy change of the following reaction at 298.15K be characterized by a ∆S o ?
C(s) + 2Cl2 (g, P = 0.1 atm) = CCl4 (`)
4.65 Entropy Change
Use a table of absolute entropies to predict the value of ∆S o for
2K(s) + F2 (g) = 2KF(s)
Give a rationalization for your answer. HINT: If the text you are using does not have a
good table of thermodynamic values, try the web page at
http://chemistrytable.webs.com/enthalpyentropyandgibbs.htm
or other sites found by searching for “enthalpy of formation” or “absolute entropy”.
Chm 118
Exercise Set 4
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4.66 Entropy Change
Use a table of absolute entropies to predict the value of ∆S o for
NH3 (g) + HBr(g) = NH4 Br(s)
Give a rationalization for your answer.
4.67 Entropy Change
Use a table of absolute entropies to predict the value of ∆S o for
CH3 CHCH2 (g) = cyclic-C3 H6 (g)
Give a rationalization for your answer.
4.68 Entropy Change
Use a table of absolute entropies to predict the value of ∆S o for
3 H2 (g) + Fe2 O3 (s) = 2Fe(s) + 3H2 O(g)
Give a rationalization for your answer.
4.69 Entropy Change and Temperature
Comment on which direction the reaction
2NO(g) + Cl2 (g) = 2NOCl(g)
will proceed if all reagents are under standard conditions and you are at some “high enough”
temperature. HINT: You need to apply your thoughts about ∆Suniv since you are being
asked about spontaneity.
4.70 Entropy Change and Temperature
Comment on which direction the reaction
SrSO4 (s) = SrO(s) + SO3 (g)
will proceed at some “high enough” temperature.
4.71 Entropy Change and Temperature
Comment on which direction the reaction
SO3 (g) = SO2 (g) + 21 O2 (g)
will proceed at some “high enough” temperature.
4.72 Entropy
The entropy of CCl4 is about 210 J/(mole K) at 65o C, 214 J/(mole K) at 75o C, but is 309
J/(mole K) at 80o C. Explain these data.
4.73 Entropy
Which would have the higher entropy at 298o K, NH3 (g) or Ne(g)? Why?
4.74 Entropy Differences
The entropy of CO2 is 213.7 J/(mole K) and that of SO2 is 248.2 J/(mole K), both at
298o K. Explain the difference.
Chm 118
Exercise Set 4
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4.75 Entropies
The entropies of NaF(s), MgO(s), and AlN(s) are, respectively, 51.5, 26.8, and 20.2 J/(mole
K) at 298o K. Give a rationalization.
4.76 Entropy with Pressure Change
Justify why one mole of a gas has a higher entropy if it is changed from a pressure of 1.0
atm to a pressure of 0.1 atm at constant temperature.
4.77 Work on a Gas
Calculate the work done on an ideal gas (in a piston and cylinder) when 1.0 moles at a
pressure of 20.0 atm. and T = 200K is expanded against an external pressure of 1.0 atm
isothermally. Find the heat.
4.78 Work Depends on Path
Do the expansion in the last problem, but in this case let the external pressure be 10.0 atm
until the piston stops moving, then reduce the external pressure to 1.0 atm. Find the work
and heat for the overall process.
4.79 Path Dependent Processes
Are work and heat dependent upon the path?
4.80 Reversible Work
Expand the gas in problem 77 from the same initial state to the same final state, but do it
reversibly. What is the work? What is the reversible heat? What can you say about the
reversible work compared to that found in problem 77 and problem 78?
4.81 Total Energy of a Collection of Particles
Write the total energy, E, in terms of the energy of the various quantum levels and the
number of particles in each level. HINT: This is a generic problem; just give the various
levels some label and sum over those that are occupied times the number of particles in
each level.
4.82 Total Energy for a Collection of Particles
Use the Boltzmann equation, equation 1, to revise your equation of the last problem so that
the number of particles in the various levels are removed.
4.83 Total Energy for a Collection of Particles
The sum in the equation you got for the last problem becomes an integral if the levels are
close enough together (as they are for translation, for instance). If we evaluate that integral
the result is the equation
3
3
E = N kb T = RT
2
2
(2)
for the translational energy of an ideal gas. In general terms, where does this temperature
dependence originate?
Chm 118
Exercise Set 4
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4.84 First Law at Constant Pressure
Use the first law of thermodynamics,
dE = δw + δq = −P dV + δq
(3)
set the pressure to a constant, and solve for the heat, called here dqP (in a clever attempt
to indicate constant pressure). Is this heat a “state” function (and hence should have a “d”
instead of a “δ”? HINT: It is if it equals a collection of state functions.
4.85 Defining Enthalpy
We define a function which we call the enthlapy, H, as
H = E + PV
Write the differential of this expression and then apply a constant pressure restriction. Does
qP from problem 84 equal dH (or ∆H for a finite change)?
4.86 Enthalpy
What is the change in enthalpy?
4.87 Enthalpy, Exothermicity, and Spontaneity
The enthalpy change for a reaction is -5.4 kJ/mole. Is the process exothermic? Is the
process spontaneous?
4.88 Enthalpy Change
Use tables of enthalpies of formation to determine the enthalpy change for the process
2SO2 (g) + O2 (g) = 2SO3 (g)
under standard conditions. What do those numbers tell you about the spontaneity of this
reaction?
4.89 Rationalizing an Enthalpy Change
Give a reason why the enthalpy change is what it is in the reaction in the last problem.
HINT: Think about what is happening.
4.90 Enthalpy of Vaporization
What is the sign of the enthalpy of vaporization of a liquid, any liquid?
4.91 Enthalpy of Vaporization
Give a molecular explanation for your answer to the last problem.
4.92 Enthalpy of Vaporization
The enthalpy of vaporization of water is 40.7 kJ/mole and that of fusion of water is 6.01
kJ/mole. Assuming these values are independent of temperature, what is the enthalpy of
sublimation of ice to water vapor at -10o C?
4.93 Enthalpy Change and Hess’s Law
State Hess’s Law. Be articulate.
4.94 Enthalpy of Combustion
Describe the process of combustion. What is the enthalpy of combustion?
Chm 118
Exercise Set 4
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4.95 Enthalpy of Combustion
The standard enthalpy of combustion of propane is -2220; of C(gr), -394; and of dihydrogen,
-286; all in kJ/mole of the substance named. Find the enthalpy change for the conversion
of graphite into propane by reaction with dihydrogen.
4.96 Enthalpy Change
Find the enthalpy change for the reaction (not balanced)
B2 O3 (s) + CaF2 (s) = BF3 (g) + CaO(s)
HINT: You will need to look up some numbers.
4.97 Enthalpy Change and Calorimetry
Describe how a calorimeter works.
4.98 Enthalpy Change and Calorimetry
How would you calibrate a calorimeter?
4.99 Quantitative Calculation of Entropy Change; Review of Concepts
What is our equation relating the change in energy to the change in the number of microstates in the predominant configuration? There were two assumptions made during the
derivation of this equation: what are they?
4.100 Quantitative Calculation of Entropy Change
We have
dE
dlnW =
kb T
How is entropy related to W? How is entropy related to dE and T?
4.101 Changes During a Rapid Expansion
To calculate the entropy change we need to evaluate dE = q when the temperature is
constant. Imagine one mole of an ideal gas in a cyclinder with a piston at a pressure of
10 atm and a temperature of 298K held in place with a pin. We remove the pin and let it
expand against an external pressure of one atm in a large temperature bath at 298K. Does
this occur quickly? How much work is done? When (rapidly or slowly) is it done? Does the
energy of the gas decrease because of the work done? Does heat flow from the temperature
bath to the system? When (rapidly or slowly)? Is the temperature of the gas constant as
this heat flows? Make an approximate plot of Tgas versus t. Can you use the heat that the
gas absorbs to calculate the entropy? Why or why not?
4.102 Entropy is a State Function
“Entropy is a state function.” Use our statistical definition in terms of the number of
microstates in the predominant configuration to justify this. HINT: A state function is
independent of path.
4.103 Finding the Correct Path to Calculate Entropy
Since entropy is a state function, if we can find a path in which we can calculate the entropy
change, and that path connects the same initial and final states, then we have the entropy
change. Our task is then to find the appropriate path. What must be true for this path,
given the issues raised in problem 101. What would that path be called.
Chm 118
Exercise Set 4
1.113
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4.104 Quantitative Calculation of Entropy Change
Do a reversible expansion of the gas in problem 101 to the same final state and determine
the heat. Use that to find the entropy change of the system.
4.105 Analysis of the Large Surroundings
Our surroundings in the expansion in problem 101 are large. If they start at T = 298K
and they are really large, say roughly the size of the Pacific Ocean, what will the final
temperature be? Do the surroundings absorb heat at constant temperature?
4.106 Quantitative Calculation of Entropy Change
Calculate the entropy change of the surroundings in the expansion in problem 101. Find
the entropy change of the universe. Is the process spontaneous?
4.107 Calculation of Entropy Change for Vaporization of Water
Calculate the entropy change of the universe when a mole of liquid water is converted to a
mole of gaseous water at a pressure of one atm against an external pressure of 1 atm and
a temperature of 298K. HINTS: Remember that the temperature does not change during
such a conversion. You will have to look up a number.
4.108 Calculation of Entropy Change for Vaporization of Water
Repeat the last problem—find the entropy change of the unverse–at a temperature of 373K
assuming that the enthalpy and entropy of the conversion are not a function of temperature.
4.109 Entropy and Pressure, a Review
Does the entropy of a mole of gas depend on the pressure? State why.
4.110 Analysis of Spontaneity in a Real System with All Species in Their Standard State
Seldom are we interested in gases expanding. In most instances of interest to a chemist, we
are focussed on a reaction. The concept of spontaneity in a reaction depends, as we shall
see, on the states of the various components. For a while we will deal with the question of
spontaneity when all chemicals are in their standard states, one atm pressure and pure. To
do this, we can proceed in one of two equivalent thinking processes: (1) We can imagine
a very large vessel of reactants and products, with all species at one atm pressure and
then allow a mole of reactants to move to products without that conversion causing any
significant change in pressure of any reagent or product; or (2) we can move a small amount
of reactants to products so that pressures do not change, compute the entropy change
for that small amount, and convert it to a “per mole” number. The description of this
system is “all species in their standard state”. Compute the enthalpy change when CaCO3
decomposes into CaO and CO2 (g) with all species in their standard state. If this occurs at
25o C, what is the change in the entropy of the surroundings?
4.111 Analysis of Spontaneity under Standard Conditions
Find the entropy change of the system for the reaction in the last problem.
4.112 Analysis of Spontaneity and Temperature with All Species in Their Standard State
Is the reaction of the last two problems spontaneous at room temperature? What would
happen at a higher temperature? Show your answer to this second question mathematically.
Chm 118
Exercise Set 4
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4.113 Entropy of Surroundings’ Change with All Species in Their Standard State
The reaction to produce formaldehyde is
H2 (g) + CO(g) = H2 CO(g)
At T = 25o C, ∆H o = 1.96 kJ/mole and ∆S o = -109.6 J/mole. Find the change in entropy
of the surroundings when one mole of formaldehyde is produced at 25o C under standard
conditions.
4.114 Entropy of Surroundings’ Change with All Species in Their Standard State
If the temperature for the formation of formaldehyde (see last problem) is raised to 50o C,
what will happen to the value of the change in entropy of the surroundings provided ∆H o
and ∆S o are approximately independent of temperature over this range.
4.115 Spontaneity with All Species in Their Standard State
Is the production of formaldehyde (see problem 113) spontaneous at 25o C?
4.116 Spontaneity with All Species in Their Standard State
Will KClO3 (s) decompose into KClO4 (s) and KCl(s) at 25o C? Consider the change in
entropy of the universe and be quantitative.
4.117 Spontaneity and Temperature with All Species in Their Standard State
What, if any, will be the effect of a higher temperature upon the decomposition of KClO3
(see last problem)?
4.118 Enthalpy Change
Find ∆H o for the reaction 2Mg(s) + O2 (g) = 2MgO(s).
4.119 Sign of Entropy Change
Estimate the sign of ∆S o for the reaction in problem 118.
4.120 Entropy Change
Find the exact magnitude of ∆S o for the reaction in problem 118.
4.121 Thermodynamic Cycle
There are a number of thermodynamic functions that are “state” functions. What does this
mean? Give some examples.
4.122 Thermodynamic Cycle
What is sublimation? Write a thermodynamic cycle that relates sublimation to two other
well-known processes. HINT: The final state of a sublimation is a gas; the initial is a solid.
Think about an indirect way to get from solid to gas.
4.123 Entropy Change of Sublimation
Predict the sign and magnitude of the entropy of sublimation relative to the entropy of
fusion (melting) of the same substance.
Chm 118
Exercise Set 4
1.131
15
4.124 Definition of a New Function of Great Use
Let’s invent a new function called G that is composed of previously defined functions:
G = E + PV − TS
(4)
Find the total differential of this equation. Then assert the conditions that T and P are
constant to get an expression for dG. Finally insert the definition of dH from problem 85
and show that
dGP,T = dHP − T dST
or, for a finite change in the standard state
∆Go = ∆H o − T ∆S o
(5)
4.125 The Equivalence of dG the Entropy of the Universe
We can take equation 5 and change the enthalpy change from a system value to one for the
surroundings, then divide all terms by T. Show that this equates -∆Go /T with ∆S o univ .
4.126 Spontaneity
What is the criterion for spontaneity in terms of WP C ?
4.127 Spontaneity
What is the criterion for spontaneity in terms of ∆S univ ?
4.128 Spontaneity
What is the criterion for spontaneity in terms of ∆G?
4.129 The Pressure Dependence of the Free Energy Change of a Substance
Go back to equation 4 and take the total differential. Now substitute into this expression
the value for dE from the first law where we specify a reversible process so that TdS =
δq/T. Show that you get
dG = V dP − SdT
(6)
4.130 Pressure Dependence of the Free Energy Change of a Substance
Take equation 6 and integrate it between the limits of pressure = 1 atm and some arbitrary
pressure, P at constant T, for an ideal gas. Note that you must substitute V for the
equivalent perfect gas expression nRT/P. Show that your answer is
P
o
(7)
G = G + RT ln
Po
where Go is the G when P = Po , the pressure in some “standard” state.
Chm 118
Exercise Set 4
1.138
16
4.131 The Change in Free Energy of a Mixture
The function G is a state property and hence must be defined by the conditions of the state.
If we consider a mixture of chemicals, say A, B, and C, then G must be a function of T and
P, as always, but also the number of moles of each of the various components. We could
write this as G = G(T, P, nA , nB , nC ). Show that at constant T and P that the change in
G would be given by
X ∂G X
dG =
dni =
µi dni
(8)
∂n i nj ,T,P
i
i
where i runs over, in our specific case, the three components A, B, and C, and nj is all of
those n except ni . In the second equality I have inserted, as is conventionally done, the
∂G
chemical potential, µi , which is simply the name for the value of ( ∂n
)nj ,T,P , which is also
i
called a partial molar free energy.
4.132 Partial Molar Free Energy
Look at the definition of µi in the last problem and see if you can justify the following
verbal description of a partial molar quantity, stated for reasons of concrete appreciation in
terms of a partial molar volume: “The partial molar volume of substance i is the change
in volume (dV) of the mixture (of which it is a component) when a small number of moles
(dni ) of substance i is added.”
4.133 Extent of Reaction
For clarity, let’s consider a specific reaction: A + B = C. and let’s define a parameter ξ
that we define as the extent of reaction
dξ =
1
dni
νi
where νi is the stoichiometric coefficient of reagent i and is a negative quantity for reactants.
Show that for the reaction above that a value of ξ = 0.5 corresponds to the correct number
of moles for the point where the reaction is half over if the initial number of moles of A and
B are both 1.
4.134 Extent of Reaction
For the reaction A + 3B = 2C, show that the extent of reaction varies from zero at the
beginning of the reaction to 1 at the end (no limiting reagent left) if the initial number of
moles of A, B, and C are respectively 1, 4, and 1.
4.135 Extent of Reaction
For the reaction A + 3B = 2C, show that the extent of reaction varies from zero at the
beginning of the reaction to 13 at the end (no limiting reagent left) if the initial number of
moles of A, B, and C are respectively 3, 1, and 2.
4.136 Change in Moles Expressed with dξ
Whatever the number of moles, ξ is a single parameter that tells us how far we have gone
from reactants toward products. We can then express the change in any component’s
number of moles with dξ. Write such an expression for dni . HINT: An exceptionally easy
problem given the definition of ξ in problem 133.
Chm 118
Exercise Set 4
1.147
17
4.137 dG for a Specific Problem
Write the second of the two equations labeled 8 for the specific reaction A + B = C.
4.138 Finding the Change in G
Use your result from the last problem for dG and your result for dni from problem 136 and
evaluate the second of the two equations labeled 8 for dG in terms of µA , etc., and dξ.
4.139 An Expression for the dG/dξ
Rearrange your result from the last problem to obtain dG/dξ, whose sign gives us the
spontaneity of a reaction at the conditions specified by the various µi .
4.140 The Pressure Dependence of the Chemical Potential
The last problem suggests our job now is to find how the µi vary with conditions. Since µi
is a partial molar quantity, the variation with conditions is the same as that for G, which
we found in problem 130 except that the variables need to refer to the individual species i.
For a gaseous example, we get
Pi
o
µi = µi + RT ln
(9)
Po
where Pi is the pressure of substance i and Po is the standard state, 1 atm. When is µi
equal to µoi ?
4.141 A Final Version of the Free Energy Change as ξ changes
Insert the value of µi from equation 9 for i = A, B, and C, respectively, into your answer
in problem 139 and rearrange. Show the result is
PC /PCo
dG
o
(10)
= ∆G = ∆G + RT ln
dξ
PA PB /(PAo PBo )
where ∆Go = µoC - µoA - µoB and where all the Po could be set to unity.
4.142 Free Energy Change
Find ∆Go for the reaction 2Mg(s) + O2 (g) = 2MgO(s).
4.143 Entropy Change in the Universe
Give an explanation of the spontaneity of the reaction in problem 142 in terms of the entropy
of the universe and its components. Be complete.
4.144 Free Energy Change
Find the free energy change for the reaction in problem 142 if the pressure of O2 (g) is
1.0 ×10−4 atm. Is the reaction spontaneous under these conditions? HINT: Since the two
solids have rather small change in volume with a change in pressure, their Go values are
nearly independent of P.
4.145 Free Energy Change
Find ∆Go for the reaction CH3 OH(g) = CO(g) + 2H2 (g) at 298K.
Chm 118
Exercise Set 4
1.155
18
4.146 Free Energy Change
Find ∆G for the reaction in the last problem if the pressure of CH3 OH(g) is 1.0 atm and the
pressures of CO(g) and H2 (g) are both (a) 1.0×10−4 atm and (b) 0.01 atm. Is the reaction
spontaneous in the forward direction under any of these conditions? in the reverse?
4.147 Free Energy Change
Find ∆G for the reaction in problem 145 if the pressure of CH3 OH(g) is 0.1 atm, that of
CO(g) is 1.0×10−2 atm, and that of H2 is 5.0×10−3 atm.
4.148 Free Energy Change and Temperature
Assuming that ∆H o and ∆S o are independent of T, find ∆Go for the reaction in problem 145
at 50o C and 150o C.
4.149 Understanding Chemistry
A long tube is prepared containing SiCl4 (g); at one end ot this tube is placed some impure
Si(s) and that end of the tube is heated. Over time at the cool end of the tube pure Si(s)
forms. Account for this result using thermodynamic arguments. HINT: Think about the
conproportionation of Si(IV) and Si(0):
SiCl4 + Si = 2SiCl2
4.150 Free Energy and Pressure
Write the chemical equation that represents benzene liquid vaporing to benzene gas at a
pressure of one atmosphere. HINT: We can express the full conditions of a gas, say X, in a
reaction via X(g, p = 0.5 atm), for instance.
4.151 Free Energy and Pressure
Write the chemical equation that represents benzene liquid vaporing to benzene gas at a
pressure of p atmosphere.
4.152 Free Energy and Pressure
We can couple the two equations from the last two problems by letting benzene liquid at
p = 1 atm. go to benzene liquid at p = p atm. What is your estimate of ∆H for this
process. HINT: Think about what energy changes take place. What is your guess about
∆S for this process? HINT: Is anything happening to the benzene liquid?
4.153 Free Energy and Pressure
To complete the cycle we have been working on, we could let benzene gas at p = 1 atm.
go to benzene gas at p = p atm. To be concrete, if p = 0.1, and we work at constant
temperature, we are changing the pressure of benzene gas at constant temperature. What
does this do to the energy under the assumption of ideal gas behavior? What does it do to
the entropy?
4.154 Free Energy and Pressure
Write the complete thermodynamic cycle from the information in the last four problems.
Let the value of ∆G at p = 1 atm. be ∆Go and the value of ∆G at p = p be ∆G. Write one
in terms of the other and additional terms as appropriate. You have just produced what
we derived more rigorously in problem 141.
Chm 118
Exercise Set 4
1.167
19
4.155 Free Energy Change and Pressure
Free energies of formation of benzene liquid and benzene gas at 298K are 124.5 and 129.7
kJ/mole, respectively. Find the value of ∆G when the pressure of benzene gas is 10 mm
Hg. HINT: 760 mm Hg is one atmosphere.
4.156 Free Energy Change and Pressure
Use data from the last problem to find the value of ∆G when the pressure of benzene vapor
is 400 mm Hg at 25o C.
4.157 Free Energy Change and Pressure
What is the value of the pressure of benzene vapor at equilibrium with benzene liquid at
25o C?
4.158 Free Energy Change and Pressure
Find the value of ∆G at 298K for the reaction in problem 145 if the pressures are: PCH3 OH(g)
= 0.15 atm. PCO(g) = 0.01 atm, PH2 (g) = 2.0 atm.
4.159 Free Energy Change and Pressure
Find the value of ∆G at 298K for the reaction in problem 145 if the pressures are: PCH3 OH(g)
= 10.0 atm. PCO(g) = 0.10 atm, PH2 (g) = 0.002 atm.
4.160 Equilibrium Constant
Find the equilibrium constant at 298K for the reaction in problem 145.
4.161 Equilibrium Constant
Find K at 298K for the reaction 2NO(g) + Cl2 (g) = 2NOCl(g).
4.162 Free Energy Change and Specified Pressure
Find ∆G at 298K for the reaction in the last problem when all pressures are 0.001 atm.
4.163 Free Energy Change and Pressure
Is the reaction
I2 (g) + Cl2 (g) = 2ICl(g)
spontaneous when all pressures are 1 atm? HINT: The free energy of formation of ICl(g) is
-6.1 kJ/mole at 298K.
4.164 Free Energy Change and Pressure
Is the reaction in problem 163 spontaneous when the pressure of ICl is 0.001 atm and the
other pressures are 1.0 atm?
4.165 Free Energy Change and Pressure
Is the reaction in problem 163 spontaneous when the pressure of ICl is 10.0 atm and the
other pressures are 1.0 atm?
4.166 Free Energy Change and Pressure
Offer a rationalization as to why the standard free energy change for the reaction in problem 163 is so small in magnitude.
Chm 118
Exercise Set 4
1.176
20
4.167 Free Energy Change and Equilibrium Constant
For the reaction
ZnF2 (s) = Zn2+ (aq) + 2F– (aq)
the value of K is 0.003. Find the value of ∆Go for this reaction.
4.168 Free Energy Change and Concentration
If you make 50 mL of a 2 M solution Zn(NO3 )2 and 50 mL of a 4.0M solution of CsF, and
mix them, what will happen? HINT: Ignore any reactions of nitrate ion and cesium ion,
but pay attention to the last problem.
4.169 Some Thermodynamic Statements
For each statement, indicate if it is true or false. If false, fix it. If the statement is just silly,
say so and fix it.
• All spontaneous reactions take place quickly.
• If a reaction is spontaneous, the reverse reaction is nonspontaneous.
• All spontaneous reactions release heat.
• The boiling of water at 100o C and 1 atm pressure is a spontaneous process.
• If a process increases the disorder of the particles of a system, the entropy of the
system increases.
• The energy of the universe is a constant. The entropy of the universe just grows and
grows.
• Both the entropy change of the system and the entropy change of the surroundings
are zero at equilibrium.
• A spontaneous process has a positive change in entropy.
4.170 Free Energy Change
List five things that ∆G is.
4.171 Enthalpy Change
What is the enthalpy change?
4.172 Entropy
What is entropy?
4.173 Boiling Point
Assuming that ∆H o and ∆S o are independent of temperature, find the boiling point of
CCl4 (`).
4.174 Boiling Point
The normal boiling point of trimethylphosphine is 38.4o C. Its vapor pressure at -45.2o C is
0.017 atm. Find ∆H o , ∆S o , and the vapor pressure at 15o C.
Chm 118
Exercise Set 4
1.184
21
4.175 Enthalpy Change
Find the enthalpy change for the reaction BaCO3 (s) = BaO(s) + CO2 (g).
4.176 Enthalpy Change
Find the enthalpy change for the reaction BaSO4 (s) = BaO(s) + SO3 (g)
4.177 Enthalpy of Combustion
The enthalpy of combustion of acetone, CH3 C(O)CH3 , is -1821.4 kJ/mole and that of
propanal, CH3 CH2 C(O)H, is -1845 kJ/mole. Which compound is more stable? By how
much?
4.178 Spontaneity and Concentration
For the reaction
HNO2 (aq) = H+ (aq) + NO–2 (aq)
determine if the process is spontaneous to the right when the concentrations are: [HNO2 ] =
0.010; [H + ] = 0.00010; [NO–2 ] = 0.0150.
4.179 Spontaneity and Concentration
For the reaction
HNO2 (aq) = H+ (aq) + NO–2 (aq)
determine if the process is spontaneous to the right when the concentrations are: [HNO2 ]
= 0.010; [H + ] = 0.0010; [NO–2 ] = 0.00150.
4.180 Spontaneity and Concentration
For the reaction
HNO2 (aq) = H+ (aq) + NO–2 (aq)
determine if the process is spontaneous to the right when the concentrations are: [HNO2 ]
= 0.0010; [H + ] = 0.0010; [NO–2 ] = 0.0150.
4.181 Mixed Equilibrium Constants
Imagine you have a chemical reaction with a ∆Go 1 and an associated K1 . You also have
a second chemical reaction with a a ∆Go 2 and an associated K2 . Show, using the additive
property of ∆Go , that the equilibrium constant for the reaction that is the sum of chemical
reactions 1 and 2 is K = K1 K2 .
4.182 Determining ∆H o and ∆S o from Equilibrium Constants
Use equation 5 and the relationship between ∆Go and lnK to show if you determine K at
various temperatures and then plot ln[K] versus 1/T that you can obtain ∆H o and ∆S o .
4.183 Review of Qualitative Microstate Reasoning
Figure 3 show two sets of energy levels. In set A the long (black if you are viewing in color)
lines represent the quantum levels of a reactant and the short (red) lines those of a product
of a reaction. Is this reaction endo- or exothermic? Will it occur at higher temperatures to
produce more product than reactant? Explain your reasoning.
Chm 118
Exercise Set 4
1.188
22
Figure 3: Figure for Problems 183 and 184. The energy level in arbitrary units is listed on
the left.
4.184 Review of Qualitative Microstate Reasoning
There is a second set of energy levels in Figure 3, set B. The long (black if you are viewing
in color) lines represent the quantum levels of a reactant and the short (red) lines those
of a product of a reaction. Is this reaction endo- or exothermic? Will it occur at higher
temperatures to produce more product than reactant? Why? Try to give an example of a
system for which these quantum levels might occur.
4.185 Units
What are the “units” of kb T?
4.186 Quantitative Application in Figure 3B
In the following table, for several values of kb T, in the arbitrary units used for the spacings
of the energy levels, are given the fractions of the total particles in “reactants” (long, black
lines) and “products” (short red lines) quantum levels illustrated in Figure 3B. How, in
general, did I get these data? Show using these values that the equilibrium constant for
reactants going to products are consistent with your qualitative observations from the last
problem.
kb T
0.2
0.5
1.0
1.4
f(A)
0.9998
0.9196
0.5267
0.3610
f(B)
0.0001
0.0874
0.4733
0.6388
4.187 Estimations Using Thermochemical Data
Make an estimate (why is it an estimate?) of the temperature at which SO3 and a mixture
of SO2 and O2 are present in equilibrium with all pressures equal to one atmosphere?
Chm 118
Exercise Set 4
1.197
23
4.188 Balancing Oxidation-Reduction Equations
Write a stoichiometrically balanced reaction for the process
Cl2 O7 + H2 O2 = ClO–2 + O2
in basic solution.
4.189 Balancing Oxidation-Reduction Equations
Write a stoichiometrically balanced reaction for the process
Cr3+ + MnO2 (s) = Mn2+ + CrO2–
4 in basic solution.
4.190 Balancing Oxidation-Reduction Equations
Write a stoichiometrically balanced reaction for the process
H2 SeO3 + ClO–3 = HSeO–4 + Cl2
in acidic solution.
4.191 Potential of Electrochemical Cell
Find the potential for the cell
Pt(s) | Fe3+ (aq),Fe2+ (aq) k Ag+ (aq) | Ag(s)
if all concentrations are 1.00 M.
4.192 Potential of Electrochemical Cell
Find the potential for the cell
Pt(s) | Fe3+ (aq),Fe2+ (aq) k Ag+ (aq) | Ag(s)
if all concentrations are 1.00 M except [Ag+ ] = 0.01M.
4.193 Potential of Electrochemical Cell
Find the potential for the cell
Ag(s) | AgI(s) | Ag+ (aq, 0.025M) k Cl– (aq, 0.67M) | AgCl(s) | Ag.
4.194 Concentration Cell
Explain what is behind the workings of a “concentration” cell.
4.195 Concentration Cells
Will the cell
Ag | Ag+ (aq, 0.0010 M) k Ag+ (aq, 1.00 M) | Ag
develop a voltage? What is it? Why does it develop a voltage?
4.196 Concentration Cells
Assume that in a nerve cell that the concentration of potassium ion inside the cell is 25
times as great as that found outside. What potential is developed across the cell wall?
4.197 Electrochemical Cells and Entropy
Assume for this problem that ∆H o and ∆S o are independent of temperature. Show that
if you measure the potential of a cell at temperatures T1 and T2 (and get E1o and E2o , that
nF (E2o −E1o )
.
you can evaluate the value of ∆S o as (T2 −T
1)
Chm 118
Exercise Set 4