Download Molar Mass by Freezing Point Depression

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PCh8a-98
Molar Mass by Freezing Point Depression
Gary L. Bertrand University of Missouri-Rolla
Cooling Curves
When a pure liquid is cooled, the temperature may drop below the
melting point without the formation of crystals - a phenomenon known as
“supercooling”. As soon as the first crystals form, however, the temperature
rises quickly to the melting point and remains constant. The heat (enthalpy)
released by the solidification process exactly compensates the heat transfer
to the surroundings, maintaining the equilibrium temperature as long as
both liquid and solid phases are present. When solidification is complete
the temperature again decreases with time.
A similar process occurs if an impurity is present in the liquid, but after
supercooling, the temperature returns to a slightly lower value as predicted
by thermodynamics. In this case, however, the formation of a pure solid
phase lowers the concentration of that component in the liquid phase and
thus lowers the equilibrium temperature. The heat of solidification
compensates for the heat transfer to the surroundings, at a slowly but
steadily deceasing temperature. The rate of cooling increases as solid is
formed.
The composition of the remaining liquid phase may eventually reach a
point such that both materials begin to solidify - the eutectic point. The
liquid composition and the temperature then remain constant until all of the
material has solidified. This region of the cooling curve is often not
observed, and may be obscured by the appearance of new solid phases.
When these complexities are not present, this region of constant
temperature will be observeable if the initial composition of the liquid is
close to the eutectic composition. This flattening of the cooling curve will
occur at the same temperature irrespective of the initial composition of the
liquid. If the initial composition of the liquid is exactly that of the eutectic,
the cooling curve will flatten at the eutectic temperature in the same manner
as a pure liquid.
Equilibrium
The equilibrium of a component (A) between a pure solid (S) phase and a
pure or mixed liquid (L) phase:
A(S,T,P) æ A(L,T,P,X A ) ; XA = mole fraction
is described in terms of the Raoult’s Law activity (aA ) as
T
ln(aA) = ln(XAnA) = ∫ (∆fusH A°/RT2 )dT
(1)
T A°
This relationship is derived in most Physical Chemistry texts in the unit on
Colligative Properties, though many only present the approximate solution.
The utility of Eq (1) derives from the fact that the quantity on the right
depends solely on the temperature and properties of the pure component.
Molar Mass of a solute by Freezing Point Depression:
The integral may be evaluated precisely using experimental values for the
enthalpy of fusion and the molar heat capacities of the pure liquid and solid,
and expanded in the freezing point depression (θ) to obtain:
ln(XAγA) = -(∆fusH A°/RTA°2 )θ(1 + b θ + ... )
(2)
with
θ = TA° - T ; b = 1/TA° - ∆fusCpA°/2 ∆fusH A°
The parameter b is normally positive and less than 1/TA°, so that it does not
contribute significantly for temperature changes up to a few degrees. The
activity coefficient of the solvent is approximated as unity
( γA = 1), and the logarithmic term is expanded in series:
ln(XA) = ln(1 - XB) = - XB - X B 2 /2 - ...
and the mole fraction of the solute is related to the molality (mB) as
XB = mBMA/(1 + mBMA) ;
MA = molar mass of solvent in kilograms ,
giving the relationship
mBMA = (∆fusH A°/RTA°2 )θ(1 + b’θ) ; b’ = b + ∆fusH A°/2RTA°2
In the limiting case of very small values of mB and θ, this becomes
θ = K fmB ,
with Kf (the molal freezing point depression constant) given as
Kf = MA RTA° 2 /∆fusH A° .
(3)
The relationship may be extended to higher concentrations with theoretical
or experimental values of b’
mB = (θ/Kf)(1 + b’θ).
(4)
If theoretical values for K f and b’ are not available, values of θ are measured
for a series of solutions of solute(s) of known molar mass and measured
molality, allowing determination of 1/Kf and b’ from the intercept and slope
of a graph of mB/θ vs θ. Values of θ are then measured for solutions
prepared with solutes of unknown molar mass (g grams of solute in G grams
of solvent). A value of mB is calculated for the unknown solution from θ, K f ,
and b’; then the molar mass of the solute (M B) is calculated as
MB = g/mBG
In some cases, particularly at higher concentrations, the calculated molar
mass may show a trend with concentration, and should be extrapolated to
θ = 0. The “best” extrapolated value is obtained as:
MB = K f/Lim(θ G/g) as θ -> 0.
Simulation:
The simulation FP_Mac or FP_Win contains a brief discussion of the
principles involved in this experiment, and a demo of the experiment. The
program asks the user to type an assigned code number which determines
the “unknown” for that session. In performing the “experiment”, the user
selects a solvent and a solute (which may be an unknown) from the menu. A
mouseclick on the “Properties” button displays the molar masses, melting
points, and enthalpies of fusion of the named components. The default
condition is 10.00 grams of solvent and 0 grams of solute, but these may be
reset to 0 - 20 grams of either, in increments of 0.10 gram. The program
chooses initial temperatures for the melting and cooling baths, but the
melting bath may be set in the range of - 10 to 180°C, in increments of 10°,
and the cooling bath may be set from - 25° to within 5° of the melting bath, in
increments of 5°. The chosen conditions may lead to warning messages of
“The sample is not completely melted” or “No solidification may be observed
under these conditions.” A mouseclick on the “cooling curve” button leads to
a slight delay, then the temperature-time curve is displayed. A mouseclick
on the graph “zooms” the vertical axis in or out. A horizontal marker line is
shown in the “zoom-in” mode. This may be dragged with the mouse to
facilitate reading temperatures. On leaving the graph, the user may select
“Retain” or “Return” to keep or erase the display. The “retain” feature allows
the display of several color-coded data sets. Color bars on the right side of
the graph may be clicked for a reminder of the compositions.
Simulated Experiment:
Textbooks rarely mention the fact that there are some limitations on the
choice of solvent for freezing point depression measurements. The solute
must have sufficient solubility in the solvent to lower the freezing point
significantly. A few “quick-and-dirty” experiments with the unknown and a
few different solvents will often save a great deal of wasted effort. A good
rule of thumb is that the freezing point should be lower with a 12% (w/w)
solution than with a 10% solution. Otherwise the solute is precipitating at a
concentration which is probably too low for an accurate molecular weight
determination by this method. This test should also be applied to the known
solutes which are being considered for this experiment.
Choose a solvent (A, benzene, naphthalene, diphenyl, etc.) and two known
solutes (B) of differing molar mass and chemical nature. Calculate the
relative masses of solvent and solute necessary to prepare solutions of mB =
0.50, 1.00, and 1.50 moles of solute per kilogram of solvent or each solute.
You will not be able to prepare these concentrations precisely, so get as
close as you can and then calculate the exact molalities of the solutions you
prepare. Measure the freezing points for these solutions as accurately as
possible, and calculate the freezing point depression (θ). Select the
unknown as solute, and measure the cooling curve for a 5% (w/w) solution.
By comparing the observed freezing point depression to the observations on
known solutes, estimate the relative masses required to prepare solutions of
the unknown with molalities near 0.50, 1.00, and 1.50. Prepare the solutions
and observe their cooling curves to determine the freezing points.
From the data for known solutes, construct a graph of mB/θ vs θ, and
determine the “best” values for K f and b’. Use these values to calculate the
“best” value of the molar mass of the unknown.
To be sure of your results, select a different solvent and repeat the entire
process.
Report:
Report the freezing point depression constant(s) you have determined, and
compare to the theoretical value(s). Report the Molecular Weight of your
unknown, along with its estimated uncertainty.
References:
1 Atkins, Peter Physical Chemistry, 6th Ed., Freeman, NY, 1998, p.179-80