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PCh8-99
Solid - Liquid Phase Diagram of a Binary Mixture:
The Question of Fatty Acids Dimers in the Liquid Phase
Gary L. Bertrand University of Missouri-Rolla
Overview
Two components will be assigned for this experiment. One will be a carboxylic acid,
and the other will be a relatively non-polar material.
Pre-Lab: Access the Computer-Simulated Experiment FPWin.STA or FPMac, and
observe the DEMO section of this simulation. If you have any questions about the
performance of the actual experiment, go through several exercises with the simulation.
Students will obtain the required physical properties of the assigned materials from
handbooks or other sources: Molecular Weight, Melting Point, and Enthalpy of Fusion.
These will be used in spreadsheet calculations to prepare graphs representing the
equilibrium temperature - composition diagram for co-existence of ideal solutions with
these two pure solid components. One of these diagrams will assume that the
carboxylic acid exists entirely as monomers in the melt, and the other will assume that it
exists entirely as dimers. By analysis of these diagrams, students will select four mass
fractions (in addition to the pure components) at which experimental measurements will
give the most definitive differentiation between the two models, and provide a reasonable
description of the real system.
Laboratory: Starting with 5-6 grams of a pure component (A) and adding two increments
of the other (B), three cooling curves will be obtained for one-half of the phase diagram.
The other half of the phase diagram will be obtained by starting over with 5-6 grams of
the other pure component (B) and adding two increments of A. It is likely that the
experimental work will be divided between two groups.
Cooling curves will be recorded with the Vernier data acquisition system on
laboratory computers. It is important that this data be transferred to a floppy disk, along
with a text file describing the composition for each recorded curve, before leaving the
laboratory. The TA will insure that this data is made available to all students.
Thermodynamic Relationships
The equilibrium of a component (A) between a pure solid (S) phase and a pure or
mixed liquid (L) phase:
A(S,T,P) → A(L,T,P,XA) ; XA = mole fraction
is described in terms of the Raoult’s Law activity (aA) as
T
ln(aA) = ln(XAγA) = ∫ (∆fus HA°/RT2)dT
(1)
TA°
This relationship is derived in most Physical Chemistry texts in the unit on Colligative
Properties, though many only present an 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. It is important to note that as the activity of the
component decreases (the Raoult’s Law activity cannot be greater than unity), the
equilibrium temperature must decrease, since the enthalpy of fusion (∆fus HA°) is a
positive quantity.
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The application of Eq (1) to an ideal liquid solution (γA = γB = 1) neglecting the
difference in the heat capacities of the solid and liquid phases gives
ln(XA) = -(∆fus HA°/R)(1/T - 1/TA°)
(2a)
and
ln(XB) = -(∆fus HB°/R)(1/T - 1/TB°)
(2b)
These equations may be improved by including terms for ∆fus Cp A° and ∆fus Cp B°, and with
estimated values for activity coefficients. They are used to model the solid-liquid phase
diagram for pure solids in equilibrium with an ideal liquid solution. Values of T are
calculated from these equations for values of XA and XB ranging from 0 to 1.0, generating
two lines - one from Eq. (2a) for pure solid A in equilibrium with the solution, and one
from Eq (2b) for pure solid B in equilibrium with the solution. These are graphed as T vs
XA(= 1 - XB). They cross at the temperature and composition known as the Eutectic Point.
Dimer
Monomer
This calculation is easily performed with a spreadsheet such as Microsoft Excel or
Corel QuattroPro. Since experimental mixtures will be prepared by weighing the
components, it is more convenient to base the calculations on mass fractions rather than
mole fractions. Create a column (Column B) with values for mass fraction of A ranging
from zero to 1 in increments of 0.05. In a column to the left of this(Column A), create a
column which will convert mass fractions to mole fractions. In Columns C and D,
calculate the equilibrium temperatures for components A and B. [Equation 3a shows an
error for XA = 0, and Equation 3b shows an error for XA = 1.] The equilibrium temperature
for solid A in equilibrium with the solution (the melt) is
T = 1/ [1/TA° - (R/∆fus HA°)ln(XA)]
(3a)
and for component B,
T = 1/[1/TB° - (R/∆fus HB°)ln(1 - XA)]
(3b)
By observing the differences between the temperatures calculated with Equations (3a)
and (3b) for the same composition, the cross-over condition (the eutectic) may be easily
identified. The spreadsheet may be expanded in this region for smaller increments of
composition to obtain a better estimate of the eutectic temperature and composition.
Equilibrium temperatures calculated below this temperature have no physical meaning,
since one of the components will precipitate before such temperatures are reached.
The spreadsheet can be used to construct a graph with a temperature scale ranging
from slightly above the higher melting point of a pure component to slightly below the
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eutectic temperature. Two separate lines are graphed with the solutions to Equations
(3a) and (3b), crossing with a discontinuity in slope at the eutectic point.
Dimer Model: Carboxylic acids are known to have a tendency to form hydrogen - bonded
dimers in the solid state and when dissolved in non-polar solvents. There are then two
different ways to model the “ideal solution” leading to Equations (2a,b) and (3a,b). These
will obviously differ in the molecular weight assigned to the carboxylic acid, since a dimer
weighs twice as much as a monomer. The enthalpy of fusion is an experimental quantity
which is determined with the primary units of energy/mass. The molar enthalpy of fusion
is reported for one gram-molecular-weight of the monomer, and the molar enthalpy of
fusion for the dimer model is then twice the molar enthalpy of fusion of the monomer.
On the spreadsheet discussed above, calculate the mole fraction (Column G) from
the weight fraction (Column B) for the Dimer Model, by dividing the weight fraction of the
acid by twice its normal molecular weight. These mole fractions are used to calculate the
equilibrium temperatures in Columns E and F with the same equations that were used in
Columns C and D, except that the enthalpy of fusion for the acid is doubled. Select the
data block in columns B through G to create two phase diagrams on a single graph.
These graphs based on the ideal solution models will be compared to experimental data
to gain insight regarding the actual state of the carboxyllic acid molecules in solution.
It is interesting to consider the effect of the model on the freezing point depresssion
constant (Kf ), which can be calculated from Eq (1).
Kf = (MARTA°2)/(1000 ∆fus HA°) .
The effect of the model on the molecular weight is cancelled by the effect on the molar
enthalpy of fusion, so the experimental evaluation of the freezing point depression
constant gives no information regarding the model.
Cooling Curves
Experimental solid - liquid phase diagrams are constructed from observations of the
cooling curves (temperature vs time) of molten mixtures through the point of solidification.
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 (the negative of the
enthalpy of fusion) 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 since the activity of the solvent is less
than unity. In this case, however, the formation of a pure solid phase lowers the
concentration (and the activity) 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. If supercooling is
minimal, the cooling curve shows a sharp discontinuity in slope at the temperature at
which the pure solid is in equilibrium with a solution of the original composition of the
melt. This temperature must be estimated if supercooling distorts the discontinuity.
As solid is removed from the solution, the concentration of that component decreases,
leading to further lowering of the equilibrium temperature. 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
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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 observable if the initial
composition of the liquid is close to the eutectic composition. If there are no new solid
phases, this horizontal region of the cooling curve will occur at the same temperature,
irrespective of the initial composition of the mixture. If the initial composition of the
mixture is exactly that of the eutectic, the cooling curve will flatten at the eutectic
temperature in the same manner as a pure component.
The phase diagram is constructed by plotting the temperature of the discontinuity for
each cooling curve vs the corresponding composition of the melt.
Experiment:
In the laboratory, observe the actual cooling curves for the calculated
compositions, based on the ideal solution graphs. If the expected temperature is above
40°C, the “bath” can be the air in the room. For lower temperatures, a beaker of cold
water may be required. If the break point is expected above 40°C and the eutectic is
expected at a lower temperature, observe the break (and the appearance of crystals) in
air then take a few more readings before transferring to a water bath. Try to observe the
eutectic halt for at least one composition near the eutectic composition.
Report:
Show the observed cooling curves, indicating the point at which crystals were first
observed on each. Note any eutectic points that were observed. Plot the experimental
points on the graphs constructed from the spreadsheets. Comment on similarities and
differences between the experimental and model phase diagrams.
References:
1 Atkins, Peter Physical Chemistry, 6th Ed., Freeman, NY, 1998, p. 204-5
2. Halpern, A. M. Experimental Physical Chemistry, 2nd Ed., Prentice-Hall, Upper
Saddle River, NJ 1997, p. 233.
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