Temperature Dependence of Equilibrium
Bruce H. Mahan University of California Berkeley
A first experiment in general chemistry
Recently increasing attention has been given to the problem of introducing thermodynamic concepts into the freshman chemistry course. One of the most useful results of thermodynamic analysis is the expression for the temperature dependence of the equilibrium constant:
This equation says that In K is a linear function of l / T if AH0 and AS0 are independent of temperature. The temperature dependence of AHo and AS0 can be expressed as
where K , and Ks are the equilibrium constants a t the temperatures Tiand Ti,AHo is the standard enthalpy change for the reaction, and R is the gas constant. This valuable equation applies not only to chemical equilibria; it also forms the basis for understanding the temperature dependence of vapor pressure, boiling point elevation, and freezing point depression phenomena. Its derivation need not involve the GibbsHelmholtz equation. Starting from the fundamental relations which relate the equilibrium constant to the standard free energy change, AFo, the standard entropy change, ASo, and the standard enthalpy change
where ACy is the difference in the heat capacities of products and reactants. Clearly if AC?,is very small, AHo and AS0 are both virtually independent of temperature and we can write
APo = AH'
In&= In K,
=
AHo --+KT* AHo
-ST,
ASn
R AS'
+R
and
- TASO
APo = -RT
In K
we can write AH'
InK=--+RT
AS0
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reagents, and whose equilibrium constant can be determined rapidly a t several temperatures without the
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use of a thermostat. The dissolution of naphthalene in diphenylamine meets all these specifications Since the reaction can be written as naphthalene (pure solid) =naphthalene (solution, cone X)
the equilibrium constant is K = X , where Xis the mole fraction of naphthalene in a solution which is in equilibrium with pure solid naphthalene The temperature dependence of the equilibrium can be found by weighing out naphthalene and diphenylamine to form mixtures of known composition, melting them in a water bath, and determinmg the temperature a t which naphthalene first precipitates by the cooling curve method A plot of log X as a function of the reciprocal temperature yields a straight line of slope - AH0/2.3 R. I t is easy to see that the A H o determined this way is the enthalpy of fusion of naphthalene. If the reaction is ttritten as two steps pule solid naphthalene ±-pure l q u ~ d¥" solution it is clear that the enthalpy changeof the first step is the enthalpy of fusion of naphthalene, while the enthalpy change of the second step is zero, since the solution is ideal.
The Experiment
Data for cooling curves (temperature us. time) are obtained on the pure solvent, naphthalene, and on at least four solutions of diphenylamine in naphthalene in the range of 1 to 0.4 mole fraction naphthalene. Conventional procedures can be used; we suggest having the students plot each cooling curve (temperature readings every 30 seconds) to identify the temperature a t winch the slope changes as well as having them attempt to identify the temperature at which the first solid appears. A 20-gram sample of solvent naphthalene is sufficient, if weighings are made to 0.10 gram. The addition of successive samples of solute to the same amount of solvent simplifies the procedure. Diphenylamine can be added until the naphthalene mole fraction is about 0.4; more diphenylamine will not remain in solution. The four values of log X as a function of 1 / T are plotted and the enthalpy of fusion of naphthalene is evaluated from the slope. Typical student results are shown in the drawing. Generally the answers fall within ±50 calories of the accepted value of 4600 calories.
solid. Defined in this way, AFo is a function of temperature, and is zero only a t the freezing point of pure naphthalene. On the other hand, the condition for equilibrium at all temperatures is" that AF = 0. This means that a t equilibrium, T A S = A H , or if A H o is independent of temperature. T A S = AHo. Thus in order for equilihrium to occur a t a lower temperature than the melting point of pure naphthalene, the entropy change associated with the solution process, AS, must he greater than ASo, the entropy of fusion. By using the molecular interpretation of entropy, we can say that the entropy of fusion, ASo, is a measure of the increased freedom of movement which the molecules in the pure liquid have compared with those in the solid. Equivalently we can say that the entropy of fusion measures the increased disorder of the pure liquid compared with the solid, since if molecules have available to them more positions and types of motion, we arc less able to predict their hehavior; thus we characterize them as more disordered. Now if a solid has the opportunity to "melt" or dissolve not into its pure liquid phase, hut into another liquid, the gain in the number of positions available to its molecules is even greater, and increases as the solution formed becomes more dilute. That is, solute molecules in a dilute solution have available to them a greater number of positions than they would in a more concentrated solution or in a pure liquid. Consequently, AS, the entropy associated with the process pure solid state -solute
in solution, cone X
increases as X decreases, and thus the condition for equilibrium between the solid and its ideal solution, TAX = AHo, can be satisfied at lower and lower temperatures
Making Thermodynamics Meaningful
The simplicity of the solubility equilibrium makes it an excellent device for pointing out the difference between AFo and AF The standard free energy change, AFo, is the difference in free energy between the pure liquid naphthalene and the pure solid, while AF is the difference in free energy between the naphthalene in solution a t a concentration X and the pure
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Typical student data for the temperature dependence of the solubilityof naphthalene in diphenylamine. The slope of the line corresponds to. AHo = 5100 calories.
