C. Thermodynamic Relationships Involving the Equilibrium Constant1

1 For the limitations and for further explanation of these relationships, see, for example, K. Denbigh, The Principles of Chemical Equilibrium, 3rd ed. (Cambridge: Cambridge University Press, 1971), p. 138.

For the gas-phase reaction

Image

1. The true (dimensionless) equilibrium constant

RT lnK = –ΔG

Image

where ai is the activity of species i

Image

Image

Image

where γi is the activity coefficient

Image

K = True equilibrium constant
Kγ = Activity equilibrium constant
Kp = Pressure equilibrium constant
KC = Concentration equilibrium constant

2. For the generic reaction (2-2), the pressure equilibrium constant KP is

Image

3. For the generic reaction (2-2), the concentration equilibrium constant KC is

Image

It is important to be able to relate K, Kγ, KC, and Kp.

4. For ideal gases, KC and Kp are related by

Image

Where for the generic reaction (2-2),

Image

5. KP is a function of temperature only, and the temperature dependence of KP is given by van’t Hoff’s equation:

Van’t Hoff’s equation

Image
Image

6. Integrating, we have

Image

Kp and KC are related by

Image

when

Image

then

KP = KC

7. KP neglecting ΔCP Given the equilibrium constant at one temperature, T1, KP(T1), and the heat of reaction, Image, the partial pressure equilibrium constant at any temperature T is

Image

8. From Le Châtelier’s principle we know that for exothermic reactions, the equilibrium shifts to the left (i.e., K and Xe decrease) as the temperature increases. Figures C-1 and C-2 show how the equilibrium constant varies with temperature for an exothermic reaction and for an endothermic reaction, respectively.

Image

Figure C-1 Exothermic reaction.

Image

Figure C-2 Endothermic reaction.

Variation of equilibrium constant with temperature

9. The equilibrium constant for the reaction (2-2) at temperature T can be calculated from the change in the Gibbs free energy using

Image
Image

10. Tables that list the standard Gibbs free energy of formation of a given species Image are available in the literature.

1) www/uic.edu:80/~mansoori/Thermodynamic.Data.and.Property_html

2) webbook.nist.gov

11. The relationship between the change in Gibbs free energy and enthalpy, H, and entropy, S, is

Image

See bilbo.chm.uri.edu/CHM112/lectures/lecture31.htm. An example on how to calculate the equivalent conversion for ΔG is given on the Web site.


Example C–1 Water-Gas Shift Reaction

The water-gas shift reaction to produce hydrogen

Image

is to be carried out at 1000 K and 10 atm. For an equimolar mixture of water and carbon monoxide, calculate the equilibrium conversion and concentration of each species.

Data: At 1000 K and 10 atm, the Gibbs free energies of formation are Image; Image; Image and Image.

Solution

We first calculate the equilibrium constant. The first step in calculating K is to calculate the change in Gibbs free energy for the reaction. Applying Equation (C-10) gives us

Calculate Image

Image

Calculate K

Expressing the equilibrium constant first in terms of activities and then finally in terms of concentration, we have

Image

where ai is the activity, fi is the fugacity, γi is the activity coefficient (which we shall take to be 1.0 owing to high temperature and low pressure), and yi is the mole fraction of species i.2 Substituting for the mole fractions in terms of partial pressures gives

2 See Chapter 9 in J. M. Smith, Introduction to Chemical Engineering Thermodynamics, 3rd ed. (New York: McGraw-Hill, 1959), and Chapter 9 in S. I. Sandler, Chemical and Engineering Thermodynamics, 2nd ed. (New York: Wiley, 1989), for a discussion of chemical equilibrium including nonideal effects.

Image
Image

In terms of conversion for an equimolar feed, we have

Image

Relate K and Xe

Image

Figure EC-1.1 Modell, Michael, and Reid, Robert, Thermodynamics and Its Applications, 2nd ed., © 1983. Reprinted and electronically reproduced by permission of Pearson Education, Inc., Upper Saddle River, NJ.

From Figure EC-1.1 we read at 1000 K that log KP = 0.15; therefore, KP = 1.41, which is close to the calculated value. We note that there is no net change in the number of moles for this reaction (i.e., δ = 0); therefore,

K = Kp = KC (dimensionless)

Taking the square root of Equation (EC-1.7) yields

Image

Calculate Xe, the equilibrium conversion

Solving for Xe, we obtain

Image

Then

Image

Calculate CCO, e, the equilibrium conversion of CO

Figure EC-1.1 gives the equilibrium constant as a function of temperature for a number of reactions. Reactions in which the lines increase from left to right are exothermic.


The following links give thermochemical data. (Heats of Formation, CP, etc.)

1) www.uic.edu/~mansoori/Thermodynamic.Data.and.Property_html

2) webbook.nist.gov

Image

Also see Chem. Tech., 28 (3) (March), 19 (1998).

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