Appendix C. Thermodynamic Relationships Involving the Equilibrium Constant1

For the gas-phase reaction

2-2

image

  1. The true (dimensionless) equilibrium constant is

    RT lnK = –ΔG

    Where G is the Gibbs Free Energy and R is the ideal gas constant.

    image

    where ai is the activity of species i

    image

    where

    fi = fugacity of species i

    image = fugacity of species i at the standard state. For gases, the standard state is 1 atm.

    image

    K = True equilibrium constant

    Kγ = Activity equilibrium constant

    KP = Pressure equilibrium constant

    KC = Concentration equilibrium constant

    where γi is the activity coefficient of species i

    image

    Pi = partial pressure of species i, atm, kPa.

    Pi = CiRT

    Kγ has units of image

    KP has units of image

    For ideal gases Kγ = 1.0 atm

  2. The pressure equilibrium constant KP is

    C-1

    image

  3. The concentration equilibrium constant KC is

    C-2

    image

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

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

    C-3

    image

    C-4

    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

    C-5

    image

    C-6

    image

  6. Integrating, we have

    C-7

    image

    KP and KC are related by

    C-8

    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

    C-9

    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.

    Variation of equilibrium constant with temperature

    Figure C.1. Exothermic reaction.

    image

    Figure C.2. Endothermic reaction.

    image

  9. The equilibrium constant at temperature T can be calculated from the change in the Gibbs free energy using

    C-10

    image

    C-11

    image

  10. Tables that list the standard Gibbs free energy of formation of a given species image are available in the literature.
    1. www.trigger.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

    C-12

    image

    See bilbo.chm.uri.edu/CHM112/lectures/lecture31.htm.

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 cal/mol; image cal/mol; image cal/mol; 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

EC-1.1

image

C-10

image

EC-1.2

image

then

Calculate K

K = 1.44

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

EC-1.3

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

EC-1.4

image

EC-1.5

image

In terms of conversion for an equimolar feed, we have

EC-1.6

image

Relate K and Xe

EC-1.7

image

From Figure EC-1.1 on the DVD-ROM 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

EC-1.8

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 on the DVD-ROM 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

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

image

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