CHEMICAL EQUILIBRIUM IN SYSTEMS OF VARIABLE COMPOSITION

9.6 CHEMICAL EQUILIBRIUM IN SYSTEMS OF VARIABLE COMPOSITION

We can apply the criterion of equilibrium expressed in Equation (9.17) to chemically reacting systems. Consider the reaction

a A (9 :47) in which all reactants and products are in the same phase. If this chemical reaction is

at equilibrium at a fixed pressure and temperature, it follows from Equation (9.15) and Equation (9.17) that

dG ¼m A dn A þm B dn B R dn R þm S dn S (9 :48) However, the various dn’s in Equation (9.48) are not independent, but, in view of the

stoichiometry of the reaction of Equation (9.47), they must be related as follows:

dn A dn B dn R dn S

As reactants disappear and products appear in the reaction, the corresponding dn’s in Equation (9.49) have opposite signs. In view of the series of equalities in this equation, let us define a quantity dj such that

in which n i is merely a generalized notation for the dimensionless stoichiometric coef- ficients , 2a, 2b, r, s, and so on. The quantity j is called the extent of reaction or the progress variable , and it has the dimensions of amount of substance (Table 2.1) and has the unit mol. From the relationships of Equation (9.50), Equation (9.48) can be rewritten as

R dj þ sm S d (9 :51) Alternatively, we can say that

A B ¼0 (9 :52) @j T ,P

¼ rm R þ sm S

is a criterion of equilibrium at constant temperature and pressure. The derivative ( @G /@j) T ,P is the slope of a plot of the Gibbs function of the system G against j the progress variable. When j ¼ 0, the system is all reactants, and when j ¼ 1, the system is all products. At equilibrium, G is at a minimum, and the slope is equal to zero. Such a graph is given in Figure 9.3.

THERMODYNAMICS OF SYSTEMS OF VARIABLE COMPOSITION

Figure 9.3. A graph of the Gibbs function G as a function of the progress variable j, which shows equilibrium at the minimum.

If Equation (9.52) is integrated with respect to j from j ¼ 0 to j ¼ 1 at constant values of the chemical potentials (fixed composition of the reacting mixture), then

we obtain, at equilibrium,

dj ¼ Xn i m i ¼0 (9 :53)

@j T ,P

in which it is understood that n i is a negative number for the stoichiometric coeffi- cients of the reactants and a positive number for the products. The result is a molar quantity, because the integration leads to a mole of reaction in the sense given in the definition of mole in Table 2.1. As the composition of the reacting mixture does not change when one mole of reaction occurs, we say that we are using an “infi- nite copy model,” which is a system so large that the conditions of constant compo- sition are satisfied.

Another way of writing Equation (9.53) is

( jn i jm i ) products (9 :54) The concept of escaping tendency also can be applied to the chemical reaction in

X ( X jn

i jm i ) reactants ¼

Equation (9.47). At equilibrium, from Equation (9.54), we can say that the sum of the escaping tendencies of the reactants is equal to the sum of the escaping tendencies of the products.

For a chemical transformation capable of undergoing a spontaneous change, it follows from Equations (9.17) and (9.50) that

Xn i m i ,0

223 or that

EXERCISES

( jn i jm i ) products (9 :56) Thus, for a spontaneous reaction, we can say that the sum of the escaping tendencies

X ( X jn

i jm i ) reactants .

for the reactants is greater than the sum of the escaping tendencies for the products. We can compare the sums of v i m i for reactants and products to arrive at a decision as to whether a transformation is at equilibrium or capable of a spontaneous change. Although we can compare escaping tendencies or m’s of a given substance under different conditions at constant temperature, it is meaningless to compare individual escaping tendencies of different substances because we have no way of determining absolute values of m. For similar reasons, we cannot compare escaping tendencies of

a single substance at different temperatures.