GIBBS FUNCTION AND THE EQUILIBRIUM CONSTANT IN TERMS OF ACTIVITY

16.3 GIBBS FUNCTION AND THE EQUILIBRIUM CONSTANT IN TERMS OF ACTIVITY

With the definition of the activity function, we could derive a general expression that relates DG8 m of a reaction to the equilibrium constant and hence to eliminate the restrictions imposed on previous relationships.

Let us consider the chemical reaction

a A(a A ) þ bB(a B ) ¼ cC(a C ) þ dD(a D ) From Equation (9.15)

dG ¼m A dn A þm B dn B þm C dn C þm D dn D (16 :20) Applying the procedure used in Equations (9.49) through (9.51) and integrating from

j ¼ 0 to j ¼ 1, we obtain DG m ¼ cm C þ dm D A B (16 :21)

Substitution for the chemical potentials from Equation (16.1) gives DG m ¼ c(m C 8 þ RT ln a C ) þ d(m D 8 þ RT ln a D )

8 B þ RT ln a B ) This equation may be written as

A a b B The expression in parentheses in Equation (16.22) is equal to DG8 m , so we can write

DG C m D ¼ DG m 8 þ RT ln a b (16 :23)

Equation (16.23) is a general relationship for the calculation of DG m for any reaction from the value for DG8 m and from the activities a A ,a B ,a C , and a D . We emphasize that Equation (16.23) refers to a system in which a mole of reaction occurs with no change in the activity of any reactant or product. Either the system is very large, the infinite copy model (as described in Section 9.6) or one in which we calculate the molar

ACTIVITIES, EXCESS GIBBS FUNCTIONS, STANDARD STATES FOR NONELECTROLYTES

change in Gibbs function from the change corresponding to an infinitesimal amount of reaction, with (dG /dj) T,P as a criterion of equilibrium [see Equation (9.52)].

At equilibrium at constant temperature and pressure, DG m ¼ 0 and

DG m 8 a c C d a D

a b (16 RT :24) a

A a B equil

From the definitions of standard states for components of solutions, it is clear that DG8 m is a function only of the temperature, because the standard state of each reactant and product is defined at a specific fixed pressure. Thus, DG8 m is a constant for a particular reaction at a fixed temperature. Hence, we can write

in which K a is the equilibrium constant in terms of activities. Consequently

B equil

If the mole fraction is a convenient variable, Equation (16.3) can be used to write Equation (16.26) as

¼K x K g (16 :29) Similarly, for reactants and products for which molality is the convenient compo-

sition variable, we can write K a ¼K m K g (16 :30) where

(m

C =m8) (m D =m8)

(m A =m8) a (m B =m8) b

An equilibrium constant for some reactions can be expressed in terms of mole frac- tions for some components and molalities for other components.

367 If the value of K a for a reaction is calculated from the value of DG8 m , we must have

16.4 DEPENDENCE OF ACTIVITY ON PRESSURE

values of the g i to substitute into Equation (16.27) or Equation (16.30) to obtain equi- librium yields in terms of m i or X i . The determination of these quantities from experi- mental data will be discussed in Chapters 17 and 19.

16.4 DEPENDENCE OF ACTIVITY ON PRESSURE As m i ¼ m8 i þ RT ln a i

@m i @m 8 i

@ ln a

(16 :32) @P T ,X

From Equation (9.25)

@m i ¼V mi @P T ,X

Although we have chosen to define the standard state at a fixed pressure, the vapor pressure of the pure solvent, the standard chemical potential is still a function of the pressure chosen, so that the first term on the right in Equation (16.32) is equal to V8 mi . Therefore,

For solvents, V8 mi is equal to V † mi , because the standard state is the pure solvent, if we neglect the small effect of the difference between the vapor pressure of pure solvent and 1 bar. As the standard state for the solute is the hypothetical unit mole fraction state (Fig. 16.2) or the hypothetical 1-molal solution (Fig. 16.4), the chemical potential of the solute that follows Henry’s law is given either by Equation (15.5) or Equation (15.11). In either case, because mole fraction and molality are not pressure dependent,

@m 2 @m 8

@P T ,m 2 ,X 2 @P T ,m 2 ,X 2

so that

V m ,2 ¼V m 8 ,2

Thus, the partial molar volume is constant along the Henry’s-law line and equal to the standard partial molar volume. The only real solution along the Henry’s-law line is the infinitely dilute solution, so

V m2 8 ¼V 1 m2

ACTIVITIES, EXCESS GIBBS FUNCTIONS, STANDARD STATES FOR NONELECTROLYTES

16.5 DEPENDENCE OF ACTIVITY ON TEMPERATURE From previous discussions of temperature coefficients of the Gibbs function (see

Section 7.2), we expect the expression for the temperature dependence of the activity to involve an enthalpy function. Hence, we need to develop relationships for the enthalpies of the standard states.

Standard Partial Molar Enthalpies For pure solids and liquids, the standard enthalpy is the enthalpy of the substance at

the specified temperature and at 1 bar.

Solvent. We have defined the pure solvent at the same temperature as the solution and at its equilibrium vapor pressure as the standard state for the solvent. It follows that

(16 :35) with the second equality valid if we neglect the trivial effect on the enthalpy of the small

H m1(standard state) ¼H m1 8 ¼H † m1

change of pressure from 1 bar to the vapor pressure of the solvent.

Solute. The standard state for the solute is the hypothetical unit mole fraction state (Fig. 16.2) or the hypothetical 1-molal solution (Fig. 16.4). In both cases, the standard state is obtained by extrapolation from the Henry’s-law line that describes behavior at infinite dilution. Thus, the partial molar enthalpy of the standard state is not that of the actual pure solute or the actual 1-molal solution.

The chemical potential on a molality scale of the solute that follows Henry’s law is

given by Equation (16.10) with g 2 ¼ 1. m 2

m 2 ¼m 8 2 þ RT ln m 2 8

If we divide each term in Equation (16.10) by T and differentiate with respect to T at constant P and m 2 , the result is

From Equation (9.57)

@(m=T)

@T P

2 ,m 2 T

Substituting from Equation (9.57) into Equation (16.36), we have

H m2

H m2 8 T 2 T 2

369 or

16.5 DEPENDENCE OF ACTIVITY ON TEMPERATURE

H m2 ¼H m2 8 (16 :37) Thus, the partial molar enthalpy along the Henry’s-law line is constant and equal to

the standard partial molar enthalpy. The only real solution along the Henry’s-law line is the infinitely dilute solution, so that

(16 :38) For this reason, the infinitely dilute solution frequently is called the reference state for

H m2 8 ¼H 1 m2

the partial molar enthalpy of both solvent and solute.

Equation for Temperature Derivative of the Activity From Equation (16.1)

From Equation (9.57)

@(m=T)

@T P ,m

If we differentiate Equation (16.39) with respect to temperature at constant molality and pressure, and substitute from Equation (9.57), the result is

From Equations (16.35), (16.37), and (16.38), we observe that for solute and solvent, H8 mi is equal to H mi 1 . Therefore, we can write Equation (16.40) for either solvent or solute as

in which H mi 1 2H mi is the change in partial molar enthalpy on dilution to an infinitely dilute solution.

ACTIVITIES, EXCESS GIBBS FUNCTIONS, STANDARD STATES FOR NONELECTROLYTES

From the definition of the activity coefficient [Equations (16.5) and (16.9)],

or

m i =m 8

we can show that for the solute

and for the solvent

16.6 STANDARD ENTROPY We have pointed out that a concentration m 2(i) of the solute in the real solution may

have an activity of 1, which is equal to the activity of the hypothetical 1-molal stan- dard state. Also, H8 m2 , the partial molar enthalpy of the solute in the standard state, equals the partial molar enthalpy of the solute at infinite dilution. We might inquire whether the partial molar entropy of the solute in the standard state S8 m2 corresponds to the partial molar entropy in either of these two solutions.

Let us compare S m2 for a real solution with S8 m2 of the hypothetical 1-molal sol- ution. For any component of a solution, from Equation (9.20), we can write

m2 8 ) ¼ (H m2 8 m2 ) þ (m 2 2 8 ) (16 :44)

At infinite dilution, that is, when m 2 ¼0

(16 :45) because

S m2 = S m2 8 (at m 2 ¼ 0)

(16 :46) even though

m 2 = m 8 2 (at m 2 ¼ 0)

H m2 ¼H m2 8 (at m 2 ¼ 0)

371 Hence, the partial molar entropy of the solute in the standard state is not that of the solute

16.6 STANDARD ENTROPY

at infinite dilution. Similarly, at the molarity m 2( j ) (Fig. 16.4), where a 2 is unity S m2 = S m2 8 (at molality where a 2 ¼ 1)

(16 :47) because

H m2 = H m2 8 (at molality where a 2 ¼ 1) (16 :48) even though m 2 ¼m 2 8 (at molality where a 2 ¼ 1)

(16 :49) Thus, S m2 can be equal to S8 m2 only for a solution with some molality m 2(k) at which (H m2 8 m2 ) ¼ (G m2 8 m2 ) ¼m 8 2 2 (16 :50) The particular value of the molality m 2(k) at which S m2(k) ¼ S8 m2 differs from solute to

solute and for different solvents with the same solute. We can summarize our conclusions about the thermodynamic properties of the solute in the hypothetical 1-molal standard state as follows. Such a solute is charac- terized by values of the thermodynamic functions that are represented by m8 2 , H8 m2 , and S8 m2 . Frequently a real solution at some molality m 2( j ) also exists (Fig. 16.4) for which m 2 ¼ m8 2 , that is, for which the activity has a value of 1. The real solution for which H m2 is equal to H8 m2 is the one at infinite dilution. Furthermore, S m2 has a value equal to S8 m2 for some real solution only at a molality m 2(k) that is neither zero nor m 2( j ) . Thus, three different real concentrations of the solute exist for which the

thermodynamic qualities m 2 ,H m2 , and S m2 respectively, have the same values as in the hypothetical standard state. For the solvent, the standard thermodynamic properties are

m 1 8 ¼m † 1 ¼m 1 1 (16 :51) and

(16 :53) so long as we neglect the effects of the pressure difference between the pure liquid at

S m1 8 ¼S † m1 ¼S 1 m1

1 bar and the pure solvent at its vapor pressure. Table 16.1 summarizes the information on the standard states of pure phases as well as those of solvents and solutes.

A more elegant (although more difficult to visualize) formulation of the procedure for the selection of the standard state for a solute may be made as follows. From Equation (16.1)

m 2 ¼m 8 2 þ RT ln a 2

372 TABLE 16.1. Standard States for Thermodynamic Calculations (For every case, it is assumed that the temperature has been specified) Physical State

Volume Pure gas

Chemical Potential

Enthalpy a Entropy

Hypothetical ideal gas at 1 bar

Hypothetical ideal gas at 1

Hypothetical ideal gas at 1 bar

Hypothetical ideal gas at 1 bar

(0.1 MPa). Also, a pressure,

bar; also real gas at zero

(0.1 MPa). Also, a pressure

(0.1 Mpa). V

m ¼(RT/P ).

usually near 1 bar, will exist

pressure. (See Exercise 1,

of the real gas will exist, not

at which the real gas has a

this chapter.)

zero and not that of unit

fugacity of unity.

fugacity, with an entropy equal to that in the standard state.

Pure liquid or pure

1 bar (0.1 MPa). solid

1 bar (0.1 MPa).

1 bar (0.1 MPa).

1 bar (0.1 MPa).

Solvent in a solution b Pure solvent at vapor pressure

Pure solvent at vapor pressure

Pure solvent at vapor pressure

Pure solvent at vapor pressure

of pure solvent. Solute in a solution b,c

of pure solvent.

of pure solvent.

of pure solvent.

Hypothetical 1-molal solution

Hypothetical 1-molal solution

Hypothetical 1-molal solution

Hypothetical 1-molal solution

obeying limiting law

obeying limiting law

obeying limiting law

obeying limiting law

corresponding to Henry’s

corresponding to Henry’s

corresponding to Henry’s

corresponding to Henry’s

law at vapor pressure of

law at vapor pressure of

law at vapor pressure of pure

law at vapor pressure of

solvent. Also, a finite

pure solvent. The value of

solvent. Also, a solution of

pure solvent. The value of

concentration may exist, not

the partial molar enthalpy

finite concentration may

V8 m2 is equal to V m2 1 .

equal to zero, at which the

in the standard state H8 m2 , is

exist, not equal to zero but

activity of the solute is unity.

always equal to that at

also not having an activity of

This may be considered as

infinite dilution. Hence, the

unity, with a partial molar

the standard state only in free

infinitely dilute solution can

entropy of solute equal to

energy calculations.

be considered the standard

that in the standard state.

state only in enthalpy calculations .

a For every case, it is assumed that the temperature has been specified. b The standard state for the heat capacity is the same as that for the enthalpy. For a proof of this statement for the solute in a solution, see Exercise 2 in this chapter. c This choice of standard state for components of a solution is different from that used by many thermodynamicists. It seems preferable to the choice of a 1-bar standard

state, however, because it is more consistent with the extrapolation procedure by which the standard state is determined experimentally, and it leads to a value of the activity coefficient equal to 1 when the solution is ideal or very dilute whatever the pressure. It is also preferable to a choice of the pressure of the solution, because that choice produces a different standard state for each solution. For an alternative point of view, see Ref. 2.

373 and from Equation (16.9)

16.7 DEVIATIONS FROM IDEALITY IN TERMS OF EXCESS THERMODYNAMIC FUNCTIONS

It follows that

m 2 8 ¼m 2 2

Therefore the state in which the solute has a partial molar Gibbs function of m8 2 can be found from the following limit, because g 2 approaches unity as m 2 approaches zero:

lim m

2 m 8 ¼m 2 !0 m 2

With this method of formulation, it also is possible to show that frequently a real solution at some molality m j exists for which m 2 ¼ m8 2 , that H8 m2 corresponds to H m2 for a real solution at infinite dilution, and that S8 m2 equals S m2 for a real solution at a molality m 2k , which is neither zero nor m 2j .