CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

CHAPTER 20 CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

Deviations from ideality in real solutions have been discussed in some detail to provide an experimental and theoretical basis for precise calculations of changes in the Gibbs function for transformations involving solutions. We shall continue our discussions of the principles of chemical thermodynamics with a consideration of some typical calculations of changes in Gibbs function in real solutions.

20.1 ACTIVITY COEFFICIENTS OF WEAK ELECTROLYTES (1) Let us consider a typical weak electrolyte, such as acetic acid, whose ionization can

be represented by the equation HC 2 H 3 O 2 ¼H þ þC 2 H 3 O 2 (20 :1) We defined the activity coefficient for strong electrolytes in Chapter 19 in Equations

(19.9) and (19.24) as

g þ ¼ m þ =m8

g ¼ m =m8

Chemical Thermodynamics: Basic Concepts and Methods, Seventh Edition . By Irving M. Klotz and Robert M. Rosenberg Copyright # 2008 John Wiley & Sons, Inc.

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

and

g + ¼ m + =m8

and disregarded the possibility of incomplete dissociation. As it is possible to measure (or closely approximate) the ionic concentrations of a weak electrolyte, it is convenient to define ionic activity coefficients for weak electro-

lytes in the same way, based on the actual ionic concentrations, m þ or m 2 . Thus,

Similarly, for the undissociated species of molality m u ,

(m u =m8)

The degree of dissociation a of a uni-univalent weak electrolyte such as acetic acid is given by the equation

a¼

where m s is the stoichiometric or total molality of acetic acid.

20.2 DETERMINATION OF EQUILIBRIUM CONSTANTS FOR DISSOCIATION OF WEAK ELECTROLYTES

Three experimental methods that are capable of determining dissociation constants with a precision of the order of tenths of 1% have been most commonly used. Each of these methods—the cell potential method (2), the conductance method (3), and the optical method (4)—provides data that can be treated approximately, assuming that the solutions obey Henry’s law, or more exactly on the basis of the methods developed in Chapter 19. We will apply the more exact procedures. As the optical method can be used only if the acid and conjugate base show substantial differences in absorption of visible or ultraviolet light, or differences in raman scattering or with the use of indicators, we shall limit our discussion to the two electrical methods.

473 From Measurements of Cell Potentials

20.2 EQUILIBRIUM CONSTANTS FOR DISSOCIATION OF WEAK ELECTROLYTES

It is possible to select a cell that contains a weak acid in solution whose potential depends on the ion concentrations in the solution and hence on the dissociation constant of the acid. As an example, we will consider acetic acid in a cell that contains

a hydrogen electrode and a silver – silver chloride electrode:

H 2 (g, P ¼ 1bar); HC 2 H 3 O 2 (m 2 ), NaC 2 H 3 O 2 (m 3 ), NaCl(m 4 ); AgCl(s), Ag(s) (20 :6)

As the reaction that occurs in this cell is [Equation (19.27)]

2 H 2 (g) þ AgCl(s) ¼ Ag(s) þ HCl(aq) the cell potential must be given by the expression [from Equations (19.30) – (19.32)]

As the molality m H þ depends on the acetic acid equilibrium, which we can indicate in

a simplified notation by the equation

(20 :7) where Ac 2 stands for the acetate ion, we can introduce the dissociation constant K for

Hac ¼H þ þ Ac

acetic acid into the equation for the cell potential. For acetic acid, K is given by

from which m H þ can be expressed in terms of the other variables and can be substi- tuted into Equation (19.32). Thus, we obtain

This equation can be rearranged to the form

ln K (20 :10) in which

g Ac F

K 0 ;K g Cl g HAc

g Ac

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

All terms on the left in Equation (20.10) are known from previous experiments (see Section 19.2 for the determination of E8) or can be calculated from the composition of the solution in the cell. Thus [see Equation (20.6)],

m Cl ¼m 4 (20 :12)

(20 :13) and

m HAc ¼m 2 H þ

Ac ¼m 3 þm H (20 :14) Generally, m H þ

2 or m 3 , so it can be estimated from Equation (20.8) by inserting an approximate value of K and neglecting the activity coefficients. Thus, it is possible to obtain tentative values of 2(RT/F) ln K 0 and hence K 0 at various concentrations

of acetic acid, sodium acetate, and sodium chloride, respectively. The ionic strength I can be estimated as

(20 :15) It can be observed from the limiting behavior of activity coefficients [Equation

Thus, if

RT

F ln K 0

or K 0 is plotted against some function of the ionic strength and extrapolated to I ¼ 0, the limiting form of Equation (20.10) is

It has been found that the ionic strength to the first power as the abscissa yields a meaningful extrapolation.

A typical extrapolation of the data for acetic acid is illustrated in Figure 20.1. At 258C the value of 1.755

25 has been found for K by this method. If the equilibrium constant is not already known fairly well, the K determined by

this procedure can be looked on as a first approximation. It then can be used to estimate m H þ for substitution into Equations (20.13) – (20.15), and a second extrapolation can be carried out. In this way a second value of K is obtained.

20.2 EQUILIBRIUM CONSTANTS FOR DISSOCIATION OF WEAK ELECTROLYTES

Figure 20.1. Extrapolation of K 0 values in the determination of the ionization constant of acetic acid at 258C. Based on data from H. S. Harned and R. W. Ehlers, J. Am. Chem. Soc. 54 , 1350 (1932).

Theprocess can then be repeated until successive estimates of K agree within the precision of the experimental data. The iterative calculation can be programmed for

a computer.

From Conductance Measurements Conductance measurements also have been used for the estimation of dissociation

constants of weak electrolytes. If we use acetic acid as an example, we find that the equivalent conductance L shows a strong dependence on concentration, as illus- trated in Figure 20.2. The rapid decline in L with increasing concentration is largely from a decrease in the fraction of dissociated molecules.

In the approximate treatment of the conductance of weak electrolytes, the decrease in L is treated as resulting only from changes in the degree of dissociation, a. On this basis, it can be shown that an apparent degree of dissociation a 0 can be obtained from

in which L 0 is the equivalent conductance of the weak electrolyte at infinite dilution. Hence, the apparent dissociation constant K 0 is obtainable from the expression

C 0 þ C 0 (a 0 C )(a 0 C )

K 0 ¼ H 0 Ac ¼

HAc

C C8 (1

)CC8

(a 0 ) 2 C (L =L 0 ) 2 C

)C8 [1

0 )]C8

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

Figure 20.2. Equivalent conductance of aqueous solutions of acetic acid at 258C. Based on data from D. A. MacInnes and T. Shedlovsky, J. Am. Chem. Soc. 54, 1429 (1932).

in which C is the total (stoichiometric) concentration of acetic acid in moles per liter. Generally, L 0 is evaluated from data at infinite dilution for strong electrolytes. Thus, for acetic acid, L 0 is obtained as follows:

L 0 (HAc) ¼L 0 (H þ þ Ac )

¼L 0 (H þ þ Cl ) þL 0 (Na þ þ AC )

(Na 0 þ þ Cl ) (20 :21)

¼L 0 (HCl) þL 0 (NaAc)

0 (NaCl)

However, for more precise calculations, it is necessary to consider that the mobi- lity (hence, the conductance) of ions changes with concentration, even when dis-

sociation is complete, because of interionic forces. Thus, Equation (20.20) is oversimplified in its use of L 0 to evaluate a, because at any finite concentration, the equivalent conductances of the H þ and Ac 2 ions, even when dissociation is

complete, do not equal L 0 .

To allow for the change in mobility resulting from changes in ion concentrations, MacInnes and Shedlovsky [5] proposed the use of a quantity L e in place of L 0 . The quantity L e is the sum of the equivalent conductances of the H þ and Ac 2 ions at the concentration C i at which they exist in the acetic acid solution. For example, for acetic acid L e is obtained from the equivalent conductances of HCl, NaAc, and NaCl at a concentration C i equal to that of the ions in the solution of acetic acid. Thus, because

) (20 :22) L NaAc

1 =2 HCl

477 and

20.2 EQUILIBRIUM CONSTANTS FOR DISSOCIATION OF WEAK ELECTROLYTES

L NaCl

1 =2 ) (20 :24) the effective equivalent conductance L e of completely dissociated acetic acid is

given by L e ¼L HCl þL NaAc

NaCl

) (20 :25) Assuming that the degree of dissociation at the stoichiometric molar concentration C

þ 166:0C i

is given by the expression

00 a L ¼

we obtain a better approximation for the dissociation constant than Equation (20.20):

C 00 H C 00 00 00 2 þ AC (a C )

(L =L ) 2 C

00 (20 )C C8 :27) [1 e )]C8 Now if we insert appropriate activity coefficients, we obtain a third approximation

K ¼ 00 ¼

C HAc C8 (1

for the dissociation constant:

This equation can be converted into logarithmic form to give

in which

g 2 + ¼g H þ g Ac (20 :30) and

g u ¼g HAc

(20 :31) To evaluate log K 00 , it is necessary to know L e and, therefore, C i . Yet to know C i

we must have a value for a, which depends on a knowledge of L e . In practice, this impasse is overcome by a method of successive approximations. To begin, we take L e ¼L 0 and make a first approximation for a 00 from Equation (20.26). With this value of a 00 , we can calculate a tentative C i , which can be inserted into Equation (20.25) to give a tentative value of L e . From Equation (20.25) and Equation (20.26), a new value of a 00 is obtained, which leads to a revised value for C i and sub- sequently for L e . This method is continued until successive calculations give substan- tially the same value of a 00 . Thus, for a 0.02000 molar solution of acetic acid, with an

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

equivalent conductance of 11.563 ohms 21 mol 21 cm 2 , a first approximation for a is

390 :59 because L 0 ¼ 390.50 ohms 21 mol 21 cm 2 . Therefore,

C i ¼a 0 C ¼ 0:029604(0:02000) ¼ 0:00059208 mol L and

¼ 0:024333 Substitution of this value of C 1 i /2 into Equation (20.25) yields a value of

1 C =2

L e ¼ 387:07 V mol cm 2

Coupling this value with 11.563 for L, we obtain

a 00 11 ¼ :563 ¼ 0:029873 387 :07

C i ¼ 0:00059746 mol L

L e ¼ 387:06 V mol cm 2

A third calculation of a 00 gives 0.029874, which is substantially the same as the result of the second approximation; hence, it can be used in Equation (20.27). As with the iterative procedure for calculating equilibrium constants from data on cell potentials, the iterative procedure for conductance data can be programmed for a computer.

Once a value of log K 00 is obtained, the value of log K can be determined by an extrapolation procedure. From Equation (19.11),

From Equation (16.4),

I m 2 !0 u !0 ¼1

lim g u ¼ lim g

479 and

20.2 EQUILIBRIUM CONSTANTS FOR DISSOCIATION OF WEAK ELECTROLYTES

Thus, the limiting form of Equation (20.29) is

in which K is the thermodynamic dissociation constant.

The best functions to use in the extrapolation can be determined from the depen-

dence of g 2 + and g u on the ionic strength. Theoretically, little is known about the dependence of g u on concentration (6), but from the Debye – Hu¨ckel theory, we should expect log g 2 + to depend on I 1 /2 , with the dependence approaching linearity with increasing dilution. The data for acetic acid, when log K 00 is plotted against the square root of the ionic strength (Fig. 20.3), provide a meaningful value for K by extrapolation. MacInnes and Shedlovsky [5] report a value for K of 1.753

25 at 258C.

An alternative method of extrapolation, in which the slope is reduced almost to zero, can be carried out by the following modification of Equation (20.29). If we separate the activity coefficients, we obtain

(20 :33) We know from experiment that log g u is a linear function of I. The value of log g + 2

log K 000 ¼ log K 00 þ log g 2 +

is described well by the Debye – Hu¨ckel limiting law in very dilute solution. Thus, we can substitute the expression

Figure 20.3. Extrapolation of ionization constants of acetic acid.

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

Figure 20.4. Alternative method of extrapolation of ionization constants of acetic acid.

into Equation (20.33) and can rearrange the resultant equation into the relationship log K 000 þ log g u ¼ log K 00 1 =2 ; log K 00 00

(20 :35) As both K 00 and I 1 /2 can be calculated from experimental data, log K 000 can be deter-

mined. A plot of log K 000 against I gives a curve with small slope, such as illustrated in Figure 20.4. The determination of the intercept in this graph is easier than it is in Figure 20.3 for uni-univalent electrolytes, and the improvement is greater when the dissociation process involves polyvalent ions.

Mesmer et al. [7] used conductance methods to determine ionization constants over a wide range of temperatures and pressures. An alternative procedure uses the Fuoss conductance – concentration function to relate the measured conductance to the ionic concentrations at equilibrium (8).

20.3 SOME TYPICAL CALCULATIONS FOR D f G88888 m

Standard Gibbs Function for Formation of Aqueous Solute: HCl We have discussed in some detail the various methods that can be used to obtain the

standard Gibbs function of formation of a pure gaseous compound such as HCl(g). As many of its reactions are carried out in aqueous solution, it also is desirable to

know D f G8 m for HCl(aq). Our problem is to find DG8 m for the reaction

HCl (g, a ¼ f ¼ 1) ¼ HCl (aq, a 2 ¼ 1) (20 :36) because to this DG8 m we always can add D f G8 m of HCl(g). Although in Equation

(20.36), a HCl is 1 on both sides, the standard states are not the same for the

481 gaseous and aqueous phases; hence, the chemical potentials are not equal. To obtain

20.3 SOME TYPICAL CALCULATIONS FOR D f G8 m

DG8 m for Reaction (20.36), we can break up the reaction into a set of transformations for which we can find values of DG m .

We can write the following three equations, whose sum is equivalent to Equation (20.36):

DG m,298 K ¼ RT ln (20 :37)

1 bar

in which 0.2364

24 bar is the partial pressure of HCl(g) (9) in equilibrium with a 4-molal solution of HCl.

24 bar) ¼ HCl(aq, m 2 ¼ 4),

DG m,298 K ¼ 0 (equilibrium reaction) (20 :38)

3. HCl(aq, m 2 ¼ 4, a 0 2 ¼ 49.66) ¼ HCl(aq, a 2 ¼ 1),

DG m,298 K ¼ RT ln ¼ RT ln (20 :39)

a 0 2 49 :66

The activity of HCl in a 4-molal solution required for the DG m in Equation (20.39) was calculated from the mean activity coefficient, 1.762, taken from tables of Harned and Owen [10], as follows:

2 + ¼ (4) 2 :762) ¼ 49:66 (20 :40) Now we can obtain the standard Gibbs function change for Equations (20.36),

a 2 ¼m 2 + g (1

because the sum of Equations (20.37) through (20.39) yields

(20 :41) Having obtained the standard Gibbs function change accompanying the transfer

HCl(g, a ¼ 1) ¼ HCl(aq, a 2 ¼ 1), DG8 m,298 K

of HCl from the gaseous to the aqueous state, we can add it to the standard Gibbs function for formation of gaseous HCl [11],

(20 :42) and we can obtain the standard Gibbs function of formation of aqueous HCl:

2 H 2 (g) þ 1 2 Cl 2 (g) ¼ HCl (g), D f G8 m,298 K

2 H 2 (g) þ 2 Cl 2 (g) ¼ HCl(aq), D f G8 m,298 K

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

Standard Gibbs Function of Formation of Individual Ions: HCl As it has been shown that the Gibbs function for formation of an individual ion has no

operational meanings [12], no way exists to determine such a quantity experimentally. However, for the purposes of tabulation and calculation, it is possible to separate

D f G8 m of an electrolyte arbitrarily into two or more parts, which correspond to the number of ions formed, in a way analogous to that used in tables of standard electrode potentials. In both cases, the standard Gibbs function for formation of aqueous H þ is defined to be zero at every temperature:

2 H 2 (g) ¼H þ (aq) þe , D f G8 m ¼0 (20 :44) With this definition it is possible to calculate the standard Gibbs function of formation

of other ions. For example, for Cl 2 ion, we proceed by adding appropriate equations to Equation (20.44). For the reaction

HCl(aq, a 2 ¼ 1) ¼ H þ (aq, a þ ¼ 1) þ Cl (aq, a ¼ 1) (20 :45) DG8 m is equal to zero because our definition of the individual ion activities [Equation

(20.5)] is

a 2 ¼ (a þ )(a )

If we add Equation (20.45) to Equation (20.43) and then subtract Equation (20.44),

2 H 2 (g) þ 1 2 Cl 2 (g) ¼ HCl(aq), D f G8 m,298 K

HCl(aq) ¼H þ (aq) þ Cl (aq) D f G8 m ¼0

H þ (aq) þe ¼ 1 2 H 2 (g) D f G8 m ¼0

we obtain the standard Gibbs function for formation of Cl 2 ion:

(20 :46) This D f G8 m corresponds to the value that can be calculated from the standard electrode

2 Cl 2 (g) þe ¼ Cl (aq), D f G8 m,298 K

potential.

Standard Gibbs Function for Formation of Solid Solute in Aqueous Solution

Solute Very Soluble: Sodium Chloride. As the standard Gibbs function for formation of NaCl(s) is available [11], the D f G8 m,298 for NaCl(aq) can be obtained

483 by a summation of the following processes:

20.3 SOME TYPICAL CALCULATIONS FOR D f G8 m

Na(s) þ 1 2 Cl 2 (g) ¼ NaCl(s)

D f G8 m,298 K

(20 :47) NaCl(s) ¼ NaCl(aq, satd:, m 2 ¼ 6:12)

D f G m ¼ 0 (equilibrium)

(20 :48) NaCl(m 2 ¼ 6:12, a 0 2 ¼ 38:42) ¼ NaCl(a 2 ¼ 1)

D f G m ¼ RT ln

NaCl(a 2 ¼ 1) ¼ Na þ (a þ ¼ 1) þ Cl (a ¼ 1) DG8 m ¼0

(20 :50) Na(s) þ 1 2 Cl (g)

2 ¼ Na (a þ ¼ 1) þ Cl (a ¼ 1)

(20 :51) The value of a 2 0 in Equation (20.49) is obtained as follows:

D f G8 m,298 K

0 (m + ) 2 (g + ) 2 (6 :12) 2 (1 :013) a 2

From the standard Gibbs function of formation of the aqueous electrolyte, we also can obtain that for the Na þ ion alone by subtracting Equation (20.46) from Equation (20.51). Thus, we obtain

Na(s) ¼ Na þ (aq) þe

(20 :53) Slightly Soluble Solute: Silver Chloride. For AgCl we can add the following

D f G8 m,298 K

equations: Ag(s) þ 1 2 Cl 2 (g) ¼ AgCl(s)

D f G8 m,298 K ¼ 109:789 kJ mol (11) (20 :54) AgCl(s) ¼ AgCl(aq, satd)

DG m ¼0 (20 :55)

AgCl(ag, satd, a 0 ¼a 0 a 2 0 þ ) ¼ Ag þ (a þ ¼ 1) þ Cl (a ¼ 1) (a þ a )

DG m ¼ RT ln 0 0 sp ¼ 55:669 kJ mol (20 :56)

(a þ a ) satd soln

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

Ag(s) þ 1 2 Cl 2 (g) ¼ Ag þ (a þ ¼ 1) þ Cl (a ¼ 1)

(20 :57) We can calculate a value for the Ag þ ion by subtracting Equation (20.46) from

D f G8 m,298 K ¼ 54:120 kJ mol

Equation (20.57). Thus, we obtain

Ag(s) ¼ Ag þ (a þ ¼ 1) þ e

(20 :58) Standard Gibbs Function for Formation of Ion of Weak Electrolyte

D f G8 m,298 K ¼ 77:266 kJ mol

As part of a program to determine the Gibbs function changes in the reactions by which glucose is oxidized in a living cell, Borsook and Schott (13) calculated the Gibbs function for the formation at 258C of the first anion of succinic acid,

C 4 H 5 O 4 2 . The solubility of succinic acid in water at 258C is 0.715 mole (kg H 2 O) 21 . In such a solution the acid is 1.12% ionized (a ¼ 0.0112) and the undissociated portion has an activity coefficient of 0.87. As we know that the first dissociation constant of succinic acid is equal to 6.4

25 , we can calculate

D f G8 m of the C 4 H 5 O 4 2 ion by adding the following equations:

3H 2 (g) þ 4C(graphite) þ 2O 2 (g) ¼C 4 H 6 O 4 (s)

D f G8 m,298 K ¼ 748:100 kJ mol

C 4 H 6 O 4 (s) ¼C 4 H 6 O 4 (aq, satd)

DG m ¼ 0 (equilibrium)

C 4 0 H 6 O 4 (aq, satd, a 2 ) ¼C 4 H 6 O 4 (a 2 ¼ 1)

DG m ¼ RT ln (a =a 0 2 2 ) ¼ 1:21 kJ mol

C 4 H 6 O 4 (a 2 ¼ 1) ¼ H þ (a þ ¼ 1) þ C 4 H 5 O 4 (a ¼ 1) DG8 m

(20 :62) 3H 2 (g) þ 4C(graphite) þ 2O 2 (g) ¼H þ (a þ ¼ 1) þ C 4 H 5 O 4 (a ¼ 1)

(20 :63) The value in Equation (20.63) is also the standard change in Gibbs function for the

D f G8 m,298 K

formation of the C 4 H 5 O 2 4 ion because, by convention, the corresponding quantity for

H þ is set equal to zero [see Equation (20.44)].

485 For the Gibbs function change in Equation (20.61), a 2 0 of the undissociated species

20.3 SOME TYPICAL CALCULATIONS FOR D f G8 m

of succinic acid in the saturated solution is obtained as follows:

a 0 2 ¼m u g u ¼m stoichiometric (1

Standard Gibbs Function for Formation of Moderately Strong Electrolyte

Moderately strong electrolytes, such as aqueous HNO 3 , generally have been treated thermodynamically as completely dissociated substances. Thus, for HNO 3 (aq), the value for D f G8 m

of 2111.25 kJ mol 21 listed in [Ref. 11] refers to the reaction

2 H 2 (g) þ 1 2 N 2 (g) þ 3 2 O 2 (g) ¼H þ (aq, a8 þ ¼ 1) þ NO 3 (aq, a8 ¼ 1) (20 :65) The activity of the nitric acid is defined by the equation

(20 :66) in which m s is the stoichiometric (or total) molality of the acid.

a HNO 3 ¼a þ a ¼m H 2 NO 3 s g 2 +

Optical and nuclear magnetic resonance methods applicable to moderately strong electrolytes have been made increasingly precise (14). By these methods, it has proved feasible to determine concentrations of the undissociated species and hence

of the dissociation constants. Thus, for HNO 3 in aqueous solution (14) at 258C, K is 24. However, in defining this equilibrium constant, we have changed the standard state for aqueous nitric acid, and the activity of the undissociated species is given by the equation

(20 :67) in which the subscript “u” refers to the undissociated species. The standard states of

a 0 HNO 3 ¼m u g u ¼a u

the ions are unchanged despite the change in the standard state of the undissociated acid. The limiting law also is the same for the ions. Therefore, Equation (20.66) is no longer applicable, and in its place, we have

With the preceding considerations clearly in mind, we can calculate D f G8 m of undissociated HNO 3 . For this purpose, we can add the following equation to Equation (20.65):

H þ (aq, a8 þ ¼ 1) þ NO 3 (aq, a8 ¼ 1) ¼ HNO 3 (aq, a8 u ¼ 1) (20 :69)

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

For this reaction

The sum of Reactions (20.65) and (20.69) is

2 H 2 (g) þ 1 2 N 2 (g) þ 3 2 O 2 (g) ¼ HNO 3 (aq, a u ¼ 1) which is the formation of molecular, undissociated, aqueous HNO 3 , and the sum

of the DG8’s for Reactions (20.65) and (20.69) is the standard Gibbs function for formation of molecular, undissociated, aqueous HNO 3

D f G8 m,298 K

Effect of Salt Concentration on Geological Equilibrium Involving Water

In Section 13.3, we discussed the gypsum – anhydrite equilibrium [Equation (13.16)] CaSO 4 2 O(s) ¼ CaSO 4 (s) þ 2H 2 O(l)

on the assumption that the liquid phase is pure water, and that DG m for the reaction is dependent only on T, P S , and P F [Equation (13.17)]. If dissolved salt is in the water, as is likely in a rock formation, the chemical potential and activity of the water (as shown in Chapter 19) depend on the salt concentration, as does DG m for Equation (13.16). The equation for DG m would be a modified form of Equation (13.17) with a term taking into account possible variation in the activity of water, as

follows (with P S and P F much greater than 1 bar): DG m (P F ,P S , T) ¼ DG m (P ¼ 1, T, a H 2 O ¼ 1)

(20 :72) þP S (DV m,S ) þP F (DV m,F ) þ 2RT ln X H 2 O g H 2 O

The osmotic coefficient of water in NaCl solutions of varying concentration can be calculated from data in Ref. 15. From the resulting values of the osmotic coefficients, the effect of NaCl concentration on the equilibrium temperature for Equation (13.16) can be determined. The results of some calculations for a constant pressure of 1 atm are shown in Figure 20.5 (16).

General Comments The preceding examples illustrate some methods that can be used to combine data for

the Gibbs functions for pure phases with information on the Gibbs function for con- stituents of a solution to calculate changes in the Gibbs function for chemical

20.4 ENTROPIES OF IONS

Figure 20.5. Effect of NaCl concentration on the equilibrium temperature of the anhydrite – gypsum reaction at 1 bar. Data from Ref. 16.

reactions of those compounds in solution. The examples discussed, together with some of the exercises at the end of this chapter, should help students apply the same principles to particular problems in which they are interested.

20.4 ENTROPIES OF IONS In dealing with solutions, it frequently may be necessary to obtain values of DG for a

process in solution for which only thermal data are available. If standard entropy data also could be obtained for solutions, then it would be possible to calculate DG8 from a calorimetric determination of DH8 and from Equation (7.26):

For aqueous solutions of electrolytes, a concise method of tabulating such entropy data is in terms of the individual ions, because entropies for the ions can be combined to give information for a wide variety of salts. The initial assembling of the ionic entropies generally is carried out by a reverse application of Equation (7.26); that

is, D f S8 m of a salt is calculated from known values of D f G8 m and D f H8 m for that salt. After a suitable convention has been adopted, the entropy of formation of the

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

cation and anion together then can be separated into the entropies for the individual ions.

The Entropy of an Aqueous Solution of a Salt From Equation (20.51), we have

Na(s) þ 1 2 þ Cl 2 (g) ¼ Na (a þ ¼ 1) þ Cl (a ¼ 1)

D f G8 m,298 K

The enthalpy of formation of NaCl(s) is 2411.153 kJ mol 21 (11). Hence, we may write

Na(s) þ 1 2 Cl 2 (g) ¼ NaCl(s)

(20 :73) To this calculation, we need to add the enthalpy change for the reaction

D f H8 m,298 K

(20 :74) As the enthalpy of the dissolved sodium chloride in its standard state according to

NaCl(s) ¼ Na þ (a þ ¼ 1) þ Cl (a ¼ 1)

Henry’s law is that of the infinitely dilute solution, DH m for the reaction in Equation (20.74) is

(20 :75) which, according to Equation (18.7), is 2L m2(s) . The relative partial molar enthalpy

DH m ¼H 1 m2

m2(s) ¼ H8 m2

m2(s)

of solid sodium chloride is 23.861 kJ mol 21 (17). Thus, for the reaction in Equation (20.51)

Entropy of Formation of Individual Ions As in the case of Gibbs function changes, we also can divide the entropy change for a

reaction [such as Equation (20.51)] into two parts and can assign one portion to each ion. As actual values of individual-ion entropies cannot be determined, we must establish some convention for apportioning the entropy among the constituent ions.

489 In treating the Gibbs functions for individual ions, we adopted the convention that

20.4 ENTROPIES OF IONS

D f G8 m of the hydrogen ion equals zero for all temperatures; that is [Equation (20.44)],

2 H 2 (g) ¼H þ (aq) þe , D f G8 m ¼0 We also have shown previously [Equation (7.49)] that

@DG @T P

If Equation (20.44) is valid at all temperatures, it follows that the entropy change in the formation of hydrogen ion from gaseous hydrogen must be zero; that is,

:77) Therefore, a consistent convention would set the standard entropy of aqueous H þ ion

2 H þ (aq) þe , D @D f 2 m (g) ¼H f S8 m @T ¼0 (20

1 G8

e (18) or 65.342 J mol 21 K 2 21 H 2 ðgÞ 2 S8 2 S8 e (11). Historically, the usefulness of ionic entropies first was emphasized by Latimer and Buffington (19), who established the convention of setting the standard entropy of hydrogen ion equal to zero; that is,

equal to 1 S

(20 :78) Therefore, to maintain the validity of Equation (20.77), we should assign a value of

S8 m,H þ ;0

65.342 J mol 21 K 21 , 1 2 S8 m (H 2 ), to S8 m for a mole of electrons. In practice, half- reactions are combined to calculate DS8 m for an overall reaction in which no net gain or loss of electrons occurs; hence, any value assumed for S8 m of the electron will cancel out. We will use Equation (20.78) and the consequent value for S8 m for the electron.

Having chosen a value for S8 m,H þ , we can proceed to obtain S8 m,298K for the Cl 2 ion from any one of several reactions [for example, Equation (20.43)] for the formation of aqueous H þ and Cl 2 21 . Using values of 2131.386 kJ mol 21 and 2167.159 kJ mol for DG8 and DH8

m , respectively (11), we can calculate 2119.98 J mol K m 21 for DS8 m at 298.15 K. If we adopt the convention stated in Equation (20.78), and if S8

for Cl (g) is taken as 223.066 J mol 21 K m,298 21 2 and that for H 2 (g) is 130.684 J mol 21 K 21 , it follows that for Reaction (20.43)

D f S8 m,298 K ¼ S8 m,H þ þ S8 m,Cl

2 S8 m,H

2 2 S8 m,Cl 2

(20 :79) Hence

S8 m,Cl ¼ 56:89 J mol K

CHANGES IN GIBBS FUNCTION FOR PROCESSES IN SOLUTIONS

D f S8 m,298 K ¼ S8 m,Na þ þ S8 m,Cl

m,Na(s)

m,Cl 2 ( gas )

(20 :81) Having a value for S8 m ;Cl , we can proceed to obtain the entropy of formation

for Na þ (aq) from DS8 m for Reaction (20.51). Consequently, with S8 m,Na(s) ¼ 51.21 J mol 21 K 21 (11),

(20 :82) By procedures analogous to those described in the preceding two examples, we can

S8 m,Na þ ¼ 58:54 J mol K

obtain entropies for many aqueous ions. A list of such values is assembled in Table 20.1.

TABLE 20.1. Entropies of Aqueous Ions at 298.15 K [11] Ion

S8 m /J mol 21 K 21 H þ

S8 m /J mol 21 K 21 Ion

ClO 2 42.0 Li þ

13.4 ClO 2 2 101.3 Na þ

59.0 ClO 2 3 162.3 K þ

ClO 4 2 182.0 Rb þ

BrO 2 3 161.71 Cs þ

IO 2 118.4 NH þ

HS 2 62.8 Ag þ

SO 22 3 3 229.0 Tl þ

72.68 HSO 2 139.7

Ag(NH ) þ 3 2 245.2

HSO 2 131.8 Mg 2 þ

NO 2 2 123.0 Sr 2 þ

Ca 253.1

NO 3 2 146.4 Ba 2 þ

9.6 H 2 PO 4 2 90.4 Fe 2 þ

Hg 2 2 þ 84.5 C O

2 a 22 4 45.6

CN 2 94.1 Pb 2 þ

H 2 AsO 4 2 117.0 Fe 3 þ

Al 3 þ

2315.9 CrO 4 22 50.21

OH 2 210.75 F 2 213.8

Cl 2 56.5 Br 2 82.4

I 2 111.3

a In 3-M NaClO 4 . For entropies of aqueous ions at high temperatures see C. M. Criss and J. W. Cobble, J. Am. Chem. Soc. 86 , 5385 (1964).

491 Ion Entropies in Thermodynamic Calculations

EXERCISES

With tables of ion entropies available, it is possible to estimate a Gibbs function change without the necessity of carrying out an experiment or seeking specific exper- imental data. For example, without seeking data for the potential of calcium electro- des, it is possible to calculate the calcium electrode potential or the Gibbs function change in the reaction

Ca(s) þ 2H þ (aq) ¼ Ca 2 þ (aq) þH 2 (g) (20 :83) from the data in Table 20.1 plus a knowledge of DH8 m of this reaction. Thus,

DS8 m,298K ¼ S8 m,Ca 2 þ þ S8 m,H 2 m,Ca(s)

m,H þ

K (20 :84) As DH8

m is 2542.83 kJ mol (11), we find a value of DG8 m of 2542.07 kJ mol . Hence, E8 for Reaction (20.83) is 2.8090 V.