CALCULATION OF CHANGE IN THE GIBBS FUNCTION FOR SPONTANEOUS PHASE CHANGE

8.4 CALCULATION OF CHANGE IN THE GIBBS FUNCTION FOR SPONTANEOUS PHASE CHANGE

Thus far we have restricted our attention to phase changes in which equilibrium is maintained. It also is useful, however, to find procedures for calculating the change in the Gibbs function in transformations that are known to be spontaneous, for example, the freezing of supercooled water at 2108C:

H 2 O(l,

At 08C and 101.325-kPa pressure, the process is at equilibrium. Hence

DG m,08C ¼0

At 2108C, the supercooled water can freeze spontaneously. Therefore,

DG m,

Now we wish to evaluate DG m numerically.

Arithmetic Method The simplest procedure to calculate the change in the Gibbs function at 2108C uses

the relationship [Equation (7.26)] for one mole,

DG m ¼ DH m

for any isothermal process. DH m and DS m at 2108C (T 2 ) are calculated from the known values at 08C (T 1 ) and from the temperature coefficients of DH m and DS m . As the procedure can be represented by the sum of a series of equations, the method may be called an arithmetic one. The series of equations is given in Table 8.1. (Here, as in Chapter 4, Equations (4.83) – (4.86) we assume that C P is con- stant for ice and supercooled water over this temperature range. See Ref. 18 in

8.4 CALCULATION OF CHANGE IN THE GIBBS FUNCTION

TABLE 8.1. Change in Gibbs Function for Freezing of Supercooled Water H 2 O (l, 08C) ¼ H 2 O (s, 08C)

H 2 O (s, 08C) ¼ H 2 O (s, 2108C)

H 2 O (l, 2108C) ¼ H 2 O (l, 08C)

DH m ¼

C P m,l dT

¼ 75(10) ¼ 750 J mol ð T 1 C m,l

DS m ¼

dT ¼ 2:797 J mol K

H 2 O (l, 2108C) ¼ H 2 O (s, 2108C) DH m ¼ 25622 J mol 21 DS m ¼ 220.556 J mol 21 K 21

Chapter 4 for recent results on C P for this system.) From the values calculated for DH m and DS m .

¼ 0:809 J mol K

263 :15 K

Analytic Method The proposed problem also could be solved by integrating Equation (7.57) for

one mole,

As in the arithmetic method, we can assume that the heat capacities of ice and water are substantially constant throughout the small temperature range under consider- ation. Thus, from Equation (8.28),

@DH m ¼ DC P m @T P

APPLICATION OF THE GIBBS FUNCTION AND THE PLANCK FUNCTION

and during integration, we obtain

DH m ¼ DH m0 þ (C P m,ice P m,water )dT

(8 :33) where DH m0 ia a constant of integration (not equal to DH m at 08C). As, at 08C, DH m

¼ DH m0

is 26008 J mol 21 , we can determine DH m0 :

DH m0

(More significant figures are retained in these numbers than can be justified by the precision of the data on which they are based. However, such a procedure is necessary in calculations that involve small differences between large numbers.) Thus,

in which I is a constant of integration. As DY m is known to be zero at 08C, the constant

I can be evaluated: 4617 :54

I ¼ þ 38:9 ln (273:15) ¼ 235:135

DY m (8 :35)

Equation (8.35) leads to DG m

(8 :36) At 2108C, the value of the Gibbs function is

DG m,

This result is the same as that obtained by the arithmetic method.

205 As

EXERCISES

¼Q system

suroundings

(8 :37) we should note that a process like the freezing of supercooled water, which is spon-

surroundings

system

taneous despite having a negative value of DS sys , can have a negative value of DG only if the value of DS surr is positive and of greater magnitude than DS sys .