CALCULATION OF A STANDARD GIBBS FUNCTION FROM STANDARD ENTROPIES AND STANDARD ENTHALPIES

12.5 CALCULATION OF A STANDARD GIBBS FUNCTION FROM STANDARD ENTROPIES AND STANDARD ENTHALPIES

Now that we can calculate standard entropies from the third law (as in Chapter 11), let us consider their primary use, which is the calculations of the standard Gibbs function.

Enthalpy Calculations As we mentioned, it is necessary to have information about the standard enthalpy

change for a reaction as well as the standard entropies of the reactants and products to calculate the change in Gibbs function. At some temperature T, DH8 mT can be

obtained from D f H m 8 of each of the substances involved in the transformation. Data on the standard enthalpies of formation are tabulated in either of two ways. One method is to list D f H m 8 at some convenient temperature, such as 258C, or at a series of temperatures. Tables 4.2 through 4.5 contain values of D f H m 8 at 298.15 K. Values at temperatures not listed are calculated with the aid of heat capacity equations, whose coefficients are given in Table 4.8.

On the basis of statistical thermodynamics, another method of tabulation, using H8 mT 2 H8 m0 , or (H8 mT 2 H8 m0 ) /T, or (H8 mT 2 H8 m298.15 ) /T, in which the subscripts refer to the Kelvin temperatures, has come into general use. This method of

TABLE 12.1. Enthalpy Increment Function (H88888 mT 2 H88888 m298.15 ) a

2 H8 21 m298.15 )] /(kJ mol ) Substance

[(H8 mT

800 1000 1500 CO(g)

D f H8 m298.15 /(kJ mol 21 )

15.177 21.690 38.850 CO 2 (g)

22.806 33.397 61.705 H 2 O(g)

18.002 26.000 48.151 HBr(g)

14.956 21.294 38.103 HCl(g)

14.835 21.046 37.508 HF(g)

14.677 20.644 36.239 NH 3 (g)

21.853 32.637 63.582 H 2 (g)

14.702 20.680 36.290 N 2 (g)

15.046 21.463 38.405 F 2 (g)

17.277 24.622 43.458 Cl 2 (g)

18.108 25.565 44.422 C(graphite)

7.637 11.795 23.253 O 2 (g)

a M. W. Chase, Jr., NIST-JANAF thermochemical tables, 4th ed., J. Phys. Chem. Ref. Data, Monograph No. 9 (1995).

APPLICATION OF THE GIBBS FUNCTION TO CHEMICAL CHANGES

TABLE 12.2. Enthalpy Increment Function (H88888 mT 2H88888 m0 ) /T a

[(H8 mT 2 H8 m0 ) /T ]/(J mol 21 K 21 ) Substance

D f H8 m0 /(kJ mol 21 ) 298.15 K 400 K 600 K 800 K 1000 K 1500 K CO(g)

2113.8 29.11 29.12 29.37 29.81 30.36 31.68 CO2(g)

2393.1 31.41 33.42 37.12 40.21 42.76 47.37 Methane(g)

266.6 33.59 34.72 38.65 43.52 48.49 59.53 Methanol(g)

2190.1 38.37 40.75 47.03 53.70 59.94 72.73 Ethyne(g)

228.8 33.57 37.09 42.89 47.38 51.04 58.11 Ethene(g)

61.0 35.28 38.51 46.43 54.21 61.18 75.05 Aceticacid(g)

2418.1 45.60 52.24 66.02 78.61 89.44 109.59 Ethane(g)

268.2 39.83 44.68 55.72 66.53 76.36 96.09 Ethanol(g)

2217.4 47.80 54.26 67.99 80.64 91.64 112.97 Propene(g)

34.7 45.44 52.30 66.48 79.57 91.05 113.44 Propane(g)

282.4 49.44 58.19 76.15 92.73 107.19 135.48 1-Butene(g)

20.4 57.43 67.53 87.87 106.29 122.22 152.93 n -Butane(g)

297.2 64.60 76.61 100.42 121.96 140.67 176.82 Benzene(g)

100.4 47.73 60.60 86.56 108.99 127.50 161.24 Cyclohexane(g)

283.8 58.85 76.18 113.36 147.99 177.76 Toluene(g)

73.3 60.51 76.19 107.41 134.67 157.44 199.49 o -Xylene(g)

46.4 78.45 97.06 132.54 163.66 190.03 239.57 m -Xylene(g)

45.9 74.20 92.53 128.63 160.45 187.31 237.47 p -Xylene(g)

46.8 73.83 92.46 129.07 161.15 188.12 238.29 H 2 O(g)

2238.9 33.21 33.39 34.01 34.88 35.91 38.78 H 2 (g)

0.0 28.40 28.57 28.80 28.96 29.15 29.84 O 2 (g)

0.0 29.12 29.27 29.88 30.65 31.39 32.85 C(graphite) b 0.0 3.53 5.23 8.32 10.86 12.85 16.20

a M. Frenkel, K. N. Marsh, R. C. Wilhoit, G. J. Kabo, and G. N. Roganov, Thermodynamics of Organic Compounds in the Gas State , Thermodynamics Research Center, College Station, TX, 1994.

b Calculated from data in NIST-JANAF Thermochemical Tables; see Table 12.1.

presentation, which is illustrated in Tables 12.1 through 12.3, does not require the use of empirical heat capacity equations and allows easy comparison of data from different sources.

The following procedure is used to calculate D f H m 8 at any temperature T from Table 12.2. Data in Tables 12.1 and 12.3 are different only in the reference temperature. The standard enthalpy of formation of a compound C refers to the reaction

B f H8 m (12.24)

element in

compound in standard state

element in

standard state at temperature T

standard state

at temperature T

at temperature T

289 TABLE 12.3. Enthalpy Increment Function (H88888

12.5 CALCULATION OF A STANDARD GIBBS FUNCTION

mT 2H88888 m298.15 ) /T [(H8 mT 2 H8 m298.15 ) /T ]/(J mol 21 K 21 ) D f H8 m298.15 /

H8 m298 2 H8 m0 /

Substance (J mol 21 K 21 ) (J mol 21 K 21 ) 400 K 600 K 800 K 1000 K 1500 K Graphite(c)

0.0 1.05 2.60 6.58 9.54 11.78 15.51 Diamond(c)

2.09 5.88 8.90 11.24 15.43 Cl 2 (g)

0.0 9.18 8.83 17.89 22.63 25.57 29.61 HCl(g)

292.3 8.64 7.40 14.70 18.55 21.06 25.00 CaSO 4 (c)

21434.4 17.30 27.87 60.83 80.98 95.52 CuSO 4 (c)

2771.4 16.87 38.57 83.11 109.44 127.68 Fe 2 SiO 4 (c)

21478.2 22.49 36.51 78.23 102.16 118.39 Mg 2 SiO 4 (c)

22173.0 17.22 32.92 71.21 93.81 109.28 133.66 CaSiO 3 (c)

21634.8 13.84 23.77 51.50 67.83 78.74 Methane(g)

9.52 21.74 30.80 38.16 51.98 NH 3 (g)

9.43 20.28 27.31 32.65 42.37 CO(g)

2110.5 8.67 7.39 14.87 18.97 21.69 25.88 CO 2 (g)

2393.5 9.36 9.98 21.49 28.51 33.40 41.13 SiO 2 (c); quartz b 2910.7

12.52 28.18 38.58 45.52 53.96 SiO 2 (c);

2908.4 7.04 12.63 29.96 38.73 44.74 53.70 cristobalite c O 2 (g)

7.54 15.40 19.79 22.70 27.06 Cu(c) d 0.0 5.00 6.34 12.88 16.39 18.71 31.63

Ca(c) e 0.0 5.707

6.71 13.98 19.65 23.07 33.24 Fe(c) f 0.0 4.51 6.69 14.35 19.45 24.39 30.38

H 2 (g)

7.41 14.69 18.38 20.68 24.20 Mg(c) g 0.0 5.00 6.50 13.39 17.37 28.87 114.72 N 2 (g)

11.60 20.25 23.39 76.82 57.57 Si(c)

S(c) h 0.0 4.412

a R. A. Robie and B. S. Hemingway, Thermodynamic properties of minerals and related substances, U.S. Geologic Survey Bulletin 2131 , 1995.

b Trigonal crystals to 844 K. b-crystals from 844 K. c Tetragonal crystals to 523 K. Cubic crystals from 523 K. d Melting point 1357.8 K. e a-crystals to 716 K. b-crystals to melting point at 1115 K.

f Body-centered cubic to 1185 K. Curie point at 1043 K. Face-centered cubic from 1158 K to 1667 K. Body-centered cubic from 1667 K to melting point.

g Hexagonal close-packed crystals to melting point at 923 K. Boiling point at 1366.1 K. h Orthombic crystals to 368.3 K. Monoclinic crystals from 368.3 K to melting point at 388.36 K. Liquid to

fictive boiling point at 882.1 K.

The sum of the following equations gives the required D f H m 8 in terms of the functions (H8 mT 2 H8 m0 ) /T:

A(0 K)

f H8 m0 (12 :25) A(T K) ¼ A(0 K), DH ¼ DH8 m ¼ (H8 m0

mT ) A (12 :26)

APPLICATION OF THE GIBBS FUNCTION TO CHEMICAL CHANGES

C(0 K) ¼ C(T K), DH ¼ DH8 m ¼ (H8 mT m0 ) C (12 :27) B(T K) ¼ B(0 K), DH ¼ DH8 m ¼ (H8 m0

mT ) B (12 :28) A(T K)

D f H8 mT ¼ DH8 mT ¼D f H8 m0 þ (H8 mT

m0 ) compound

m0 ) elements

D f H8 mT =T ¼D f H8 m0 =T þ (H8 mT

m0 ) compound =T

m0 ) elements =T (12 :30) Each quantity in Equation (12.30) can be obtained from tables such as Table 12.2.

X (H8

mT

The equations required for use with Tables 12.1 and 12.3 will be provided as an exercise at the end of the chapter.

Entropy Calculations Standard entropies for many substances are available in tables such as Tables 11.2

through 11.6. Generally, the values listed are for 298.15 K, but many of the original sources, such as the tables of the Thermodynamics Research Center, the JANAF tables, or the Geological Survey tables, give values for other temperatures also. If heat capacity data are available, entropy values for one temperature can be converted to those for another temperature by the methods discussed in Section 11.4.

In a reaction such as that represented by Equation (12.23), the standard entropy change DS8 mT at the temperature T is given by the expression

DS8 X

mT ¼ S8 mT(compound)

S8 mT(elements) (12 :31)

Change in Standard Gibbs Function When adequate enthalpy and entropy data are available, the calculation of DG8 mT is a

matter of substitution into Equations (7.26).

DG8 mT ¼ DH8 mT

mT

Generally, data for DH8 m and DS8 m are available at least at one temperature. The conversion of the data for the Gibbs function from one temperature to another can

be carried out by the methods outlined in Chapter 7. Statistical thermodynamic methods and the use of spectroscopic information lead to the function (G8 mT 2 H8 m0 ) /T, and this function is tabulated in Table 12.4. Tables 12.5 and 12.6 give the alternative function (G8 mT 2 H8 m298.15 ) /T, which is equal to (Y8 mT 2 Y8 m298.15 ). This method of tabulation avoids the use of empirical

12.5 CALCULATION OF A STANDARD GIBBS FUNCTION

TABLE 12.4. Gibbs Function Increment (G88888

mT

2 H88888 m0 ) /T

[(G8 mT 2 H8 m0 ) /T ]/(J mol 21 K 21 ) Substance

1000 K 1500 K CO(g)

2197.48 2204.21 2216.80 CO 2 (g)

2217.26 2226.52 2244.80 Methane(g)

2189.34 2199.56 2221.41 Methanol(g)

2245.21 2257.87 2284.73 Ethyne(g)

2206.87 2217.85 2239.96 Ethene(g)

2226.29 2239.15 2266.74 Acetic acid(g)

2296.72 2315.47 2355.84 Ethane(g)

2239.47 2255.37 2290.33 Ethanol(g)

2293.70 2312.91 2354.36 Propene(g)

2280.44 2299.46 2340.89 Propane(g)

2287.83 2310.15 2359.13 1-Butene(g)

2327.81 2353.28 2409.04 n -Butane(g)

2333.46 2362.75 2427.19 Benzene(g)

2294.97 2321.35 2379.92 Cyclohexane(g)

2333.57 2369.89 Toluene(g)

2352.04 2384.62 2457.01 o -Xylene(g)

2389.81 2429.25 2516.35 m -Xylene(g)

2394.71 2433.49 2519.62 p -Xylene(g)

2388.94 2427.89 2514.36 H 2 O(g)

2188.83 2196.72 2211.81 H 2 (g)

2130.59 2137.07 2149.01 O 2 (g)

C(graphite) b

a M. Frenkel, K. N. Marsh, R. C. Wilhoit, G. J. Kabo, and G. N. Roganov, Thermodynamics of Organic Compounds in the Gas State , Thermodynamics Research Center, College Station, TX, 1994.

b Calculated from values in NIST-JANAF Thermochemical Tables; see Table 12.6.

equations, with their associated constants, and allows direct comparison of data from different sources. Although we will not discuss the methods for calculating this new function from experimental data by statistical thermodynamic methods, we will use tables of these functions to obtain the change in the Gibbs function for

a reaction.

To calculate the Gibbs function for the formation of a compound C, we use Equations (12.32) through (12.36), which are analogous to the equations used to calculate D f H m 8 . In each case we use the relationship [Equation (11.9)] at 0 K

G8 m0 ¼ H8 m0

to obtain A(0 K)

f G8 m0 ¼D f H8 m0 (12 :32) C(0 K) ¼ C(T), DG ¼ (G8 mT

m,0K ) C ¼ (G8 mT m,0K ) C (12 :33)

APPLICATION OF THE GIBBS FUNCTION TO CHEMICAL CHANGES

TABLE 12.5. Gibbs Function Increment, 2(G88888 a

mT

2 H88888 m298.15 ) /T [2 (G8 mT 2 H8 m298.15 ) /T ]/(J mol 21 K 21 )

1000 K 1500 K C(graphite)

6.12 7.97 10.29 12.67 18.22 Carbon(diamond)

2.09 5.88 8.90 11.24 15.43 Cl 2 (g)

241.20 252.44 HCl(g)

201.84 211.20 CaSO 4 (c)

169.92 CuSO 4 (c)

127.68 Fe 2 SiO 4 (c)

230.77 Mg 2 SiO 4 (c)

134.32 Methane(g)

CaSiO 3 85.26 100.71

209.27 227.53 NH 3 (g)

213.82 229.03 CO(g)

212.43 222.11 CO 2 (g)

235.88 251.04 SiO 2 (c); a quartz b 43.33 51.64 61.25 70.73 90.96 SiO 2 (c); a cristobalite c 45.31 54.64 64.55 73.87 93.91

220.86 230.99 N 2 (g)

206.73 216.30 H 2 (g)

145.54 154.66 Mg(c) d 33.66 37.76 42.20 47.11 67.49 Cu(c) e 34.11 38.08 42.31 46.23 55.58 Ca(c) f 43.93 48.17 52.99 57.75 70.13

Si(c) 19.63 23.08 26.84 30.38 37.98 Fe(c) g 28.10 32.41 37.27 42.11 53.67 S(c) h 33.19 39.82 46.14 58.59 85.56

a R. A. Robie and B. S. Hemingway, Thermodynamic properties of minerals and related substances, U.S. Geologic Survey Bulletin 2131 , 1995.

b Trigonal crystals to 844 K. b-crystals from 844 K. c Tetragonal crystals to 523 K. Cubic crystals from 523 K. d Hexagonal close-packed crystals to melting point at 923 K. Boiling point at 1366.1 K. e Melting point 1357.8 K.

f a-Crystals to 716 K. b-crystals to melting point at 1115 K. Body-centered cubic 1667 K to melting point. g Body-centered cubic to 1185 K. Curie point at 1043 K. Face-centered cubic from 1158 K. h Orthombic crystals to 368.3 K. Monoclinic crystals from 368.3 K to melting point at 388.36 K. Liquid to

fictive boiling point at 882.1 K.

B(T) ¼ B(0 K), DG ¼ (G m,0K

m,0K ) B (12 :34) A(T) ¼ A(0 K), DG ¼ (G8 m,0K

m,T ) B m,T

m,T ) A m,T m,0K ) A (12 :35) The summation of Equations (12.32) through (12.35) leads to the expression

A(T)

DG8 m,T ¼D f H8 m,0 þ (G8 m,T

m,0K ) compound

X (G8

m,T

m,0K ) elements

293 TABLE 12.6. Gibbs Function Increment, 2(G88888 a

EXERCISES

mT 2H88888 m298.15 ) /T

[2(G8 21 2 H8 m298.15 ) /T ]/(J mol 21 K ) Substance

mT

1000 K 1500 K H 2 (g)

145.536 154.652 O 2 (g)

220.875 231.002 N 2 (g)

206.708 216.277 F 2 (g)

219.930 230.839 Br 2 (g)

264.143 275.595 C(graphite)

12.662 18.216 CO(g)

212.848 222.526 CO 2 (g)

235.901 251.062 H 2 O(g)

206.738 218.520 HBr(g)

213.737 223.228 HF(g)

188.631 197.733 NH 3 (g)

213.849 229.054 Cl 2 (g)

241.203 252.438 HCl(g)

a M. W. Chase, Jr., NIST-JANAF Thermochemical Tables, 4th ed., J. Phys. Chem. Ref. Data, Monograph No. 9 (1995).

To express the result in terms of the statistical thermodynamic expression, we use (G8 mT 2 H8 m0 ) /T instead of (G8 mT 2 H8 m0 ). Hence, we have the following expression for the Gibbs function for the formation D f G8 mT of any compound at some temperature T:

D f G8 m,T

¼ m,0K þ

D f H8 m,0K

If the change in the Gibbs function for the formation of each substance in a reac- tion is known at the desired temperature, the change in the Gibbs function for any

reaction involving these substances can be calculated from the equation

DG mT ¼ SD f G mT(products)

f G8 mT(reactants) (12 :39)