## Gibbs Free Energy

### Definitions

The following definitions are useful in considering free energy relationships:

• The heat of formation, ΔHf, of an element in its standard state (298K, 1 atmosphere pressure) is zero.
• The free energy of formation, ΔFf, of an element in its standard state is zero.
• F = H - TS.
• ΔFf  = ΔHf- TΔS.

### Application to some Simple Reactions

Consider the reaction:   H2 + Cl2 -> 2HCL

From Table 1, the ΔHf change can be calculated using:

 HCL - (1/2)H2 - (1/2)Cl2 -22.06 - 0.00 - 0.00

to be -22.06 kcal/mol.

Table 1: Thermodynamic Properties of some Compounds

 Compound Formula ΔHf   (kcal/mol) ΔFf    (kcal/mol) S   (cal/mol/K) Hydrogen Chloride (g) HCl -22.06 -22.77 44.62 Water (g) H2O -57.80 -54.64 45.11 Water (l) H2O -68.32 -56.69 16.72 Methane (g) CH4 -17.89 -12.14 44.50 Oxalic acid (s) C2H2O4 -197.6 -166.8 28.70

The corresponding entropy change, ΔS, requires using the entropies for the elements in their standard state, Table 2, and can be calculated using:

 HCL - (1/2)H2 - (1/2)Cl2 44.62 - (1/2)31.21 - (1/2)53.29

to be +2.37 cal/mol/K.

Using these results, the free energy change, ΔFf, can be calculated:

 ΔFf = ΔHf - TΔS -22.77 = -22.06 - 298x2.37/1000

which agrees with the value given in the CRC handbook (Table 1, here)

In like manner, the free energy of water, as the vapor, can be calculated:

ΔFf(H2O) = ΔHf(H2O) -TΔS(H2O-H2-(1/2)O2)

as

-54.64 = -57.80 - 298x( 45.11 - 31.21 - 0.5x49.00)/1000

Similarly, the free energy of water, as the liquid, can be calculated:

ΔFf(H2O) = ΔHf(H2O) -TΔS(H2O-H2-(1/2)O2)

as

-56.70 = -68.32 - 298x( 16.72 - 31.21 - 0.5x49.00)/1000

which is close to the reported ΔFf of -56.69 kcal/mol.

For oxalic acid, the equivalent calculation is

ΔFf(C2H2O4) = ΔHf(C2H2O4) -TΔS(C2H2O4-H2-2O2-2C)

or

-166.8 = -197.6 - 298x( 28.70 - 31.21 - 2x49.00 - 2x1.36)/1000

Table 2: Entropies for the Elements at 298K (cal/K/mol)

 I II Transition Metals III IV V VI VII H 31.21 Li 6.7 Be 32.53 B 1.56 C 1.36 N 45.77 O 49.00 F 48.6 Na 12.2 Mg 7.77 Al 39.30 Si 4.47 P 10.6 S 7.62 Cl 53.29 K 15.2 Ca 9.95 Ti 7.24 V 7.05 Cr 5.68 Mn 7.59 Fe 6.49 Co 6.8 Ni 7.20 Cu 7.96 Zn 9.95 Ga Ge 10.14 As 8.4 Se Br 36.4 Rb 16.6 Sr 13.0 Zr 9.18 Mo 6.83 Pd 8.9 Ag 10.21 Cd 12.3 In Sn 12.3 Sb 43.06 Te 11.88 I 27.9 Cs 19.8 Ba 40.70 Pt 10.0 Hg 18.5 Tl 15.4 Pb 15.51 Bi 13.6

("CRC Handbook of Chemistry and Physics," 60th Edition, R. C. Weast, (Ed.), CRC Press, Boca Raton, FL, 1980.)

In MOPAC, the quantities ΔHf and S are calculable.  ΔHf is the calculated heat of formation coming out of the SCF calculations, S, the entropy, is printed near the end of a THERMO calculation.  In the following output from a methane calculation, the entropy terms is in bold and italicized:

CALCULATED THERMODYNAMIC PROPERTIES
*
TEMP. (K)   PARTITION FUNCTION  H.O.F.     ENTHALPY   HEAT CAPACITY  ENTROPY
KCAL/MOL   CAL/MOLE    CAL/K/MOL   CAL/K/MOL

298    VIB.     0.1006D+01                    24.0439     0.5429     0.0927
ROT.     0.3850D+02                   888.2854     2.9808    10.2356
INT.     0.3874D+02                   912.3293     3.5238    10.3283
TRA.     0.6212D+26                  1480.4756     4.9680    34.2611
TOT.                     -8.790      2392.8049     8.4918    44.5894

The entropies of the pure elements should be taken from Table 2.  A comparison of the three gaseous systems in Table 1 is presented in Table 3:

Table 3: Comparison of Observed and Calculated Thermodynamic Quantities

 Observed (Table 1) Calculated (AM1) Compound Formula ΔHf   (kcal/mol) ΔFf    (kcal/mol) S   (cal/mol/K) ΔHf    (kcal/mol) ΔFf    (kcal/mol) S   (cal/mol/K) Hydrogen Chloride (g) HCl -22.06 -22.77 44.62 -24.61 -25.32 44.63 Water (g) H2O -57.80 -54.64 45.11 -59.25 -56.09 45.09 Methane (g) CH4 -17.89 -12.14 44.50 -8.79 -3.04 44.59