Gibbs Free Energy

 

Definitions

The following definitions are useful in considering free energy relationships:

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