Gibb's Energy

Q

     {C}c {D}d
    -----------  = Q
     {A}a {B}b

Experience shows that, over time, the quotient Q tends to approach a constant K for a given reaction in a closed system. Such a state is called equilibrium,

Q -> K

The constant K depends on temperature and the nature of the reactants and products. Thus, K is called the equilibrium constant. This is known as the mass action law. In other words, there is a tendency for the reaction to reach a equilibrium such that

     {C}c {D}d
    -----------  = K
     {A}a {B}b

Using activities covers a wider range of concentrations than using concentrations to define the reaction quotient or equilibrium constant K. Concentrations were used in freshman chemistry for simplicity.

Standard Gibbs Free Energy Change, DG^o

Energy is the driving force for reactions. The tendency for a reaction to reach a equilibrium is driven by the Gibbs free energy, symbol DG^o and the relationship has been defined by the relationship

DG^o = - R T ln K

At the standard condition, activities of all reactants and products are unity (all equal to 1). In this system, Q = 1. If K > 1, the the forward reaction is spontaneous,

Q -> K.

The Gibb's free energy DG^o is a negative quantity for such a system. Since this is always true, we can generalize the condition. When the Gibb's energy is negative, the reaction is spontaneous. A reaction (change) is the result of a system trying to minimize its Gibb's free energy, DG^o changes from 0 to - R T ln K as the system changes from standard condition to an equilibrium state.

In contrast to the decrease in Gibb's energy, the entropy increases as an isolated system undergoes an spontaneous reaction.

Gibbs Free Energy Change, DG

The generalized statement can be represented by a generalized Gibb's free energy change, DG, for a system not at standard condition, but whose reaction quotient is Q. Obviously, the formulation is

DG = DG^o + R T ln Q.

As the system strives to reach an equilibrium state, (no longer any net change),

Q -> K

we have the following results,

DG^o = - R T ln K
DG^o + R T ln K = 0
DG = 0.

The previous discussion leads to the following conclusion. When DG is positive, the reverse reaction is spontaneous. When DG is negative, the forward reaction is spontaneous, when DG is zero, the system has reached an equilibrium state.

Gibb's Energy in Terms of Enthalpy and Entropy

So far, the Gibb's free energy is defined as the driving force for a system to reach a chemical equilibrium. The energy comes from the enthalpy and entropy of reaction in the system, and DG has been define in terms of enthalpy and entropy changes, DH and DS, at temperature T as:

DG = DH - T DS.

Since Gibb's energy, enthalpy, and entropy are state functions, they have been treated as the functions in thermodynamics, and as a result, the delta D is omitted. The relation is simply,

G = H - T S

The generalized equation is very useful, and it can be differentiacted with respect to other thermodynamical variables. However, we will not discuss it any further at this point.

Gibb's Energy and Electric Energy

For redox reactions, Gibb's energy is the electric energy, which, when properly setup in an electric cell, is the charge transferred (q in Coulomb) times the potential E (in V). Each mole of electron has a charge of 1 faraday (1 F = 96458 C), and n moles of electron have a charge of n F. Since the voltage is usually a positive value, we have,

DG = - n F E

However, lines from the Thermodynamic Data Table are reproduced below for you to compare: Note that data for diamond come from another source.

Substance   DHfo          DSo
            kJ/mol       J/(mol K)
H2(g)        0           130.680
I2(s)        0           116.14
O2(g)        0           205.152
C(graphite)  0             5.74
C(diamond)   1.985         2.377
H2O(l)    -285.83         69.95
HI(g)       26.50        206.590

DISCUSSION
Note units for the two quantities. The standard entropies in J/(mol K) are also the absolute entropies.

Therefore,

         1710 J mol-1
ln K = ------------------------- = 0.6904
       8.312 J (mol K)-1 * 298 K

Thus, K = e^0.6904 = 2.0

DISCUSSION
Examples 1, 2, and 3 illustrate how DG_f can be derived from a table of thermodynamic data. In some tables, the value of DG_f are also given.

The equilibrium constant is evaluated by

            237160 J mol-1
ln K = ------------------------- = 95.746
       8.312 J (mol K)-1 * 298 K

Thus, K = e^95.746
= 13.82*10⁴¹,
a very large number indeed indicating a reaction to almost exhaust at least one of the reactants.

DISCUSSION
This example illustrates how you may use a thermodynamic data table.

The data required are:

Substance   DHfo          DSo
            kJ/mol       J/(mol K)
H2(g)        0           130.680
N2(g)        0           191.609
NH3(g)       -45.94           192.77

Again, we write the standard entropies below the formula

3/2 H₂(g) + ½ N₂(g) -> NH₃(g)
3/2*130.680 ½*191.609 192.77

DS^o_f = SS^o(products) - SS^o(reactants)
= 192.77 - (³/₂*130.680 + ½*191.609)
= -99.125 kJ/mol
The standard Gibb's free energy of formation is,
DG^o_f = DH^o_f - T DS^o_f
= -45.94 - 298*(-0.099125) kJ/mol = -16.40 kJ/mol

DISCUSSION
Results from the previous and this examples are used in the next example.

Gibb's energy is the maximum electric energy derived from a battery.

Contributors and Attributions

• Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo)