# GIBBS ENERGY AND HELMHOLTZ ENERGY. THE CHEMICAL EQUILIBRIUM

The general criterion for equilibrium,ΔStotal = ΔSsystem + ΔSsur = 0, is the basis for deriving oth equilibrium criteria. It is usually inconvenient to consider changes in the surroundings. Under given conditions we can derive equilibrium criteria which contain thermodynamic functions of the syste alone. At constant pressure and constant temperature, the Gibbs energy is a convenient thermodynam function, while the Helmholtz energy is used when volume and temperature are constant. For bo functions we find the equilibrium criterion. Chemical reactions frequently take place under constant pressure (e.g., in an open beaker atmospheric pressure), and the Gibbs energy is a tool for studying reactions taking place und constant pressure. The Gibbs energy of formation and the Gibbs energy of reaction are parallel enthalpy of formation and enthalpy of reaction respectively. Useful equations for chemical equilibrium are developed based on the equilibrium criteric dG = O. Real gases do not behave ideally, and to describe the behaviour of a real gas we introduc fugacity, which may be considered as the effective pressure of the gas. There is a relation betwee fugacity and the compressibility factor. By a mathematical treatment of the basic equations many useful relations are found. TI1 derivation of the Maxwell relations is studied.

## THE GIBBS ENERGY AND THE HELMHOLTZ ENERGY

The fundamental equilibrium criterion was derived in Entropy lesson 5, see eq. (5.5). In the differentic form the equation is:

dStotal = dSsystem + dSsur = 0 (1.1)

while for an irreversible process we have:

dStotal = dSsystem + dSsur > 0 (1.2)

We shall consider a process taking place at constant pressure and constant temperature. The system may be of any kind, gas, liquid, solid or a mixture of these. Suppose the System undergoes a small change under equilibrium conditions. Pressure and temperature difference between the system and its surroundings are infinitely small. Pressure-volume work is the only work. An infinitesimal amount of heat, dq, is absorbed from the surroundings. The change in the entropy of the surroundings is given by:

dSsur = - dq/T (1.3)

since pRssure is constant, the heat Rceived by the system is:

dq = dHsystem (1.4)

compare eq. (2.6) in the 1st law of thermodynamics lesson 2. From eqs(1.1, 1.3, 1.4) we obtain:

dStotal = dSsystem - dHsystem /T (1.5)

Thus we have obtained dStotal expressed only by properties of the system. Omitting the subscript "system" we obtain the equilibrium criterion:

TdS - dH= 0 (1.6)

If the process is irreversible, eq. (1.2) is valid. The entropy change of the surroundings is still equal to - dq/T. As in the previous case dSsur = - dHsystem /T, and from eq(1.2) we obtain:

dStotal = dSsystem - dHsystem /T > 0 (1.7)

For an irreversible process we have:

TdS - dH> 0 (1.8)

Now we shall define a new function, the Gibbs energy, G:

G=H-TS (1.9)

Since H, T and S all are state functions, then G must be a state function. For any cha at constant temperature we have:

dG = dB - TdS (1.10)

Since dB = - TdSsur,

we have for constant P and T

dGsystem = - TdStotal

The Gibbs energy function is very useful for studying a chemical process at constant pressure, which is very often the case in practical work. The second condition for its validity is that the temperature is the same before and after the reaction. Under these conditions we have:

dG < 0 Spontaneous, irreversible process (1.11)

dG = 0 Equilibrium, reversible process

dG > 0 Process in opposite direction spontaneous and irreversible

It is very common to have constant temperature and pressure' in a process. Therefore the equilibrium criterion dG = 0 is very useful. In some processes, however, volume and temperature are kept constant.' For discussion of equilibrium under these conditions a new function is introduced, the Helmholtz energy.' The symbol used for the Helmholtz energy is A. We shall consider a process taking place at constant volume and constant temperature. Again the system may be of any kind, and again we shall consider a small change under equilibrium conditions. The temperature difference between the system and its surroundings is infinitely small. The volume is kept constant and there is no work. The infinitesimally small heat absorbed from the surroundings is;

dq = dUsystem (1.12)

compare eq. (3.7). Since dSsur = - dq/T (see eq. 1.3), we obtain from eqs (1.1, 1.12) for a reversible process;

dStotal = dSsystem - dUsystem /T (1.13)

The dSloW is expressed only by properties of the system. As before we omit the subscript "system" and obtain the equilibrium criterion for a reversible process at constant T and V:

'TdS - dU = 0 (1.14)

In a similar way, using eq. (1.2) instead of eq. (1.1), we obtain for an irreversible process at constant T and V:

TdS - dU > 0 (1.15)

Now we can define the new function, the Helmholtz energy, A, by the equation:

A=U-TS (1.16)

Since U, T and S are state functions, then A is a state function. For any change in a system at constant T and V we have:

dA = dU - TdS (1.17)

From eqs (1.14, 1.15, 1.17) we obtain for a process at constant T and V:

dA < 0 Spontaneous irreversible process

dA = 0 Equilibrium, reversible process

dA > 0 Process in opposite direction spontaneous and irreversible (1.18)

The Gibbs energy is used much more than the Helmholtz energy for equilibrium calculations, j values of Gibbs energies are readily found in tables.