Thermodynamic Relations: Helmholtz and Gibbs Function (With Derivation)

In this article we will discuss about the Helmholtz and Gibbs function used to express thermodynamic Relations.

1. Helmholtz Function or Work Function:

Helmholtz function (A) is a combination of properties and is mathematically expressed as –

A = U – TS …(1)

For unit mass, specific Helmholtz function is given by –

a = u – TS

Where, u = Specific internal energy, and

s = Specific entropy

From the first law, for a closed system, the work dW is given by –

δW = δQ – du …(2)

Thus, the energy converted to work is provided in part by the system, whose internal energy decreases by -du, and in part by the heat reservoirs with which the system is in contact and which give up a quantity of heat dQ.

The expressions for maximum amount of work can be derived when a system undergoes a process between two equilibrium states. It is assumed that system exchanges heat only with single heat reservoirs at temperature T_o.

From the principle of increase of entropy, we know that –

Thus, the above equation implies that in any process between two equilibrium states at the same temperature during which the system exchanges heat only with the reservoir, the work done is equal to or less than the decrease in Helmholtz function of the system during the process.

The maximum work is done when the process is reversible.

2. Gibbs Function (Thermodynamic Potential):

Gibbs function (G) is a property of a system, which is given as –

Consider a system that can do work in other form e.g., electrical, magnetic works etc., in addition to PdV

It is observed that the difference between Gibbs function of a system between two equilibrium states sets the maximum limit to the work that can be performed by the system in addition to the Pdv work. This is observed when the two equilibrium states are at the same temperature and pressure and when the system exchanges heat with a single heat reservoir.

The work done is maximum when the process is reversible and is less for an irreversible process.

Note:

Gibbs function is also known as Free Energy Function.