Third Law of Thermodynamics - Mathematical Formulation

Mathematical Formulation

Consider a closed system in internal equilibrium. As the system is in equilibrium there are no irreversible processes so the entropy production is zero. During the heat supply temperature gradients are generated in the material, but the associated entropy production can be kept low enough if the heat is supplied slowly. The increase in entropy due to the added heat δQ is then given by the second part of the Second law of thermodynamics which states that the entropy change of a system undergoing a reversible process is given by

The temperature rise δT due to the heat δQ is determined by the heat capacity C(T,X) according to

The parameter X is a symbolic notation for all parameters (such as pressure, magnetic field, liquid/solid fraction, etc.) which are kept constant during the heat supply. E.g. if the volume is constant we get the heat capacity at constant volume C_V. In the case of a phase transition from liquid to solid, or from gas to liquid the parameter X can be the fraction of one of the two components. Combining relations (1) and (2) gives

Integration of Eq.(3) from a reference temperature T₀ to an arbitrary temperature T gives the entropy at temperature T

We now come to the mathematical formulation of the third law. There are three steps:

1: in the limit T₀→0 the integral in Eq.(4) is finite. So that we may take T₀=0 and write

2. the value of S(0,X) is independent of X. In mathematical form

So Eq.(5) can be further simplified to

Equation (6) can also be formulated as

In words: at absolute zero all isothermal processes are isentropic. Eq.(8) is the mathematical formulation of the third law.

3: as one is free to chose the zero of the entropy it is convenient to take

so that Eq.(7) reduces to the final form

The physical meaning of Eq.(9) is deeper than just a convenient selection of the zero of the entropy. It is due to the perfect order at zero kelvin as explained before.