Answer to Question #129858 in Molecular Physics | Thermodynamics for Abhishek

Question

State the third law of thermodynamics and show that the heat capacities of a substance converge as T - 0 .

Answer 1

The third law of thermodynamics (or Nernst theorem) relates to the behavior of systems as they approach the absolute zero of temperature. Nernst suggested that the entropy of physical systems would tend to zero as $T → 0$ .

$\lim_{T\to0}S=0$ (1)

The consequences of this law are that most temperature derivatives of thermodynamic quantities go to zero at least as fast as T, and in particular the specific heat should go to zero with T. For changes at constant volume, we know that

$S(T_{2}, V) – S (T_{1}, V) = \int_{T_{1}}^{T_{2}} \frac{C_{V}(T)}{T}dT.$ (2)

The condition (1) implies that in the limit $T_{1} → 0$ , the integral in (2) must be finite, and hence we require that $C_{V}(T) → 0$ as $T → 0$ . Similarly, we can argue that $C_{P} → 0$ as $T → 0$ . Note that these conditions about the low temperature behavior of $C_{V}$ and $C_{P}$ are independent of the nature of the system.