# Consequences of Third Law

This section is Optional.

Objectives
 This section proves two interesting consequences of the third law.

There are two important consequences of the Third Law: the behavior of heat capacities as temperature goes to zero and that we cannot get to absolute zero.

## Heat capacities

We wish to know how heat capacities behave as the temperature goes to zero.

Let us consider a reversible path R, according to the second law

$dS=\frac{dQ}{T}$

or

$dS=\frac{C_R}{T}dT$

where CR is the heat capacity along path R. Integrating from T = 0 to T = T1 gives

$S_1=\int^{T_1}_{0}\frac{C_R}{T}dT+S\left ( T=0\right )$

S at T = 0 is by the third law equals zero, therefore

$S_1=\int^{T_1}_{0}\frac{C_R}{T}dT$

The third law requires that S1 → 0 as T>sub>1</sub> → 0. The integral can only go to zero if CR also goes to zero. Otherwise the integral becomes unbounded.

Therefore,

CR → 0 as T → 0 CP → 0 as T → 0 CV → 0 as T → 0

All substances measured so far have obeyed this property.

## Unattainability of Absolute Zero

The unattainability of absolute zero says that we can ever reach absolute zero experimentally.

To prove this let us consider a process where we vary parameter X from an initial state (X1, T1) to a final state (X2, T2). Then by the second law:

$S_1\left (T=0\right ) + \int^{T_1}_0 \frac{C_1}{T}dT \leq S_2\left (T=0\right ) + \int^{T_2}_0 \frac{C_2}{T}dT$

By the third law, S1 (T = 0) = S2 (T = 0), therefore,

$\int^{T_1}_0 \frac{C_1}{T} \leq \int^{T_1}_0 \frac{C_2}{T}$

Let us now cool the system from a positive T1 to absolute zero, that is T2 = 0. Then the integral on the right is zero. Then,

$\int^{T_1}_0 \frac{C_1}{T} \leq 0$

However, the integral on the left is positive since T1 %neq; 0.

Therefore, we cannot reach absolute zero.