Thermodynamics/Third Law/Third Law Consequences

 This section is Optional.

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 &rarr; 0 as T>sub>1 &rarr; 0. The integral can only go to zero if CR also goes to zero. Otherwise the integral becomes unbounded.

Therefore, CR &rarr; 0 as T &rarr; 0 CP &rarr; 0 as T &rarr; 0 CV &rarr; 0 as T &rarr; 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.