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

[math]dS=\frac{dQ}{T}[/math]

or

[math]dS=\frac{C_R}{T}dT[/math]

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

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

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

[math]S_1=\int^{T_1}_{0}\frac{C_R}{T}dT[/math]

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

Therefore,

C_R → 0 as T → 0 C_P → 0 as T → 0 C_V → 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:

[math]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[/math]

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

[math]\int^{T_1}_0 \frac{C_1}{T} \leq \int^{T_1}_0 \frac{C_2}{T}[/math]

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,

[math]\int^{T_1}_0 \frac{C_1}{T} \leq 0[/math]

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

Therefore, we cannot reach absolute zero.