Question

Use a Maxwell relation from the previous problem and the third law of thermodynamics to prove that the thermal expansion coefficient (defined in Problem 1.7) must be zero at T=0.

Short answer

Coefficient of Expansion becomes Zero at T=0.

Step-by-step solution

Given Information

Maxwell relation: ${\left(\frac{\partial V}{\partial T}\right)}_P=-{\left(\frac{\delta S}{\delta P}\right)}_T$

Explanation

We know that " The thermal expansion coefficient is defined as the fractional change in volume per unit temperature change".

$$\begin{gathered} \beta =\frac{\Delta V/V}{\Delta T} \\ \beta =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_P \end{gathered}$$

From the Maxwell relation we have$-{\left(\frac{\delta S}{\delta P}\right)}_T$

SO,

$$\begin{gathered} \beta =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_P \\ \beta =-\frac{1}{V}{\left(\frac{\delta S}{\delta P}\right)}_T \end{gathered}$$

From the the third law of thermodynamics as T→ 0 , the entropy approaches to zero or some constant value which is independent of pressure.

This means ${\left(\frac{\delta S}{\delta P}\right)}_T$becomes Zero as T→0 and $\beta$becomes 0 as T→0.