Question

Use a Maxwell relation from the previous problem and the third law of thermodynamics to prove that the thermal expansion coefficient $\beta$(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

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

Explanation

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

This means

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

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

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 $T\to 0$, 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 $T\to 0.$and $\beta$becomes 0.

We can conclude that coefficient of expansion becomes zero at $T\to 0.$