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Expert-verified Found in: Page 159 ### An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279 # 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.

Coefficient of Expansion becomes Zero at T=0.

See the 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

$\beta =\frac{\Delta V/V}{\Delta T}\phantom{\rule{0ex}{0ex}}\beta =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{P}$

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

So,

$\beta =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{P}\phantom{\rule{0ex}{0ex}}\beta =-\frac{1}{V}{\left(\frac{\delta S}{\delta P}\right)}_{T}$

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 $T\to 0.$ ### Want to see more solutions like these? 