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5.13

Expert-verifiedFound in: Page 159

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.

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

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 $T\to 0.$

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