How Do You Derive Expansivity and Isothermal Compressibility for an Ideal Gas?

[LeePhilip01]

Homework Statement

Show that:-
a) the expansivity \beta = \frac{1}{T}
b) the isothermal compressibilty \kappa = \frac{1}{P}

Homework Equations

P\upsilon = RT where \upsilon = molar volume

The Attempt at a Solution

A big mess!

 

[Mapes]

Hi LeePhilip01, welcome to PF. Do you know how the expansivity and isothermal compressibility are defined in general? (Hint: it will involve derivatives.)

 

[LeePhilip01]

Yes, however i wasn't sure whether they were important because they weren't given in th question.

\beta = \frac{1}{V} . \frac{dV}{dT}

\kappa = - \frac{1}{V} . \frac{dV}{dP}

 

[Mapes]

To be precise, we should say

\beta=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P[/itex]<br /> <br /> \kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T[/itex]&lt;br /&gt; &lt;br /&gt; to acknowledge that &lt;i&gt;V&lt;/i&gt; is a function of multiple variables and that we are taking the partial derivative with respect to one of the variables while holding the others constant.&lt;br /&gt; &lt;br /&gt; Now use&lt;br /&gt; &lt;br /&gt; Pv=RT[/itex]&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; \beta=\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P=\frac{1}{v}\,\frac{\partial }{\partial T}\left(\frac{RT}{P}\right)\right)_P[/itex]&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; and so on.