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

[tex]\beta=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P[/itex]<br /> <br /> [tex]\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T[/itex]<br /> <br /> to acknowledge that <i>V</i> 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.<br /> <br /> Now use<br /> <br /> [tex]Pv=RT[/itex]<br /> <br /> [tex]\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]<br /> <br /> and so on.[/tex][/tex][/tex][/tex]