How to find the thermal compressibility of a gas

[PKM]

Homework Statement

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A gas obeying the equation of state $PV=RT$ undergoes a hypothetical reversible process $$PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1$$ Can we prove that the thermal compressibility of the gas undergoing this process tends to a $constant$ value at very high temperature? Here, $E_0$ and $c_1$ are constants with dimensions.

Homework Equations

The thermal compressibility of a gas is given as $$\kappa = \frac{-1}{V} \frac{\delta V}{\delta P}$$

The Attempt at a Solution

First I tried to find the thermal compressibility using the above differential equation, considering the reversible process given. I made use of the equation of state $PV = RT$, to substitute $RT$ for $PV$. My result contains a term P, which cannot be cancelled, or substituted to yield a constant value, at very high temperatures.
Any solution, or comment?

 

[Chestermiller]

What did you get for $\kappa$? I'll tell you if I confirm.

 

[PKM]

What did you get for $\kappa$? I'll tell you if I confirm.

I proceeded in two ways:
First, in the equation $PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1$, I put $PV=RT$. Then partially differentiated it w.r.t. $P$. It yields $$\frac{1}{\sqrt[3] {V}}\frac{dV}{dP} = const$$
Secondly, I differentiated the equation
$PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1$ w.r.t. $P$. The result appears to be $$-\frac{1}{V}\frac{\delta V}{\delta P} = \kappa = \frac{1-E_0}{P}+\frac{5}{3V}$$
Maybe somewhere I've gone wrong (it would be much kind of you if you point it out); but if my calculations are correct, how may I proceed?

 

[Chestermiller]

I proceeded in two ways:
First, in the equation $PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1$, I put $PV=RT$. Then partially differentiated it w.r.t. $P$. It yields $$\frac{1}{\sqrt[3] {V}}\frac{dV}{dP} = const$$
Secondly, I differentiated the equation
$PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1$ w.r.t. $P$. The result appears to be $$-\frac{1}{V}\frac{\delta V}{\delta P} = \kappa = \frac{1-E_0}{P}+\frac{5}{3V}$$
Maybe somewhere I've gone wrong (it would be much kind of you if you point it out); but if my calculations are correct, how may I proceed?

I get $$\kappa=\frac{1}{P}\frac{\left(\frac{PV}{E_0}-1\right)}{\left(\frac{PV}{E_0}-\frac{5}{3}\right)}$$At high temperatures, this approaches $\kappa=1/P$.