Question

A diatomic is expanded according to ${PV}^3=$ constant under very high temperature (Assume vibration mode active). Calculate the molar heat capacity of gas (in cal/mol K) in this process:

Answer 1

Molar heat capacity ($C_p$) is the amount of heat required to change the temperature of 1 mol of the substance by unit degree.

$C_p=\frac{dH}{dT}$

and $dH=q=dU-w$

Also, $\Rightarrow \frac{dH}{dT}=\frac{dU}{dT}-\frac{w}{dT}$

or, $C_P=C_V-\frac{w}{dT}$

For 1 mole of diatomic gas at high temperature

$C_V=\frac{dU}{dT}=\frac{7}{2}R$

and $dw=-PdV$

Given, $P{V}^3=k$

$\therefore dw=-k.V^{-3}dV$

$w=-k{\int }_{V_1}^{V_2}{V}^{-3}dV=\frac{1}{2}k(V_2^{-2}-V_1^{-2})$

$w=\frac{1}{2}(P{V}_2-P{V}_1)=\frac{1}{2}RdT$

On substitution, $C_P=C_V-\frac{1}{2}R=\frac{7}{2}R-\frac{1}{2}R=3R$

Answer is: $C_P=5.96calmo{l}^{-1}K$