Question

An ideal gas is taken through a process in which the pressure and the volume are changed according to the equation p = kV. Show that the molar heat capacity of the gas for the process is given by C = Cv + R2.

Answer 1

Explanation:

Step 1:

The ratio of pressure to volume of a gas is P = kV.

Ideal equation for gas is PV = nRT.

$\begin{aligned}&n R T V=k V |\\&n R T=k V^{2}\end{aligned}$

Step 2:

To make things simpler, take the number of moles of a gas n = 1.

RdT = 2 kVdV

$\frac{R d T}{2 k V}=d V$

Step 3:

From the law of thermodynamics first,

dQ = dU + dW

$\mathrm{n} C_{p} \mathrm{dT}=\mathrm{C}_{\mathrm{v}} \mathrm{dT}+\mathrm{PdV}$

$n C_{p} d T=C_{v} d T+\frac{P R d T}{2 k V}$

$1 \times C_{p}=\mathrm{C}_{\mathrm{v}}+\frac{P R}{2 k V}$

$\therefore c_{p}=\mathrm{C}_{\mathrm{v}}+\frac{R}{2}$

Answer 2

${\bold{\huge{\red{\underline{\green{ANSWER}}}}}}$

Please refer the above attachment