Question

An ideal gas is taken through a process in which the pressure and the volume are chnaged according to the equation P = kV. Show that the molar heat capacity of the gas for the process is given by $C=C_v+\frac{R}{2}$.

Answer 1

P = KV

$\Rightarrow \frac{nRT}{V}=KV\Rightarrow RT=K{V}^2$

$\Rightarrow R\Delta T=2KV\Delta V$

$\Rightarrow \frac{R\Delta V}{2KV}=\Delta V$

dQ = dU +dW

$\Rightarrow msdT=C_{v}dT+\frac{PRdT}{2KV}$

$\Rightarrow ms=C_v+\frac{RKV}{2KV}$

$\therefore c_p=C_v+\frac{R}{2}$