An ideal gas undergoes a process where its pressure and volume vary...

The equation of the process is given by $p = kV$, which can be written as $pV^{-1} = \text{constant}$.

This represents a polytropic process of the form $pV^n = \text{constant}$, where $n = -1$.

The molar heat capacity for a polytropic process is given by $C = C_V + \dfrac{R}{1-n}$.

Substituting $n = -1$, we get:

$C = C_V + \dfrac{R}{1 - (-1)}$

$C = C_V + \dfrac{R}{2}$

Alternatively, using the ideal gas law $pV = RT$ (for one mole)$and p = kV$, we get $kV^2 = RT$.

Differentiating both sides, $2kV dV = R dT$, which gives $2p dV = R dT$, so $p dV = \dfrac{R dT}{2}$.

From the first law of thermodynamics, $dQ = dU + dW$.

$C dT = C_V dT + p dV$

$C dT = C_V dT + \dfrac{R dT}{2}$

$C = C_V + \dfrac{R}{2}$