One mole of an ideal diatomic gas undergoes a thermodynamic process...

The given process equation is $PT = \text{constant}$.

From the ideal gas equation, $T = \dfrac{PV}{nR}$. Substituting this into the process equation gives:
$P\left(\dfrac{PV}{nR}\right) = \text{constant}$
$P^2V = \text{constant}'$
$P V^{1/2} = \text{constant}''$

Comparing this with the standard polytropic process equation $PV^x = \text{constant}$, we get the polytropic index $x = \dfrac{1}{2}$.

The molar heat capacity for a polytropic process is given by:
$C = C_v + \dfrac{R}{1-x}$

For a diatomic gas, the molar heat capacity at constant volume is $C_v = \dfrac{5}{2}R$.

Substituting the values of $C_v$ and $x$:
$C = \dfrac{5}{2}R + \dfrac{R}{1 - 1/2}$
$C = \dfrac{5}{2}R + \dfrac{R}{1/2}$
$C = \dfrac{5}{2}R + 2R = \dfrac{9}{2}R$.

Answer: $\dfrac{9}{2}R$