The diatomic gas has five degrees of freedom i.e., f = 5. Thus, the internal energy per mole, specific heat at constant volume, specific heat at constant pressure, and the ratio of specific heats for a diatomic gas, are given by

$$U = (f/2)RT = (5/2)RT$$, $${C_v} = dU/dT = 5R/2$$, $${C_p} = {C_v} + R = 7R/2$$, $$\gamma = {C_p}/{C_v} = 7/5$$.

For an adiabatic process, $$T{V^{\gamma - 1}}$$ = constant

$${T_i}V_i^{\gamma - 1} = {T_f}V_f^{\gamma - 1}$$

Substituting the given values, we get

$${T_i}V_i^{\gamma - 1} = a{T_i}{\left( {{{{V_i}} \over {32}}} \right)^{\gamma - 1}} \Rightarrow a = {32^{\gamma - 1}}$$

For diatomic gas, $$\gamma = {7 \over 5}$$

$$a = {32^{{7 \over 5} - 1}} = {32^{2/5}} = {2^2} = 4$$