In an adiabatic process, the density of a diatomic gas becomes 32 times of its initial value. The final pressure of the gas is found to be n times the initial pressure. The value of n is

Given that density of the gas becomes 32 times and we know that density $\rho$ is inversely proportional to volume V.

So, volume becomes $\frac{1}{32}$ times of initial volume. In adiabatic process.

${\mathrm{pV}}^{\mathrm{\gamma }}=\text{ constant }\Rightarrow {\mathrm{p}}_1{\mathrm{V}}_1^{\mathrm{\gamma }}={\mathrm{p}}_{\mathrm{f}}{\mathrm{V}}_{\mathrm{f}}^{\mathrm{\gamma }}\text{ Now },\mathrm{\gamma }=\frac{7}{5}$ for diatomic gas

$\Rightarrow \frac{{\mathrm{p}}_{\mathrm{f}}}{{\mathrm{p}}_{\mathrm{i}}}=\left(\frac{{\mathrm{V}}_{\mathrm{i}}}{{\mathrm{V}}_{\mathrm{f}}}\right)=(32{)}^{\frac{7}{5}}=128$