In an adiabatic process, the density of a diatomic gas becomes 32 n...

Given,

Final density of diatomic gas is ${ρ}_{f} = 32n{ρ}_{i}$

Final pressure is ${P}_{f} = n{P}_{i}$

In adiabatic process, equation of state is given by,

$P{V}^{\gamma} = \text{constant} . . . \left(1\right)$

Where, $V$ is the volume, $P$ is the pressure and $\gamma = \dfrac{7}{5}$ is the ratio of specific heat of diatomic gas

$P{\left(\dfrac{m}{ρ}\right)}^{\gamma} = \text{constant} . . . \left(2\right)$, As $V = \dfrac{m}{ρ}$

and mass is constant, then the pressure is given by

$∴ P ∝ {ρ}^{\gamma}$

$⇒ \dfrac{{P}_{f}}{{P}_{i}} = {\left(\dfrac{{ρ}_{f}}{{ρ}_{i}}\right)}^{\gamma}$

$⇒ \dfrac{n{P}_{i}}{{P}_{i}} = {\left(\dfrac{32{ρ}_{i}}{{ρ}_{i}}\right)}^{{}^{7 / 5}} = ( 32{)}^{7 / 5}$

$∴ n = {2}^{7} = 128$