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

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Given,

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

Final pressure is $P_{f} = n P_{i}$

In adiabatic process, equation of state is given by,

$P V^{γ} = constant . . . (1)$

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

$P {(\frac{m}{ρ})}^{γ} = constant . . . (2)$ , As $V = \frac{m}{ρ}$

and mass is constant, then the pressure is given by

$∴ P \propto ρ^{γ}$

$\Rightarrow \frac{P_{f}}{P_{i}} = {(\frac{ρ_{f}}{ρ_{i}})}^{γ}$

$\Rightarrow \frac{n P_{i}}{P_{i}} = {(\frac{32 ρ_{i}}{ρ_{i}})}^{^{7 / 5}} = (32)^{7 / 5}$

$∴ n = 2^{7} = 128$

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