The pressure and density of a diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes adiabatically from…
The pressure and density of a diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes adiabatically from $(\mathrm{P}, \rho)$ to $\left(\mathrm{P}^{\prime}, \rho^{\prime}\right)$. If $\frac{\rho^{\prime}}{\rho}=32$ then $\frac{\mathrm{P}^{\prime}}{\mathrm{P}}$ should be
$\frac {1}{128}$
$128$
$32$
$64$
Solution
For adiabatic process, $\mathrm{PV}^\gamma=\mathrm{a}$ constant
$\therefore \quad \frac{\mathrm{P}^{\prime}}{\mathrm{P}}=\left(\frac{\mathrm{V}^{\prime}}{\mathrm{V}}\right)^\gamma=\left(\frac{\rho^{\prime}}{\rho}\right)^\gamma$
Given that, $\frac{\rho^{\prime}}{\rho}=32$ and $\gamma=\frac{7}{5}$
$\therefore \quad \frac{\mathrm{P}^{\prime}}{\mathrm{P}}=(32)^{\frac{7}{5}}=128$