The pressure and density of a diatomic gas ($\gamma$ = 7/5) change from (P1, d1) to (P2, d2) adiabat

Question

The pressure and density of a diatomic gas ($\gamma$ = 7/5) change from (P1, d1) to (P2, d2) adiabatically. If d2/d1 = 32, then what is the value of P2/P1?

Accepted Answer

$P_{1}V_{1}^{\gamma} = P_{2}V_{2}^{\gamma}$

$\Rightarrow \frac{P_{1}}{d_{1}^{\gamma}} = \frac{P_{2}}{d_{2}^{\gamma}} $$$\Rightarrow \frac{P_{2}}{P_{1}} = \left( \frac{d_{2}}{d_{1}} \right)^{\gamma} = (32)^{7/5} = 128$$

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