The pressure and density of a given mass of a | vedclass

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(D) For an adiabatic process,the relationship between pressure $P$ and density $\rho$ is given by $P \propto \rho^{\gamma}$.Therefore,we can write the

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$128$

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(D) For an adiabatic process,the relationship between pressure $P$ and density $\rho$ is given by $P \propto \rho^{\gamma}$.Therefore,we can write the ratio as $\frac{P^{\prime}}{P} = \left( \frac{d^{\prime}}{d} \right)^{\gamma}$.Given $\frac{d^{\prime}}{d} = 32$ and $\gamma = \frac{7}{5}$.Substituting these values,we get $\frac{P^{\prime}}{P} = (32)^{7/5}$.Since $32 = 2^5$,we have $\frac{P^{\prime}}{P} = (2^5)^{7/5} = 2^7$.Calculating $2^7$,we get $2^7 = 128$.Thus,the ratio $\frac{P^{\prime}}{P} = 128$.

The pressure and density of a given mass of a diatomic gas $\left(\gamma = \frac{7}{5}\right)$ change adiabatically from $(P, d)$ to $(P^{\prime}, d^{\prime})$. If $\frac{d^{\prime}}{d} = 32$,then $\frac{P^{\prime}}{P}$ is $(\gamma = \text{ratio of specific heats})$

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