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(D) When a compression process is performed suddenly or instantly,there is not enough time for heat transfer; therefore,the process is an adiabatic co

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$8:1$

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(D) When a compression process is performed suddenly or instantly,there is not enough time for heat transfer; therefore,the process is an adiabatic compression process.For an adiabatic process,the relation between pressure and volume is given by $P V^\gamma = 	ext{constant}$.Thus,$P_i V_i^\gamma = P_f V_f^\gamma$.Rearranging for the ratio of final pressure to initial pressure: $\frac{P_f}{P_i} = \left( \frac{V_i}{V_f} ight)^\gamma$.Given $V_f = \frac{V_i}{4}$ and $\gamma = 1.5 = \frac{3}{2}$,we substitute these values:$\frac{P_f}{P_i} = \left( \frac{V_i}{V_i / 4} ight)^{1.5} = (4)^{1.5} = (4)^{3/2} = (2^2)^{3/2} = 2^3 = 8$.Therefore,the ratio of final pressure to initial pressure is $8:1$.

$A$ gas $(\gamma = 1.5)$ is suddenly compressed to $(1/4)^{th}$ of its initial volume. Find the ratio of its final pressure to its initial pressure.

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