A monoatomic gas having $ \gamma = \frac{5}{3} $ is stored in a thermally insulated container and the gas is suddenly compressed to $ \left( \frac{1}{8} \right)^{\text{th}} $ of its initial volume. The ratio of final pressure and initial pressure is:
($\gamma$ is the ratio of specific heats of the gas at constant pressure and at constant volume)
Solution
<p>To find the ratio of final pressure ($ P_f $) to initial pressure ($ P_i $), we use the adiabatic condition:</p>
<p>$ P_i V_i^\gamma = P_f V_f^\gamma $</p>
<p>We aim to express the pressure ratio:</p>
<p>$ \frac{P_f}{P_i} = \left(\frac{V_i}{V_f}\right)^\gamma $</p>
<p>Given that the final volume $ V_f $ is $ \frac{1}{8} $ of the initial volume $ V_i $, this simplifies to:</p>
<p>$ \left(\frac{V_i}{V_f}\right) = 8 $</p>
<p>Thus, the pressure ratio becomes:</p>
<p>$ \frac{P_f}{P_i} = 8^\frac{5}{3} $</p>
<p>Calculating $ 8^\frac{5}{3} $ gives:</p>
<p>$ \frac{P_f}{P_i} = 32 $</p>
About this question
Subject: Physics · Chapter: Thermodynamics · Topic: Thermodynamic Processes
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