A given ideal gas with gamma = frac{C_p}{C_v} | vedclass

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(C) For an adiabatic process,the relationship between temperature $T$ and volume $V$ is given by $T V^{\gamma-1} = 	ext{constant}$.Let the initial te

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$2$

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(C) For an adiabatic process,the relationship between temperature $T$ and volume $V$ is given by $T V^{\gamma-1} = 	ext{constant}$.Let the initial temperature be $T_1 = T$ and initial volume be $V_1 = V$.The final volume is $V_2 = \frac{V}{4}$ and the adiabatic index is $\gamma = 1.5$.Using the adiabatic relation:$T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$Substitute the given values:$T (V)^{1.5-1} = T_2 (\frac{V}{4})^{1.5-1}$$T (V)^{0.5} = T_2 (\frac{V}{4})^{0.5}$Divide both sides by $V^{0.5}$:$T = T_2 (\frac{1}{4})^{0.5}$$T = T_2 (\frac{1}{2})$Therefore,the final temperature $T_2 = 2T$.

$A$ given ideal gas with $\gamma = \frac{C_p}{C_v} = 1.5$ is at a temperature $T$. If the gas is compressed adiabatically to one-fourth of its initial volume,the final temperature will be ..... $T$.

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