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One mole of an ideal gas expands adiabatically from an initial state $\left(T_A, V_0\right)$ to final state $\left(T_f, 5 V_0\right)$. Another mole of the same gas expands isothermally from a different initial state ( $T_{\mathrm{B}}, \mathrm{V}_0$ ) to the same final state $\left(T_{\mathrm{f}}, 5 V_0\right)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$ ?
$5^{\gamma-1}$
$5^{1-\gamma}$
$5^\gamma$
$5^{-1+\gamma}$
Solution
$\mathrm{T}_{\mathrm{A}} \mathrm{V}_0^{\gamma-1}=\mathrm{T}_{\mathrm{f}}\left(5 \mathrm{~V}_0\right)^{\gamma-1}$
$\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{f}}}=5^{\gamma-1}=\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}$
