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(A) For an adiabatic process, the relationship between temperature $T$ and volume $V$ is given by $TV^{\gamma - 1} = 	ext{constant}$.Given that the t

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$PV^{3/2} = 	ext{constant}$

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(A) For an adiabatic process, the relationship between temperature $T$ and volume $V$ is given by $TV^{\gamma - 1} = 	ext{constant}$.Given that the temperature increases to $\sqrt{2}$ times, we have $T_2 = \sqrt{2}T_1$.The volume is compressed to half, so $V_2 = V_1 / 2$.Using the relation $\frac{T_1}{T_2} = \left( \frac{V_2}{V_1} ight)^{\gamma - 1}$, we substitute the values:$\frac{T_1}{\sqrt{2}T_1} = \left( \frac{V_1/2}{V_1} ight)^{\gamma - 1}$$\frac{1}{\sqrt{2}} = \left( \frac{1}{2} ight)^{\gamma - 1}$Since $\frac{1}{\sqrt{2}} = (1/2)^{1/2}$, we equate the exponents:$\gamma - 1 = 1/2 \implies \gamma = 3/2$.The adiabatic equation is $PV^{\gamma} = 	ext{constant}$, so $PV^{3/2} = 	ext{constant}$.

The temperature of a hypothetical gas increases to $\sqrt{2}$ times when compressed adiabatically to half the volume. Its equation can be written as

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