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(C) For an adiabatic process, the relationship between temperature $T$ and volume $V$ is given by $TV^{\gamma-1} = 	ext{constant}$.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 $TV^{\gamma-1} = 	ext{constant}$.Let the initial temperature be $T_1 = T$ and the initial volume be $V_1 = V$.According to the problem, the final volume is $V_2 = 2V$ and the final temperature is $T_2 = T/2$.Substituting these values into the adiabatic relation:$T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$$T \cdot V^{\gamma-1} = (T/2) \cdot (2V)^{\gamma-1}$Dividing both sides by $T \cdot V^{\gamma-1}$:$1 = (1/2) \cdot 2^{\gamma-1}$$2 = 2^{\gamma-1}$Since the bases are equal, the exponents must be equal:$1 = \gamma - 1$$\gamma = 2$.

When a gas expands adiabatically, its volume is doubled while its absolute temperature is decreased by a factor of $2$. The value of the adiabatic constant $\gamma$ is

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