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(B) For a polytropic process $PV^n = 	ext{constant}$, the molar heat capacity is $C = C_V + \frac{R}{1-n}$.Given $n = 3$ and $C_V = \frac{3R}{2}$ for

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$P_1 V_1 \left( \frac{V_1^2}{V_2^2} - 1 \right)$

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(B) For a polytropic process $PV^n = 	ext{constant}$, the molar heat capacity is $C = C_V + \frac{R}{1-n}$.Given $n = 3$ and $C_V = \frac{3R}{2}$ for a monoatomic gas.Thus, $C = \frac{3R}{2} + \frac{R}{1-3} = \frac{3R}{2} - \frac{R}{2} = R$.From the ideal gas law $PV = RT$ (for $1$ mole), we have $T = \frac{PV}{R}$.Initial state: $T_1 = \frac{P_1 V_1}{R}$.Final state: Since $P_1 V_1^3 = P_2 V_2^3$, we have $P_2 = P_1 \left( \frac{V_1}{V_2} ight)^3$.$T_2 = \frac{P_2 V_2}{R} = \frac{P_1 V_1^3}{R V_2^2}$.The heat absorbed is $Q = n C \Delta T = 1 \cdot R \cdot (T_2 - T_1)$.$Q = R \left( \frac{P_1 V_1^3}{R V_2^2} - \frac{P_1 V_1}{R} ight) = P_1 V_1 \left( \frac{V_1^2}{V_2^2} - 1 ight)$.

One mole of an ideal monoatomic gas expands along the polytropic process $PV^3 = \text{constant}$ from volume $V_1$ to $V_2$. The molar specific heat capacity for this process is given by $C = C_V + \frac{R}{1-n}$. The total heat absorbed during the process can be expressed as:

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