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Question

(A) The cycle consists of four reversible processes: Process $1$: Isothermal expansion at $T_1$ from $(P_1, V_1)$ to $(P_2, V_2)$. Work $w_1 = -nRT_1

Vedclass · Accepted Answer

$w_1 + w_3 = -2T_1 \ln \frac{V_2}{V_1} - 2T_2 \ln \frac{V_4}{V_3}$

Vedclass · Answer

(A) The cycle consists of four reversible processes: Process $1$: Isothermal expansion at $T_1$ from $(P_1, V_1)$ to $(P_2, V_2)$. Work $w_1 = -nRT_1 \ln(V_2/V_1) = -2T_1 \ln(V_2/V_1)$. Process $2$: Adiabatic expansion from $(P_2, V_2, T_1)$ to $(P_3, V_3, T_2)$. Work $w_2 = \Delta U_2 = nC_v(T_2 - T_1)$. Process $3$: Isothermal compression at $T_2$ from $(P_3, V_3)$ to $(P_4, V_4)$. Work $w_3 = -nRT_2 \ln(V_4/V_3) = -2T_2 \ln(V_4/V_3)$. Process $4$: Adiabatic compression from $(P_4, V_4, T_2)$ to $(P_1, V_1, T_1)$. Work $w_4 = \Delta U_4 = nC_v(T_1 - T_2) = -\Delta U_2$. Summing the work for the isothermal processes $1$ and $3$: $w_1 + w_3 = -2T_1 \ln(V_2/V_1) - 2T_2 \ln(V_4/V_3)$. Thus, option $A$ is correct.

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