Question

A sample of ideal gas is taken through the cyclic process shown in the figure. The temperature of the gas at state A is $T_A = 200 K$. At states B and C the temperature of the gas is the same.

• A. 2$P_AV_A$
• B. 4$P_AV_A$
• C. 6$P_AV_A$
• D. 8$P_AV_A$

Answer 1

Answer: (A)

$2P_AV_A$

Answer 2

The net work done by the gas in a cyclic process is the area enclosed by the cycle in the P-V diagram. The process ABC is a triangle. The net work done is the area of this triangle. Area $= \frac{1}{2} \times \text{base} \times \text{height}$. From the P-V diagram, state A is at $(V_A, P_A)$, state B is at $(V_A, P_0)$, and state C is at $(3V_A, P_0)$. The base of the triangle can be taken along the isobaric line BC, with length $V_C - V_B = 3V_A - V_A = 2V_A$. The height of the triangle is the difference in pressure $P_0 - P_A$. Therefore, the net work done $W_{net} = \frac{1}{2} \times (2V_A) \times (P_0 - P_A) = V_A (P_0 - P_A)$.

The problem states that $T_B = T_C$. Using the ideal gas law $PV = nRT$, we have $\frac{P_B V_B}{T_B} = \frac{P_C V_C}{T_C}$. Since $T_B = T_C$, we get $P_B V_B = P_C V_C$. From the diagram, $V_B = V_A$, $P_B = P_0$, $V_C = 3V_A$, $P_C = P_0$. Substituting these values, we get $P_0 V_A = P_0 (3V_A)$, which implies $V_A = 3V_A$, leading to $V_A = 0$ (or $P_0=0$). This indicates an inconsistency in the problem statement or the diagram.

However, if we assume that the intended relationship between pressures and volumes, along with the options provided, implies a specific ratio, we can work backwards. If we assume $P_0 = 3P_A$, then the net work done becomes: $W_{net} = V_A (3P_A - P_A) = V_A (2P_A) = 2P_A V_A$. This matches option (A). Let's check if this assumption is consistent with the condition $T_B=T_C$. If $P_0 = 3P_A$, then $T_B = \frac{P_0 V_A}{nR} = \frac{3P_A V_A}{nR}$ and $T_C = \frac{P_0 (3V_A)}{nR} = \frac{3P_A (3V_A)}{nR} = \frac{9P_A V_A}{nR}$. For $T_B = T_C$, we would need $3P_A V_A = 9P_A V_A$, which implies $P_A=0$ or $V_A=0$. This confirms the inconsistency.

Despite the inconsistency, given the multiple-choice options, the most plausible intended answer is derived by assuming a relationship that leads to one of the options. The assumption $P_0 = 3P_A$ yields $W_{net} = 2P_A V_A$.

The condition $T_A = 200 K$ is not used for calculating the net work done, which is independent of temperature for a given cycle in the P-V diagram.