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In a P-V diagram, AB and CD are isothermal processes at temperatures T1 and T2 respectively, while BC and DA are adiabatic curves. The cyclic process ABCDA is carried out with 2 moles of an ideal diatomic gas. Which of the following relations among the volumes at points A, B, C, and D does NOT hold?

1. V_A * V_B = V_C * V_D
2. V_A * V_C = V_B * V_D
3. V_A * V_D = V_B * V_C
4. V_A * V_C² = V_B * V_D²

Correct answer: V_A * V_B = V_C * V_D

Solution

Adiabatic BC: T1 * V_B^(gamma-1) = T2 * V_C^(gamma-1) → (V_B/V_C)^(gamma-1) = T2/T1. Adiabatic DA: T2 * V_D^(gamma-1) = T1 * V_A^(gamma-1) → (V_D/V_A)^(gamma-1) = T1/T2. Multiply: (V_B*V_D/(V_C*V_A))^(gamma-1) = 1 → V_B*V_D = V_A*V_C. This confirms option B holds. Option C: V_A*V_D = V_B*V_C → V_A/V_B = V_C/V_D. From V_B/V_A = V_C/V_D (from above): V_A*V_C = V_B*V_D means V_A/V_D = V_B/V_C ≠ V_A*V_D = V_B*V_C in general. Let's verify C: we have V_A/V_D = V_B/V_C (ratio from adiabatic). Cross multiply: V_A*V_C = V_B*V_D (which is option B). Option C says V_A*V_D = V_B*V_C, which requires V_A/V_B = V_C/V_D. This is actually a different relation and is also true if the cycle is a Carnot cycle: V_A/V_B = T ratio... Actually V_A*V_C = V_B*V_D is the correct relation. B holds. C may not hold. A says V_A*V_B = V_C*V_D. There's no direct derivation giving this. Option A does NOT necessarily hold — it is the incorrect relation.

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