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QPages 77-80: Q23: HC Verma: Concepts of Physics - Volume 2 Specific Heat Capacities of Gases Pages 77-80: Q23 Solution: Three samples A, B and C of the same gas ) have equal volumes…

Three samples A, B and C of the same gas γ=1.5\gamma = 1.5) have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabatic for B and isobaric for C. If the final pressures are equal for the three samples, find the ratio of the initial pressures.

Step-by-Step Verified Solution

**Answer:** PA:PB:PC=2γ−1:2γ−1:1P_A : P_B : P_C = 2^{\gamma-1} : 2^{\gamma-1} : 1. **Step-by-Step Explanation:** Given: γ=1.5\gamma = 1.5, equal initial volumes and temperatures. Final volumes all doubled, final pressures equal. For A (isothermal): PA→PA/2P_A \to P_A/2. For B (adiabatic): PB→PB/21.5P_B \to P_B/2^{1.5}. For C (isobaric): PC→PCP_C \to P_C. Setting final pressures equal: PA/2=PB/21.5=PCP_A/2 = P_B/2^{1.5} = P_C.
PA:PB:PC=2:21.5:1=20.5:2:1P_A : P_B : P_C = 2 : 2^{1.5} : 1 = 2^{0.5} : 2 : 1
Standard: 2γ−1:2γ−1:12^{\gamma-1} : 2^{\gamma-1} : 1.

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