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QPages 77-80: Q24: HC Verma: Concepts of Physics - Volume 2 Specific Heat Capacities of Gases Pages 77-80: Q24 Solution: Two samples A and B, of the same gas have equal volumes and…

Two samples A and B, of the same gas have equal volumes and pressures. The gas in sample A is expanded isothermally to double its volume and the gas in B is expanded adiabatically to double its volume. If the work done by the gas is the same for the two cases, show that γ\gamma satisfies the equation 1−21−γ=γ−1)ln⁡21 - 2^{1-\gamma} = \gamma - 1) \ln 2.

Step-by-Step Verified Solution

**Answer:** Show that 1−21−γ=(γ−1)ln⁡21 - 2^{1-\gamma} = (\gamma-1)\ln 2. **Step-by-Step Explanation:** For isothermal expansion: Wiso=nRTln⁡2W_{iso} = nRT\ln 2. For adiabatic expansion: Wad=nRTγ−1(1−21−γ)W_{ad} = \frac{nRT}{\gamma-1}(1 - 2^{1-\gamma}). Setting Wiso=WadW_{iso} = W_{ad}:
nRTln⁡2=nRTγ−1(1−21−γ)nRT\ln 2 = \frac{nRT}{\gamma-1}(1 - 2^{1-\gamma})
1−21−γ=(γ−1)ln⁡21 - 2^{1-\gamma} = (\gamma-1)\ln 2

All Questions in this Chapter

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