The Atmospheric Processing Station · AI Generator #GEN-9C911F9B

Physics · thermodynamics

Problem

A Mars atmospheric processing station uses a novel thermodynamic cycle to extract water vapor from the thin Martian atmosphere. The station operates with a working gas that behaves according to the van der Waals equation of state: (P+aV2)(V−b)=RT\left(P + \frac{a}{V^2}\right)(V - b) = RT where a=0.364  Pa⋅m6⋅mol−2a = 0.364\;\text{Pa}\cdot\text{m}^6\cdot\text{mol}^{-2} and b=4.27×10−5  m3⋅mol−1b = 4.27 \times 10^{-5}\;\text{m}^3\cdot\text{mol}^{-1}. The cycle consists of four processes operating on n=2.0  moln = 2.0\;\text{mol} of working gas: 1. Isothermal compression at T1=220  KT_1 = 220\;\text{K} from volume V1=0.050  m3V_1 = 0.050\;\text{m}^3 to V2=0.020  m3V_2 = 0.020\;\text{m}^3 2. Isochoric heating from T1=220  KT_1 = 220\;\text{K} to T3=300  KT_3 = 300\;\text{K} 3. Isothermal expansion at T3=300  KT_3 = 300\;\text{K} from V3=V2V_3 = V_2 to V4=V1V_4 = V_1 4. Isochoric cooling back to the initial state The working gas has a constant volume heat capacity CV=52RC_V = \frac{5}{2}R (where R=8.314  J⋅mol−1⋅K−1R = 8.314\;\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}). (a) Calculate the work done by the gas during the isothermal expansion process (process 3). (b) Determine the efficiency of this thermodynamic cycle. (c) How does this efficiency compare to the theoretical maximum efficiency for a heat engine operating between the same temperature reservoirs?

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