A Carnot cycle is run on molecules of nitrogen

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · The laws of thermodynamics · The laws of thermodynamics

Thermal Physics, Ch Heat engines, refrigerators and the Carnot cycle

Problem

A Carnot cycle is run on N=1.50×1023N = 1.50\times10^{23} molecules of nitrogen, treated as an ideal gas with constant heat capacities and γ=7/5\gamma = 7/5. The cycle consists of an isothermal expansion at Th=600 KT_h = 600\,\mathrm{K} from V1=2.00 LV_1 = 2.00\,\mathrm{L} to V2=5.00 LV_2 = 5.00\,\mathrm{L}, a quasistatic adiabatic expansion to V3V_3 at Tc=300 KT_c = 300\,\mathrm{K}, an isothermal compression at TcT_c to V4V_4, and a quasistatic adiabatic compression back to state 1. Every leg is quasistatic and frictionless, and heat crosses only between bodies at the same temperature. Take kB=1.381×10−23 J K−1k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}, and report heats and work as magnitudes. (a) Using TVγ−1=constantTV^{\gamma-1} = \text{constant} on the two adiabats, show that V3/V4=V2/V1V_3/V_4 = V_2/V_1, and hence that the efficiency of the cycle does not contain γ\gamma at all. (b) Find V3V_3 and V4V_4. (c) Find QhQ_h, QcQ_c, the net work delivered per cycle and the efficiency, and confirm Qh/Qc=Th/TcQ_h/Q_c = T_h/T_c.

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