Suppose that 167 moles of a monatomic ideal gas is initially contained in a pist

Question

Suppose that 167 moles of a monatomic ideal gas is initially contained in a piston with a volume of 0.53 m3 at a temperature of 400 K. The piston is connected to a hot reservoir with a temperature of 1134 K and a cold reservoir with a temperature of 400 K. The gas undergoes a quasi-static Stirling cycle with the following steps: The temperature of the gas is increased to 1134 K while maintaining a constant volume. The volume of the gas is increased to 2.56 m3 while maintaining a constant temperature. The temperature of the gas is decreased to 400 K while maintaining a constant volume. The volume of the gas is decreased to 0.53 m3 while maintaining a constant temperature. It may help you to recall that CVCV = 12.47 J/K/mole for a monatomic ideal gas, and that the number of gas molecules is equal to Avagadros number (6.022 × 1023) times the number of moles of the gas. 1) What is the pressure of the gas under its initial conditions? 1047877 Pa 2) How much energy is transferred into the gas from the hot reservoir? 3527516 J 3) How much energy is transferred out of the gas into the cold reservoir? 1528670 J 4) How much work is done by the gas during this cycle? J 5) What is the efficiency of this Stirling cycle? Submit 6) What is the maximum (Carnot) efficiency of a heat engine running between these two reservoirs?

Explanation / Answer

1) For ideal gas PV = n RT
R is constant = 8.31
P = n RT/V = 167* 8.31*400/0.53 = 1047374 Pa

2) Energy transfer in constant volume process = n Cv (delta T) = 167*12.47*(1134 - 400) = 1528548
Energy transfer in constant tempreture process = nRT Ln(V2/V1) = 167*8.31*1134 * ln(2.56/0.53) = 2478446 J
Total energy transfer from hot reservoir Q1 = 4006994 J

3) Energy tranfer from gas to clod reservois is also, two step process.
Energy transfer in constant volume process = n Cv dletaT 167*12.47*(1134 - 400) = 1528548
Energy transfer in constant tempreture process = nRT Ln(V1/V2) = 167*8.31*400* ln(2.56/0.53) = 874231
Total energy transfer to cold reservoir Q2 = 2402779 J

4) In cyclic process work done by gas = Q(in) - Q(out) = 4006994 - 2402779 = 1604215

5 ) Efficiency of cycle = 100 W/Q1 = 1604215 *100/ 4006994 = 40%

6) Carnot engine efficiency = 100 ( Th - Tc )/Th = 100 (1134 -400 )/ 1134 = 64.7%

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