A piston contains 210 moles of an ideal monatomic gas that initally has a pressu

Question

A piston contains 210 moles of an ideal monatomic gas that initally has a pressure of 1.1 × 105 Pa and a volume of 4.2 m3. The piston is connected to a hot and cold reservoir and the gas goes through the following quasi-static cycle accepting energy from the hot reservoir and exhausting energy into the cold reservoir. The pressure of the gas is increased to 4.1 × 105 Pa while maintaining a constant volume. The volume of the gas is increased to 11.2 m3 while maintaining a constant pressure. The pressure of the gas is decreased to 1.1 × 105 Pa while maintaining a constant volume. The volume of the gas is decreased to 4.2 m3 while maintaining a constant pressure. It may help you to recall that CVCV = 12.47 J/K/mole and CPCP = 20.79 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-How much energy is transferred into the gas from the hot reservoir?

2-How much energy is transferred out of the gas into the cold reservoir?

3-How much work is done by the gas?

4-What is the efficiency of this cycle?

Explanation / Answer

number of moles,n=210 moles

initial pressure,p1= 1.1 × 105 Pa

Initial volume,v=4.2 m3

Final pressure of the gas,p2=4.1 × 105 Pa

Final Volume,v2=11.2 m3

Final pressure,p3=1.1 × 105 Pa

Final volume,v3=4.2 m3

Let us take temperature is constant and it is the room temperature=298 K

a) Energy tansfered into the gas for hot reservoir,Q=nRTln(v2/v1)=210*10973731.6 /m*298(ln(11.2 /4.2)=6.73*1011J

b) Energy tansfered into the gas,Q=nRTln(v2/v1)=210*10973731.6 /m*298(ln(4.2 /11.2)=- 6.73*1011J

c) Work done by the gas,L=Q=nRT(v2/v1)=210*10973731.6 /m*298(11.2 /4.2)=1.83*1012J

d) Carnottt's Efficiency=1-T(hot)/Tcold

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