A piston contains 790 moles of an ideal monatomic gas that initally has a pressu

Question

A piston contains 790 moles of an ideal monatomic gas that initally has a pressure of 1.67 × 105 Pa and a volume of 4.5 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 4.67 × 105 Pa while maintaining a constant volume. The volume of the gas is increased to 11.5 m3 while maintaining a constant pressure. The pressure of the gas is decreased to 1.67 × 105 Pa while maintaining a constant volume. The volume of the gas is decreased to 4.5 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? 1440000 J 2) How much energy is transferred out of the gas into the cold reservoir? J 3) How much work is done by the gas? J 4) What is the efficiency of this cycle?

Explanation / Answer

given number of moles ,n = 790
initial pressure P1 = 1.67*10^5 Pa
initial Volume V1 = 4.5 m^3
using PV = nRT
T1 = P1V1/nR = 114.4724 K

step 1. Constant volume heat addition
P2 = 4.67*10^5 Pa
V2 = V1 = 4.5 m^3
so T2 = P2V2/nR = 320.1115 K
heat transferred to the gas = nCvdT = 790*12.47*(T2 - T1) = 2025812.486 J
work done by the gas = 0 J ( constant volume)

step 2: constant pressure expansion
P3 = 4.67*10^5 Pa
V3 = 11.5 m^3
T3 = P3V3/nR = 818.0627 K
so heat added to the gas = nCp(T3 - T2) = 8178400.756 J
work done by the gas = P3(V3 - V2) = 3269000 j

STEP 3: CONSTANT volume heat removal
P4 = 1.67*10^5 Pa
V4 = 11.5 m^3
so T4 = P4V4/nR = 292.540 K
heat removed = nCv(T4 - T3) = -5177075.541 J
Work done by gas = 0 J

Step 4: constant pressure heat removal
P1, V1, T1
heat removed = nCp(T1 - T4) = -2924600.0691 j
WORK DONE BY GAS = p1(VF1 - V2) = -1169000 J

net heat transferred into gas = 2025812.486 + 8178400.756 = 10204213.242 J
net heat removed from the system = 5177075.541 + 2924600.0691 = 8101675.6101 J
net work done = 3269000 -1169000 = 2100000 J
efficiency = work/heat = 0.99

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