A piston-cylinder assembly contains air as shown below. The air is cooled slowly

Question

A piston-cylinder assembly contains air as shown below. The air is cooled slowly, and the piston is free to move smoothly in the cylinder with minimal friction between the piston and the cylinder wall. The piston has an area of 0.073 m^2. The mass of air is 2.26 times 10^-3 kg, and other data are provided on the figure. Atmospheric pressure is 1 bar. (a) Using a force balance on the piston, determine the mass of the piston, in kg. (b) For the air as the system, determine the amounts of energy transfer by heat and work, each in kJ.

Explanation / Answer

2-8. Area of piston, A = 0.073 m^2

mass of air, m = 2.26*10^-3 kg

a. for ideal gas equation of the form

P = rho*RT

R is the gas constant

for air R = 287.058

from the diagram, state 1, 1/rho = v1 = 0.985 m^3/kg

so from the equation we get

P1 = 287.058T1/0.985 = 291.429T1

similiarly from second state

P2 = 287.058T2/0.8442 = 340.035T2

but P2 = P1

so, T1/T2 = 1.1667

now Cp for air = 1.005 kJ / kg K

Cv = 0.718 kJ/kg K

hence (u2 - u1)mass of air = (214.07 - 250.02)*2.26*10^-3 = mCvdT = 2.26*10^-3*0.718*(T2 - T1)

T2 - T1 = -50.06 K

so, T2(1 - 1.1667) = -50.06 K

T2 = 300.357755

T1 = 500.696 K

so P1 = P2 = P = 291.429T1 = 145917.444 Pa

Weight of piston = (P - Patm)*A = 3278.973 N

Mass of piston = weight/g = 334.248 kg

b. change in internal energy, dU = (214.07 - 250.02)*2.26*10^-3 = -81.4052 J

Heat supplied = m*Cp*dT = 2.26*10^-3*1.005*10^3*(300 - 500) = -454.26 J

Work done by the gas = heat supplied to the gas - change in internal energy = -372.8548 J

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