Problem #4 A 200 g piston can move frictionlessly in a horizontal cylinder with

Question

Problem #4 A 200 g piston can move frictionlessly in a horizontal cylinder with cross-sectional area 4.0 cm and length 20 cm. One end of the cylinder is open, and the other is closed. n moles of air are enclosed between the piston and the closed end of the cylinder. (Assume that there is no way for the air to leak around the piston.) Initially, the piston is 10 cm from the closed end of the cylinder, and all air (both inside and outside of the enclosure) is at atmospheric pressure and T = 300 K. (a) How many moles of air are enclosed by the piston? (b) The piston is held in place, and heat Q is added to the air enclosed air, raising its temperature to 1000 K. Find O. (The temperature and pressure of the air to the right of the piston do not change.) (c) The piston is released, allowing the enclosed air to expand adiabatically. How much work does the enclosed air do on the piston as it pushes the piston to the open end of the cylinder? (d) How much work does the air to the right of the piston do to the piston as the piston is pushed to the open end of the cylinder? (e) What is the speed of the piston as it is pushed outside the cylinder? piston 20 cm 10 cm

Explanation / Answer

(a)

from ideal gas equation

P*V = n*R*T

n = P*V/(R*T)

P = pressure == 1 atm = 1.013*10^5 Pa

V = A*l = 4*10^-4*0.2 = 8*10^-5 m^3

T = 300 K

n = 1.013*10^5*8*10^-5/(8.314*300)

n = 0.00325 moles

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b)

at constant volume

Q = n*Cv*dT

Q = n*R*dT/(gamma-1)

Q = 0.00325*8.314*(1000-300)/(1.4-1) = 42.3 J

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c)

W = Q = 42.3 J

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work done by air outside the piston = -42.3 J

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(e)

from work energy relation

work done by gas = chang in KE of piston

W = (1/2)*m*(vf^2-vi^2)

42.3 = (1/2)*0.2*(v^2-0)

v = 20.6 m/s

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