A thin metal cylinder contains an ideal gas. The cylinder has a cross sectional

Question

A thin metal cylinder contains an ideal gas. The cylinder has a cross sectional area of 100 cm^2. The surroundings outside the cylinder are maintained at 25 degree C and 101 kPa. A piston caps one side of the cylinder. The piston is attached to a spring and is also exposed to the atmosphere. Initially, the ideal gas has a pressure of 300 kPa and volume of 1000 cm^3. A catch holds the piston so that the spring sits in its resting position. The catch is removed. There is a lot of friction between the cylinder wall and the piston head. The gas slowly expands, eventually compressing the spring 5 cm. What is the final pressure of the ideal gas What is the spring constant and how much work did the ideal gas do to compress the spring How much work did the ideal gas do to move the atmosphere out of the way How much of the work was dissipated via friction?

Explanation / Answer

(a) Final Volume = 1000 + 100 * 5   = 1500 cm^3

      Final Pressure   = 1000 * 300 / 1500   = 200 kPa

b) Final Pressure = Po + k*x /A

    200 = 101 + k * 0.05 / 100 * 10^-4

    99 * 10^3 = k * 5

    k = 19.8 kN/m

    Work = 0.5 * k * x^2

               = 0.5 * 19800 * 5 * 10^-4    = 4.95 J

c) Work done on atmosphere = Po * A * x

                                             = 101 * 10^3 * 100 * 10^-4 * 0.05    = 50.5 J

d) Work done by gas in expanding

      W =   P1 * V1 * ln(V2/V1 )    = 300 * ln(1500 / 1000) = 121.64 J

      Work on friction   = W - Wspring - Watmosphere

                                = 121.64 - 4.95 - 50.5    = 66.19 J

                     

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