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A vertical cylinder contains an ideal monoatomic gas trapped by a frictionless, heavy piston of mass 200 ~kg and cross-sectional area 0.02 ~m ^2 . The piston is in equilibrium at a height of 0.5 ~m above the base of the cylinder. If the atmospheric pressure is 10^5 ~Pa and the acceleration due to gravity is 10 ~m/s ^2 , the internal energy of the trapped gas is _______ kJ .

Correct answer

3

Step-by-step solution

The gas is in mechanical equilibrium with the piston and the atmosphere. The absolute pressure P of the gas is the sum of atmospheric pressure P₀ and the pressure exerted by the weight of the piston. P = P₀ + Mg A P = 10^5 + 200 10 0.02 P = 10^5 + 10^5 = 2 10^5 ~Pa The volume V of the gas is the area of the base times the height: V = A h = 0.02 0.5 = 0.01 ~m ^3 For a monoatomic gas, the degree of freedom is f = 3 . The internal energy U is given by: U = f 2 PV = 3 2 PV Substituting the values of P and V : U = 3 2 (

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