Question

A perfect gas, enclosed by an insulated (upright) cylinder and piston, is at equilibrium at conditions p1, t1, v1. A weight w is placed on the piston. After number of oscillations, the motion subsides and the gas reaches a new equilibrium at conditions p2, t2, v2. Find the temperature ratio t2/t1 in terms of the pressure ratio = p2/p1. Show that the change of entropy is given by

Answer 1

perfect gas enclosed by an insulated, upright cylinder with a weightless piston on top is at equilibrium conditions p1, V1, and T1. A weight is placed on the piston. After a number of oscillations, the motion subsides and the gas reaches a new equilibrium at conditions p2, V2, and T2. Find the temperature ratio T2/ T1 in terms of the pressure ratio   p2/ p1.

a. Show that the change in entropy is given by

b. Show that, if the initial disturbance is small (such that  = 1 + , where  << 1), then

c. Using isentropic relations, show that the frequency of small oscillations about the equilibrium position is

d. Justify the isentropic assumption in part c using your result from part b. How does the result in part c compare with the frequency of a pendulum?

where L is the height of the cylinder and W is the weight of the mass on the piston.