Two vertical cylinders are placed in a vacuum chamber and contain ideal gases. They are fitted with heavy, frictionless movable pistons. The piston of cylinder A has mass m and radius r , while the piston of cylinder B has mass 2m and radius 2r . Equal amounts of heat are supplied to both gases, and they undergo an expansion such that the change in their internal energies is the same. The ratio of the upward displace

Step-by-step solution

From the First Law of Thermodynamics, Q = U + W . Since Q_A = Q_B and U_A = U_B , the work done by both gases must be equal ( W_A = W_B ). In a vacuum chamber, the external atmospheric pressure is zero. The pressure of the gas is purely due to the weight of the piston: P = mg r^2 . The work done by the gas during an upward displacement x is W = P V = ( mg r^2 ) ( r^2 x) = mgx . Notice that the work done is simply the change in gravitational potential energy of the piston, which is independent of the piston's radius