Two ideal gases A and B are contained in separate horizontal cylinders fitted with movable pistons. The ratio of the pressures of the gases is P_A : P_B = 2 : 1 , and the ratio of the radii of their pistons is r_A : r_B = 1 : 2 . When an equal amount of heat is supplied to both systems, their internal energies increase by the same amount. What is the ratio of the displacements of the pistons, x_A : x_B ?

Step-by-step solution

According to the first law of thermodynamics, Q = U + W . Given that Q_A = Q_B and U_A = U_B , the work done by both gases must be equal: W_A = W_B The work done by a gas expanding at constant pressure is W = P V . For a cylinder with piston radius r and displacement x , the change in volume is V = r^2 x . Thus, P_A ( r_A^2 x_A) = P_B ( r_B^2 x_B) Rearranging for the ratio of displacements: x_A x_B = P_B P_A ( r_B r_A )^2 Substituting the given ratios P_A P_B = 2 1 and r_A r_B = 1 2 : x_A x_B = ( 1 2 ) ( 2 1 )^2 =