A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness. The piston is free to move along the length of the cylinder. The length of the cylinder above the piston is $$\ell $$₁, and that below the piston is $$\ell $$₂, such that $$\ell $$₁ > $$\ell $$₂. Each part of the cylinder contains n moles of an ideal gas at equal temperature T. If the piston is stationary, its mass, m, will be given by :
(R is universal gas constant and g is the acceleration due to gravity)

An ideal gas occupies a volume of 2m³ at a pressure of 3 $$ \times $$ 10⁶ Pa. The energy of the gas is :

A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K₁ and the of the outer cylinder is K₂. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is :

For the given cyclic process CAB as shown for a gas, the work done is :