An ideal gas is confined within a thermally isolated cylinder. It consists of N atoms initially at a pressure of p_0. A movable piston seals the right end of the cylinder. A given amount of heat Q is slowly added to the gas, while the piston allows the gas to expand in such a way that the gas's temperature remains constant at T_0. Is the internal energy of the gas the same before and after Q is added? Is this true since it has to follow the law of the conservation of energy? Does the second law of thermodynamics forbid converting all of the absorbed heat Q into work done by the piston? Is the movable piston a cycle? I think it is, so it must follow the second law? Is the total work done by the gas independent of the area of the piston? Well, the total work is dW = p*dV, and since V = A*h, then work is dependent and not independent?