first law of thermodynamics explained
When heat enters a system and the system does work, the difference stays inside. This is the 1st law of thermodynamics, and the change in internal energy is the heat in minus the work done, dU = Q - W. Energy is not created or destroyed. It only changes form. What comes in as heat and leaves as work leaves the remainder as internal energy.
Heat is energy that crosses because of a temperature difference, and work is energy that crosses as the volume changes. Both are transfers of energy. So the 1st law puts the two on one balance against the internal energy. The difference between heat in and work out is what remains inside.
However the signs are chosen, the books have to balance. The heart is that in every case dU = Q - W holds exactly. A device that takes in heat and puts out work keeps inside only the difference. A perpetual machine of the 1st kind, which keeps putting out work while taking in nothing, is impossible from the start.
Between the same start and end, heat and work differ from path to path, but their difference Q - W, the change in internal energy, is the same on both paths. Internal energy is a state, so the point now is enough, while heat and work require the path that was taken.
The 1st law grows sharper in special cases. Insulate the walls completely and Q = 0, so dU = -W, and the internal energy falls by the work the gas does. Confine it in a closed tank whose volume cannot change and W = 0, so dU = Q, and the heat that enters piles up inside. So for heating at constant volume, dU = Q is the rise in temperature. Each time one term becomes 0, the 1st law grows clearer.
In a cycle that comes back to the start, a property of the state does not change. Follow a closed loop on the P-V plane with the slider and the change in internal energy of one lap is 0. Then the 1st law, dU = Q - W, says the net heat of one cycle equals the net work. The story that there is a limit to turning heat into work continues into the 2nd law. Direction matters too. Clockwise on the P-V plane the system sends work out, and the other way it receives work.