Energy Is Conserved: dU = Q - W
Let us add heat to the gas in a box. Drag the heat Q and the molecules move faster, while at the same time the gas pushes the piston out, sending work W to the outside. The heat that came in and the work that went out, where did the difference go? It does not vanish. It piles up as energy inside the gas. This is the first law of thermodynamics. The change in a system's internal energy is the heat in minus the work out: dU = Q - W. In the end it is the law of energy conservation, rewritten in the language of heat and work. Heat and work are just two channels through which energy moves, and when you balance the books, it always comes out exactly even.
First, look at the two channels energy moves through. Drag the heat Q in. As heat enters from the left, the molecules grow agitated and the color glows hot. At the same time, on the right, the gas pushes the piston and sends work W to the outside. Heat is energy that flows in on its own because of a temperature difference, and work is energy delivered by a force acting through a distance. They look different but are equally the movement of energy. So the first law puts the two on one scale. The heat in and the work out, the balance between them sets the energy inside the system. Stirring water warms it exactly as adding heat would, and this fact, shown by Joule's experiment, proved that heat and work are two ways of trading one and the same currency, energy.
Now balance the books yourself. With the two sliders, set the heat in Q and the work out W separately. The change in internal energy dU is always Q minus W. The bar chart makes it clear at a glance. Put in a lot of heat and let out little work, and the internal energy climbs a lot; let out a lot of work and the internal energy is shaved down by that much. Add heat yet do even more work, and dU can be negative, meaning the gas can actually cool. The key is that in every case dU = Q - W balances exactly. Energy is neither created nor destroyed. This is why a perpetual-motion machine of the first kind, one that keeps putting out work with no energy fed in, is impossible from the start, because the books must always balance to the penny.
But one thing is special. Going from the same start state to the same end state, the heat Q and the work W each vary with the path taken. Use the buttons to switch between two routes. One route takes in a lot of heat and does a lot of work; the other takes in little heat and does little work. Q differs and W differs. Yet their difference Q - W, which is dU, is the same on both routes. This is what it means for internal energy to be a property of state. U is fixed by the present state alone and does not ask how you got there. Last chapter we saw that work depends on the path; heat does too. But their difference, the internal energy, forgets the path. Because U depends only on the state, the internal energy of a substance can be tabulated once and reused for any process, and this is exactly what makes engineering steam tables and property charts possible.
The first law gets especially clean in special cases. Use the buttons to see two. An adiabatic process exchanges no heat. Insulate the box perfectly and Q = 0, so dU = -W. The internal energy drops by exactly the work the gas does, so an adiabatic expansion cools the gas. The opposite, an isochoric process, fixes the volume so no work is done. Lock it in a rigid container and W = 0, so dU = Q. The heat that comes in becomes internal energy outright. For reference, in an ideal gas the internal energy depends only on temperature, so the dU = Q of isochoric heating shows up directly as a temperature rise. Each time one term goes to zero, the first law stands out more sharply. This is why air bursting from a spray can feels cold as it expands and why rising air cools until clouds condense, while conversely a quick push on a bicycle pump heats the trapped air as it is compressed.
The last piece is the cycle. Use the slider to go once around a closed loop on the P-V plane. When you return exactly to the state you started from, the internal energy, being a property of state, is back to its initial value. The dU over one loop is zero. Then in the first law dU = Q - W, the net heat taken in over one cycle and the net work done over one cycle become exactly equal. The net work shows up as the area the loop encloses. This is precisely the principle of an engine. Keep running a gas around a loop, and on every lap you can extract part of the heat taken in as work. In the next chapter, we follow this heat engine, and the second law that says you cannot turn all the heat you take in into work. The direction you go around the loop matters too: clockwise on the P-V plane yields net work as an engine, while going the other way consumes work to pump heat, becoming a refrigerator or heat pump.