The First Law of Thermodynamics

The first law is energy conservation applied to gases. Internal energy U is the total kinetic energy of all the jittering molecules inside the gas, and it can only change two ways: heat flows in or out, or the gas does work on its surroundings (or has work done on it). Write it as dU = Q - W, where Q is heat added to the gas and W is work done BY the gas. Nothing escapes this bookkeeping. That is also why a perpetual motion machine is impossible: running forever with no fuel means producing work with no energy input, and the first law forbids pulling W out of nothing. Every joule of work a machine delivers has to come from somewhere, either heat supplied or a drop in its own internal energy, and once that reserve is spent, the machine stops.

Sign conventions are where students lose points, so anchor them to physical direction, not memorized rules. Q is positive when heat enters the gas, negative when heat leaves. W is positive when the gas expands and pushes outward on its surroundings, negative when the surroundings compress the gas and push in on it. So during compression, the gas has work done ON it, meaning W is negative, and that negative W subtracted in dU = Q - W actually adds to internal energy. That is also the key to the misconception that heat always raises temperature: if the gas is expanding while heat enters, some or all of that heat exits again as work, leaving temperature unchanged or even falling.

This connects forward to specific processes: isochoric (W = 0, so dU = Q), isothermal (dU = 0, so Q = W), and adiabatic (Q = 0, so dU = -W), each a special case of the same master equation.