First Law of Thermodynamics

The first law of thermodynamics is the precise statement, for a thermodynamic system, of the general principle of conservation of energy: energy can be converted from one form into another (here, specifically, between heat, work and internal energy) but the total amount of energy involved is always conserved. It is written as

with the following sign convention, standard in this chapter: $Q$ is taken positive when heat is absorbed by the system from its surroundings, and negative when the system gives out heat to its surroundings; $W$ is taken positive when the system does work on its surroundings (as when a gas expands and pushes back a piston), and negative when the surroundings do work on the system (as when a gas is compressed by an external force); and $\Delta U = U_f - U_i$ is the resulting change in the system's internal energy, positive if internal energy increases and negative if it decreases.

Physically, the equation says: whatever heat is put into a system must go somewhere -- either it raises the system's internal energy, or it is given back out as work done by the system, or (in the general case) some combination of both. Rearranging the law as $\Delta U = Q - W$ makes this bookkeeping especially clear: the change in a system's stored (internal) energy equals the heat put in, minus the work taken out.

The first law holds for absolutely every thermodynamic process, with no exceptions -- reversible or irreversible, fast or slow, at constant temperature, constant volume, constant pressure, or none of these. Four special cases recur constantly through the rest of this unit and are worth noting immediately:

• Isochoric process (constant volume): no work is done, $W = 0$ (since $dV = 0$ throughout), so $Q = \Delta U$ -- all the heat supplied goes directly into raising the internal energy.
• Cyclic process (system returns to its exact original state after a series of steps): since internal energy is a state function, $\Delta U = 0$ over the complete cycle, so $Q = W$ -- the net heat absorbed over the whole cycle exactly equals the net work done by the system over the whole cycle. This fact underlies the analysis of every heat engine (Section 11.11). …