3.2 The first law: heat, work, and internal energy

The first law of thermodynamics is the conservation of energy, written to include heat. A system holds energy in its internal energy $U$ — the kinetic and potential energy of its molecules — and that store changes by two routes: heat added, and work done.

Statement

For a closed system (no mass crossing its boundary),

\boxed{\;dU \;=\; \delta Q - \delta W \;=\; T\, dS - p\, dV.\;}

The internal energy rises by the heat added $\delta Q = T\,dS$ and falls by the work it does $\delta W = p\,dV$ . The notation distinguishes two kinds of quantity. $U$ , $S$ , $V$ are state functions: $dU$ , $dS$ , $dV$ are exact differentials, and their change around a cycle is zero. Heat and work are not state functions — there is no “heat content” of a body — so they are written $\delta Q$ and $\delta W$ , path-dependent quantities that have meaning only as flows during a process. The first law’s content is that although $Q$ and $W$ each depend on the path, their difference $Q - W = \Delta U$ does not.

The four standard processes

For an ideal gas the equation of state and the first law together fix the energy budget of any process. Four idealised paths recur because along each one a different one of the four quantities ( $\Delta U$ , $Q$ , $W$ , or a state variable) is held fixed:

Isochoric ( $V$ fixed). No work, $W = 0$ , so all heat goes to internal energy: $Q = \Delta U$ . This defines the heat capacity at constant volume.
Isobaric ( $p$ fixed). The work is simply $W = p\,\Delta V$ ; the heat must cover both $\Delta U$ and that work.
Isothermal ( $T$ fixed). For an ideal gas $U$ depends only on $T$ , so $\Delta U = 0$ and $Q = W$ : every joule of heat in leaves as work.
Adiabatic ( $Q = 0$ ). No heat crosses the boundary, so $\Delta U = -W$ : the gas does work at the cost of its own internal energy, and cools as it expands.

Process:

Progress along path: 50%

Each path realises a different bargain among ΔU (internal energy), W (work done by the gas), and Q (heat in). Isothermal: ΔU = 0, so Q = W. Adiabatic: Q = 0, so ΔU = −W. Isobaric: W = p ΔV. Isochoric: W = 0, so Q = ΔU. The first law dQ = dU + dW is the universal balance behind all four.

Pick a process and traverse it on the $p$ – $V$ diagram. The shaded area under the path is the work $W = \int p\,dV$ ; the readouts accumulate $W$ , $\Delta U$ , and $Q$ along the way. Because work is the area under the path, a cycle — a closed loop — does net work equal to the area it encloses, even though every state variable returns to its start. That is the principle of the heat engine, and the entry to the second law in 3.5.