The first law of thermodynamics, explained

Closed systems

For a fixed mass, ΔU = Q − W, with Q the heat added and W the work done by the system. U is internal energy: the energy stored in molecular motion and bonds. Sign conventions vary between textbooks, so write yours down once and keep it.

Heat and work are energy in transit, not properties. A system does not contain heat; it contains energy. That distinction is why Q and W depend on the path and ΔU does not.

Open systems and enthalpy

Turbines, pumps, compressors, nozzles and heat exchangers have mass flowing through them. Pushing fluid in and out takes work, pv per unit mass, and it is convenient to bundle it with internal energy as enthalpy, h = u + pv.

The steady flow energy equation then balances heat and shaft work against the change in enthalpy, kinetic energy and potential energy of the stream. For most machines the kinetic and potential terms are small, and the equation shrinks to Q̇ − Ẇ = ṁ(h₂ − h₁).

What it does not tell you

The first law accounts for energy but says nothing about direction. It allows heat to flow from cold to hot, as long as the books balance. Ruling that out is the job of the second law and entropy.

A worked number

An adiabatic compressor takes 0.5 kg/s of air from 300 K to 450 K. With c_p ≈ 1.005 kJ/kg·K, the enthalpy rise is 1.005 × 150 ≈ 151 kJ/kg, so the shaft power is 0.5 × 151 ≈ 75 kW.

Common mistakes

• Mixing sign conventions for work between two equations in the same solution.
• Using c_v instead of c_p for a flowing stream.
• Dropping the kinetic energy term in a nozzle, where it is the whole point.