First law: where does the heat go?

A gas cylinder given heat under four process types; bars Q, ΔU, W with signs show how the first law splits the heat; monatomic/diatomic changes the split.

In an isobaric heating of a monatomic gas, find what fraction of the heat becomes work (2/5) — then make ΔU = 0 with heat flowing in.

Sign errors in ΔQ = ΔU + ΔW; no heat flows in an isothermal process.

• Let ΔQ be the heat supplied to the system by its surroundings, ΔW the work done by the system on its surroundings, and ΔU the change in its internal energy.
• First law: ΔQ = ΔU + ΔW. The heat put in partly raises the internal energy and partly goes out as work. It is conservation of energy for a system that exchanges energy with its surroundings.
• Written as ΔQ − ΔW = ΔU: a gas can go from (P₁, V₁) to (P₂, V₂) by many routes, e.g. first at constant pressure to (P₁, V₂) and then at constant volume, or the other way round. ΔQ and ΔW generally depend on the route, but ΔQ − ΔW = ΔU does not, because U is a state variable.
• If a process has ΔU = 0 (an ideal gas expanding isothermally, say), then ΔQ = ΔW: all the heat supplied is used as work on the surroundings.
• Work by a gas pushing a piston at constant pressure P: force = P × area and area × displacement = volume change, so ΔW = PΔV. Then ΔQ = ΔU + PΔV.
• Worked example, boiling 1 g of water: latent heat 2256 J/g, so ΔQ = 2256 J. Under atmospheric pressure this gram fills 1 cm³ as liquid and 1671 cm³ as vapour.
• Work against the atmosphere: ΔW = P(V_g − V_l) = 1.013 × 10⁵ × (1671 × 10⁻⁶) = 169.2 J. So ΔU = 2256 − 169.2 = 2086.8 J.
• Most of the heat supplied in boiling raises the water's internal energy; only a small part pushes back the atmosphere.
• Sign convention: Q > 0 when heat is added to the system, Q < 0 when heat is taken out; W > 0 when the system does work, W < 0 when work is done on it.

Teaching this idea? Put the simulation on your own page or LMS with this code.