IB Physics HLThermodynamicsPaper 1 & 2Engines & Efficiency~10 min read
Heat Engines
A heat engine is any machine that turns heat into useful work — a car engine, a power station, a jet. The idea is always the same: run a gas round a repeating cycle that takes heat from something hot, does some work, and dumps the leftover into something cold. The second law guarantees it can never use all the heat, so understanding an engine is really about understanding its waste.
📚 What you need to know
A heat engine turns thermal energy into mechanical work by running a gas through a closed cycle (it returns to its start each time)
Each cycle: take heat QH from a hot reservoir → do useful work W → dump waste heat QC to a cold reservoir
Energy is conserved: QH = W + QC, so the net work is W = QH − QC
On a p–V graph the cycle is a closed loop; the area inside = net work per cycle
Efficiency: η = W/QH = 1 − QC/QH
Efficiency is always less than 1 — the second law forbids 100%, so there’s always waste heat
What a heat engine does
Picture a gas sealed in a cylinder with a movable piston. Heat it and it expands, pushing the piston out and doing work. But to do this over and over, the gas must return to its starting state each time — so an engine runs a repeating cycle. Every engine needs three things: a hot reservoir to draw heat from, a cold reservoir to dump waste heat into, and the working gas in between.
Heat QH flows in from the hot reservoir. The engine turns some of it into useful work W and dumps the rest as waste heat QC into the cold reservoir. Energy is conserved: QH = W + QC.
A single cycle runs in four moves: (1) take in heat QH from the hot side; (2) let the gas expand, doing work on the piston; (3) release waste heat QC to the cold side; (4) compress the gas back to its start — then repeat, forever.
Here’s the crucial bit: an engine needs both a hot and a cold reservoir. It runs on the difference in temperature, just as a waterwheel runs on a difference in height. No temperature gap, no work — which is exactly the Kelvin form of the second law in action.
Net work and the p–V loop
Because the gas returns to its exact starting state after a full cycle, its internal energy is unchanged (ΔU = 0 over the loop). By the first law that means all the leftover heat becomes work — the net work out is simply the heat in minus the heat dumped:
Net work per cycleW = QH − QC
On a pressure–volume graph, a cycle traces a closed loop, and the area enclosed by that loop equals the net work done each cycle. The gas does work as it expands (the upper path) and gets some work back as it’s compressed (the lower path); the difference — the area trapped inside — is what you keep.
The engine loops clockwise: expand along the top (work done by the gas), compress along the bottom (work done on it). The shaded area between them is the net work output for one cycle.
WE 1
Each cycle, an engine takes 900 J of heat from the hot reservoir and dumps 600 J into the cold reservoir. Find (a) the net work output, and (b) the efficiency.
(a) Net work: W = QH − QCW = 900 − 600 = 300 J(b) Efficiency: η = W / QHη = 300 ÷ 900 = 0.33η ≈ 33%Only a third of the heat becomes useful work; the other 600 J is wasted to the cold reservoir.
Efficiency: how much do you actually get?
The whole point of an engine is to turn as much of the heat input into work as possible. Its efficiency is just that fraction — useful work out divided by heat in:
Efficiency of a heat engineη = W⁄QH = (QH − QC)⁄QHη = 1 − QC⁄QH
Think of the heat input as one bar. The green slice is the useful work W; the grey slice is the wasted heat QC. Efficiency is simply how big the green slice is compared with the whole bar.
Efficiency is a number between 0 and 1 (multiply by 100 for a percentage), and it’s always less than 1. Getting to 100% would mean QC = 0 — turning all the heat into work — which the Kelvin form of the second law flatly forbids. The less heat you waste (smaller QC), the closer you get.
WE 2
An engine is 40% efficient and does 200 J of useful work each cycle. Find (a) the heat it takes from the hot reservoir, and (b) the heat it dumps to the cold reservoir.
(a) Rearrange η = W / QH for QHQH = W ÷ η = 200 ÷ 0.40 = 500 J(b) Waste heat: QC = QH − WQC = 500 − 200 = 300 JQH = 500 J, QC = 300 JTo get 200 J of work at 40% efficiency, you must pull in 500 J and throw away 300 J.
WE 3
An engine runs 20 cycles every second. Each cycle it takes 500 J from the hot reservoir and rejects 350 J to the cold reservoir. Find (a) its efficiency, and (b) its power output.
(a) Work per cycle: W = 500 − 350 = 150 J; η = 150/500η = 0.30 = 30%(b) Power = work per cycle × cycles per secondP = 150 × 20 = 3000 Wη = 30%, P = 3.0 kWPower ties the per-cycle work to how fast the engine runs — more cycles per second means more watts.
Heat in QH
− waste QC
Net work W = QH − QC
÷ QH
Efficiency η = W/QH
💡 Top tips
Net work is a difference:W = QH − QC. On a p–V loop it’s the enclosed area.
Know both efficiency forms: η = W/QH and η = 1 − QC/QH.
η is a fraction (×100 for a %) and is always less than 1.
Less waste = more efficient: the smaller QC, the higher η.
Power = work per cycle × cycles per second.
⚠ Common mistakes
Using QH as the work output — the work is QH − QC
Forgetting the engine needs a cold reservoir to dump waste heat into — no cold sink, no engine
Quoting an efficiency of 100% or more — impossible under the second law
Swapping QH and QC in η = 1 − QC/QH
Forgetting the cycle is closed — over one loop ΔU = 0, which is why net heat = net work
Quick recap: A heat engine loops a gas through a closed cycle: it takes QH from a hot reservoir, does net work W = QH − QC (the area of the p–V loop), and dumps waste heat QC to a cold reservoir. Its efficiency η = W/QH = 1 − QC/QH is always less than 1 — there’s always some waste heat.
You now know every real engine wastes some heat — but how little can it possibly waste? Is there a “best” engine that no design can beat? There is: the Carnot cycle, an idealised engine built from the very processes you’ve just learned, which sets the ultimate ceiling on efficiency. That’s the grand finale of the topic, coming up next.
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