Heat engines — efficiency η = 1 − Q_C - Q_H

What is a heat engine?

WHY the cycle matters: A one-shot expansion could convert lots of heat to work, but then the gas is "used up" (expanded, cooled). To run continuously you must reset the engine — and resetting always costs you some dumped heat.

Deriving the efficiency formula from scratch

We use the First Law (energy conservation) applied over one full cycle.

Step 1 — Write the First Law for a cycle. $\Delta U = Q_{\text{net}} - W_{\text{by gas}}$ Why this step? The First Law always holds; we just specialise it to one complete loop.

Step 2 — Use that internal energy is a state function. Over a closed cycle the system returns to its start, so $\Delta U_{\text{cycle}} = 0$ Why this step? $U$ depends only on the state (e.g. $T$ for an ideal gas), not the path. Same start = same end = zero change.

Step 3 — Identify the net heat. Heat in is $Q_H$, heat out is $Q_C$ (take $Q_H, Q_C > 0$ as magnitudes): $Q_{\text{net}} = Q_H - Q_C$

Step 4 — Combine. With $\Delta U = 0$: $0 = (Q_H - Q_C) - W \quad\Longrightarrow\quad \boxed{W = Q_H - Q_C}$ Why this step? All the heat that didn't get dumped became work. Energy is conserved.

Step 5 — Define efficiency as "what you get / what you pay": $\eta \equiv \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H}$

Step 6 — Simplify. $\boxed{\eta = 1 - \frac{Q_C}{Q_H}}$

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

• First Law of Thermodynamics — gives $W = Q_H - Q_C$.
• Second Law of Thermodynamics — forbids $\eta = 1$ (Kelvin–Planck).
• Carnot Cycle — the reversible engine setting the efficiency ceiling.
• Entropy — explains why heat must be dumped ($\Delta S_{\text{universe}} \ge 0$).
• Refrigerators and Heat Pumps — the engine run in reverse; COP instead of $\eta$.
• PV diagrams and Work — area of the cycle loop = net work $W$.

Concept Map

Hinglish (regional understanding)