Heat engines — efficiency η = 1 − Q_C - Q_H: Foundations

This page builds every symbol in the parent topic from absolute zero, in an order where each idea rests on the one before it. Nothing is used before it is drawn.

1. Energy — the thing that gets moved around

Why the topic needs it: Everything in a heat engine — the heat coming in, the work going out, the heat dumped — is just energy wearing different hats. The whole efficiency formula is one big statement about where the coins went.

2. Heat $Q$ — energy moving because of a temperature difference

The letter $Q$ is just a name-tag for "an amount of this flowing energy". When we write a subscript like $Q_{H}$ , the little $H$ tells us which heat we mean — nothing more.

Figure — Heat engines — efficiency η = 1 − Q_C - Q_H

Because heat is "energy on the move", we always take $Q$ as a magnitude (a positive amount) in this chapter and use words ("in" / "out") to say which way it points. So $Q_{H} > 0$ and $Q_{C} > 0$ ; the direction lives in the subscript, not in a minus sign.

3. Temperature $T$ and the two reservoirs $T_{H}$ , $T_{C}$

Why the topic needs it: an engine needs a fixed hot side and a fixed cold side to run over and over. The subscripts match the heats: $Q_{H}$ comes from the reservoir at $T_{H}$ , $Q_{C}$ goes to the reservoir at $T_{C}$ .

Why kelvin and not Celsius? The Carnot formula divides two temperatures, $T_{C} / T_{H}$ . A ratio only makes physical sense when $0$ means "no temperature at all". On the Celsius scale $0° C$ is just "freezing water" — an arbitrary spot — so ratios there are meaningless. Kelvin's zero is real zero, so ratios behave.

4. Work $W$ — organised energy that pushes

Why the topic needs it: work is the useful output — the thing the engine exists to produce. Heat is messy, disorganised energy; work is neat, usable energy (a turning shaft, a moving car). Turning $Q$ into $W$ is the engine's entire job.

The figure shows the full flow: $Q_{H}$ enters from the hot side (top), the engine (red box) drains off $W$ as work, and the leftover $Q_{C}$ falls to the cold side. Read the picture as an accountant: everything that comes in the top must leave — some as work out the side, the rest as heat out the bottom.

5. Internal energy $U$ and state functions

Why the topic needs it: an engine runs in a cycle — it returns to its starting state every round. Because $U$ is a state function, a full loop gives $Δ U = 0$ . That single fact is what lets us say all the net heat became work.

6. The delta symbol $Δ$ and "cycle"

Put these together: over one cycle, since the gas ends where it started, $Δ U_{cycle} = U_{end} - U_{start} = 0$ . This is the keystone step in the parent's derivation.

7. The First Law — the accounting rule

Why the topic needs it: apply this law to a cycle where $Δ U = 0$ , plug in $Q_{net} = Q_{H} - Q_{C}$ , and you immediately get $W = Q_{H} - Q_{C}$ — the heart of the engine. Every symbol above was needed to make this one line readable.

8. Efficiency $η$ and the ratio idea

Why $η$ has no units: it is joules divided by joules, so the units cancel. That is why efficiency is a pure number (or a percentage). Because $W = Q_{H} - Q_{C}$ , dividing through by $Q_{H}$ turns the definition into the famous $η = 1 - Q_{C} / Q_{H}$ .

The bar in the figure splits $Q_{H}$ into two pieces: the useful slice $W$ (red) and the wasted slice $Q_{C}$ (black). Efficiency is just the fraction of the bar that is red. You can now see instantly why $η$ can never reach $1$ : the black slice $Q_{C}$ can shrink but never fully disappear.

How these feed the topic

Equipment checklist

Cover the right side and test yourself — you are ready for the parent note when every line is easy.

What are the units of energy, heat, and work?

The joule,

J

— they are all the same kind of quantity (energy).

What does the subscript in

Q_{H}

tell you?

Which heat it is — here, heat exchanged with the hot reservoir at

T_{H}

In which direction does heat flow on its own?

From hot to cold, never the reverse (one-way, like water downhill).

What is a reservoir?

A body so large its temperature stays fixed while giving or taking heat.

Why must the Carnot formula use kelvin, not Celsius?

Ratios of temperature only make sense from a true zero; kelvin's zero is real, Celsius's is arbitrary.

How does work differ from heat?

Work is ordered, usable energy (a push through a distance); heat is disordered energy flowing from a temperature difference.

What is a state function, and which quantity here is one?

A quantity depending only on the current state, not the path taken — internal energy

U

Why is

Δ U = 0

over a cycle?

The gas returns to its start state, and

U

(a state function) is the same there.

State the First Law in symbols.

Δ U = Q_{net} - W

What does

η

mean in one phrase, and what are its units?

Useful work out divided by heat paid in; it is a pure number (unitless).

Connections

First Law of Thermodynamics — the energy-accounting rule built here.
Second Law of Thermodynamics — why the wasted slice $Q_{C}$ can never vanish.
Carnot Cycle — needs the kelvin idea from this page.
Entropy — the deeper reason heat must flow one way.
Refrigerators and Heat Pumps — the same symbols, engine run backwards.
PV diagrams and Work — where the work $W$ shows up as a loop area.