Heat engines

Take a gas round a closed cycle and measure the area of the loop. How much of the heat you pay for comes back as work — and why can none of them do better than Carnot?

Gases & thermodynamics — Pressure, volume and temperature, and the particles whose motion they average.

Open it in The Lab The equations behind it

Experiments already set up on it

One trip round the loop
Watch the gas go round a Carnot cycle and the area of the loop build up.
- the work done in a cycle is the area enclosed by the loop on a p–V diagram
- ΔU is zero round a whole cycle, so W = Q_h − Q_c
- heat goes in along the hot isothermal and comes out along the cold one
Efficiency against the cold reservoir
Hold the hot side and cool the cold one. Efficiency against T_c is straight.
- η = 1 − T_c/T_h for a Carnot engine: a straight line against T_c of gradient −1/T_h
- the only way to reach 100% is a cold reservoir at absolute zero
- the efficiency depends on the two temperatures and on nothing else about the engine
Why engines squeeze hard
A petrol engine’s efficiency against how far it compresses the mixture.
- an Otto cycle manages η = 1 − r^(1−γ): squeezing harder helps, with diminishing returns
- ln(1 − η) against ln r is a straight line of gradient 1 − γ
- it is always below the Carnot limit for the same two temperatures, because heat goes in over a range of temperatures rather than all at the hottest
How far short of Carnot?
A Stirling engine between the same two temperatures, and the heat it has to pay for twice.
- no engine beats Carnot, and most are well below it
- a Stirling cycle takes heat in at constant volume as well as along the hot isothermal, and that extra heat is paid for but does no extra work
- that is what a regenerator is for: it stores the heat given out on the way down and gives it back on the way up

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)
Pressurep (Pa)
VolumeV (L)
TemperatureT (K)
Internal energyU (J)
Work done by the gasW (J)
Work in the last cycleW_c (J)
Heat taken inQ_h (J)
Heat given outQ_c (J)
Efficiency measuredη (1)
Carnot limit for these two temperaturesη_c (1)
PowerP (W)
Cycles completedN (1)
How far the last cycle missed closingε (K)

What you can change

Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.

Compression ratior (1)
Smallest volumeV_1 (L)
Time for one cyclecycleTime (s)
Hot reservoirT_h (K)
Cold reservoirT_c (K)
Amount of gasn (mmol)

The equations behind it

What heat engines is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

The ideal gas

$p V = n R T = N k_{B} T$
Ideal gas law
$\frac{p _{1} V _{1}}{T _{1}} = \frac{p _{2} V _{2}}{T _{2}}$
Combined gas law
$p = \frac{1}{3} ρ v^{2}$
Pressure from molecular motion
$E_{k} = \frac{3}{2} k_{B} T$
Average kinetic energy of a molecule — Temperature *is* this, which is why it has an absolute zero.
$n = \frac{N}{N _{A}}$
Number of moles
$c_{rms} = \frac{3 k _{B} T}{m} = \frac{3 R T}{M}$
Root mean square speed of a molecule — A light gas is faster at the same temperature: at 300 K helium averages about 1370 m/s and argon about 430.

Thermodynamics

$Q = Δ U + W$
First law of thermodynamics
$W = p Δ V$
Work done by an expanding gas — At constant pressure.
$Δ U = \frac{3}{2} N k_{B} Δ T$
Internal energy of an ideal monatomic gas
$η = \frac{W _{out}}{Q _{in}}$
Efficiency of a heat engine
$η_{Carnot} = 1 - \frac{T _{C}}{T _{H}}$
Best possible efficiency — Temperatures in kelvin.
$p V^{γ} = constant, T V^{γ - 1} = constant$
Adiabatic change — When no heat enters or leaves. γ is 5/3 for a monatomic gas and 7/5 for a diatomic one.
$η_{Otto} = 1 - r^{1 - γ}$
Efficiency of a petrol engine — r is the compression ratio — which is why engines squeeze as hard as they can.
$Δ S = \frac{Δ Q}{T}$
Change in entropy

Also in Gases & thermodynamics

Experiments already set up on it

One trip round the loop

Watch the gas go round a Carnot cycle and the area of the loop build up.

the work done in a cycle is the area enclosed by the loop on a p–V diagram
ΔU is zero round a whole cycle, so W = Q_h − Q_c
heat goes in along the hot isothermal and comes out along the cold one

Efficiency against the cold reservoir

Hold the hot side and cool the cold one. Efficiency against T_c is straight.

η = 1 − T_c/T_h for a Carnot engine: a straight line against T_c of gradient −1/T_h
the only way to reach 100% is a cold reservoir at absolute zero
the efficiency depends on the two temperatures and on nothing else about the engine

Why engines squeeze hard

A petrol engine’s efficiency against how far it compresses the mixture.

an Otto cycle manages η = 1 − r^(1−γ): squeezing harder helps, with diminishing returns
ln(1 − η) against ln r is a straight line of gradient 1 − γ
it is always below the Carnot limit for the same two temperatures, because heat goes in over a range of temperatures rather than all at the hottest

How far short of Carnot?

A Stirling engine between the same two temperatures, and the heat it has to pay for twice.

no engine beats Carnot, and most are well below it
a Stirling cycle takes heat in at constant volume as well as along the hot isothermal, and that extra heat is paid for but does no extra work
that is what a regenerator is for: it stores the heat given out on the way down and gives it back on the way up

What you can measure

Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.

Timet (s)

Pressurep (Pa)

VolumeV (L)

TemperatureT (K)

Internal energyU (J)

Work done by the gasW (J)

Work in the last cycleW_c (J)

Heat taken inQ_h (J)

Heat given outQ_c (J)

Efficiency measuredη (1)

Carnot limit for these two temperaturesη_c (1)

PowerP (W)

Cycles completedN (1)

How far the last cycle missed closingε (K)

The equations behind it

What heat engines is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.

The ideal gas

$p V = n R T = N k_{B} T$
Ideal gas law
$\frac{p _{1} V _{1}}{T _{1}} = \frac{p _{2} V _{2}}{T _{2}}$
Combined gas law
$p = \frac{1}{3} ρ v^{2}$
Pressure from molecular motion
$E_{k} = \frac{3}{2} k_{B} T$
Average kinetic energy of a molecule — Temperature *is* this, which is why it has an absolute zero.
$n = \frac{N}{N _{A}}$
Number of moles
$c_{rms} = \frac{3 k _{B} T}{m} = \frac{3 R T}{M}$
Root mean square speed of a molecule — A light gas is faster at the same temperature: at 300 K helium averages about 1370 m/s and argon about 430.

Thermodynamics

$Q = Δ U + W$
First law of thermodynamics
$W = p Δ V$
Work done by an expanding gas — At constant pressure.
$Δ U = \frac{3}{2} N k_{B} Δ T$
Internal energy of an ideal monatomic gas
$η = \frac{W _{out}}{Q _{in}}$
Efficiency of a heat engine
$η_{Carnot} = 1 - \frac{T _{C}}{T _{H}}$
Best possible efficiency — Temperatures in kelvin.
$p V^{γ} = constant, T V^{γ - 1} = constant$
Adiabatic change — When no heat enters or leaves. γ is 5/3 for a monatomic gas and 7/5 for a diatomic one.
$η_{Otto} = 1 - r^{1 - γ}$
Efficiency of a petrol engine — r is the compression ratio — which is why engines squeeze as hard as they can.
$Δ S = \frac{Δ Q}{T}$
Change in entropy

Heat engines

Experiments already set up on it

One trip round the loop

Efficiency against the cold reservoir

Why engines squeeze hard

How far short of Carnot?

What you can measure

What you can change

The equations behind it

The ideal gas

Thermodynamics

Also in Gases & thermodynamics

Ideal gases

Heating and cooling

Thermal radiation

Heat engines

Experiments already set up on it

One trip round the loop

Efficiency against the cold reservoir

Why engines squeeze hard

How far short of Carnot?

What you can measure

What you can change

The equations behind it

The ideal gas

Thermodynamics

Also in Gases & thermodynamics

Ideal gases

Heating and cooling

Thermal radiation