Ideal gases

Squeeze, heat and cool a box of atoms. The pressure is measured from the hammering on the walls and the temperature from how fast they are going.

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

Boyle’s law
Squeeze the gas at a steady temperature and watch the pressure climb.
- p is inversely proportional to V at constant temperature
- p against 1/V is a straight line through the origin, and its gradient is nRT
- the pressure comes from atoms hitting the walls, not from the gas "pushing"
Pressure and temperature
Hold the volume and change how hot the walls are. Extrapolate back to p = 0.
- p is proportional to T in kelvin at constant volume
- the line extrapolates to zero pressure at absolute zero, which is what fixes the kelvin scale
- a gas takes its temperature from the walls it is in contact with
Squeezing a gas heats it
Insulate the walls and drive the piston in. Nothing heats the gas but the piston.
- work done on a gas raises its internal energy, and so its temperature
- with no heat flow, TV^(γ−1) stays constant, so ln T against ln V is a straight line of gradient 1 − γ
- for a monatomic gas γ = 5/3, because all of its internal energy is translation
Where the energy goes
Compress the gas against walls held at room temperature: the work goes straight out as heat.
- ΔU = Q + W, with every term measured separately
- an isothermal compression leaves the internal energy alone: the work done goes out as heat
- temperature is internal energy, so holding the temperature means holding the energy
How fast are the atoms?
Leave the gas alone and watch the spread of speeds settle into the same shape every time.
- the atoms do not all travel at the same speed: there is a distribution with a long fast tail
- c_rms = √(3RT/M), so a light gas is faster at the same temperature
- the mean kinetic energy of an atom is (3/2)kT whatever the gas is

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)
TemperatureT (K)
VolumeV (L)
One over the volumeinvV (1/L)
Piston positionL (m)
Pressure on the pistonp_p (Pa)
pVpV (J)
pV over TpV/T (J/K)
Adiabatic invariantA (1)
Root mean square speedc (m/s)
Mean speedc̄ (m/s)
Collisions with the wallsn_w (1)
Collisions between atomsn_a (1)
Internal energyU (J)
Work done on the gasW (J)
Heat into the gasQ (J)
Change in internal energyΔU (J)
ΔU − (Q + W)ε (J)

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.

Amount of gasn (mmol)
Starting temperatureT_0 (K)
VolumeV (L)
Piston moves toV_f (L)
Piston speedu (m/s)
Walls held atT_w (K)
Atoms simulatedN (1)
Gauge averaging timegaugeWindow (ms)

The equations behind it

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

Thermal energy transfers

$Q = m c Δ T$
Heating something up — c is the specific heat capacity.
$Q = m L$
Changing state — L is the specific latent heat; the temperature does not change.
$ρ = \frac{m}{V}$
Density
$T [K] = T [^{\circ} C] + 273$
Kelvin from Celsius

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.

Also in Gases & thermodynamics

Experiments already set up on it

Boyle’s law

Squeeze the gas at a steady temperature and watch the pressure climb.

p is inversely proportional to V at constant temperature
p against 1/V is a straight line through the origin, and its gradient is nRT
the pressure comes from atoms hitting the walls, not from the gas "pushing"

Pressure and temperature

Hold the volume and change how hot the walls are. Extrapolate back to p = 0.

p is proportional to T in kelvin at constant volume
the line extrapolates to zero pressure at absolute zero, which is what fixes the kelvin scale
a gas takes its temperature from the walls it is in contact with

Squeezing a gas heats it

Insulate the walls and drive the piston in. Nothing heats the gas but the piston.

work done on a gas raises its internal energy, and so its temperature
with no heat flow, TV^(γ−1) stays constant, so ln T against ln V is a straight line of gradient 1 − γ
for a monatomic gas γ = 5/3, because all of its internal energy is translation

Where the energy goes

Compress the gas against walls held at room temperature: the work goes straight out as heat.

ΔU = Q + W, with every term measured separately
an isothermal compression leaves the internal energy alone: the work done goes out as heat
temperature is internal energy, so holding the temperature means holding the energy

How fast are the atoms?

Leave the gas alone and watch the spread of speeds settle into the same shape every time.

the atoms do not all travel at the same speed: there is a distribution with a long fast tail
c_rms = √(3RT/M), so a light gas is faster at the same temperature
the mean kinetic energy of an atom is (3/2)kT whatever the gas is

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)

TemperatureT (K)

VolumeV (L)

One over the volumeinvV (1/L)

Piston positionL (m)

Pressure on the pistonp_p (Pa)

pVpV (J)

pV over TpV/T (J/K)

Adiabatic invariantA (1)

Root mean square speedc (m/s)

Mean speedc̄ (m/s)

Collisions with the wallsn_w (1)

Collisions between atomsn_a (1)

Internal energyU (J)

Work done on the gasW (J)

Heat into the gasQ (J)

Change in internal energyΔU (J)

ΔU − (Q + W)ε (J)

The equations behind it

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

Thermal energy transfers

$Q = m c Δ T$
Heating something up — c is the specific heat capacity.
$Q = m L$
Changing state — L is the specific latent heat; the temperature does not change.
$ρ = \frac{m}{V}$
Density
$T [K] = T [^{\circ} C] + 273$
Kelvin from Celsius

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.

Ideal gases

Experiments already set up on it

Boyle’s law

Pressure and temperature

Squeezing a gas heats it

Where the energy goes

How fast are the atoms?

What you can measure

What you can change

The equations behind it

Thermal energy transfers

The ideal gas

Also in Gases & thermodynamics

Heating and cooling

Heat engines

Thermal radiation

Ideal gases

Experiments already set up on it

Boyle’s law

Pressure and temperature

Squeezing a gas heats it

Where the energy goes

How fast are the atoms?

What you can measure

What you can change

The equations behind it

Thermal energy transfers

The ideal gas

Also in Gases & thermodynamics

Heating and cooling

Heat engines

Thermal radiation