The Boltzmann constant

Q: What is the value of the Boltzmann constant?

1.380649 × 10⁻²³ J/K. Since 2019 k_B is exactly 1.380649 × 10⁻²³ J/K. The best measurements before that, mostly acoustic gas thermometers, agreed to better than one part per million.

What is temperature, really, and how hot is one molecule?

k_B = 1.380\,649 \times 10^{-23}\ \text{J/K}

Exact, by definition, since 2019 12 min read, with an experiment to run

Temperature is jiggling. The air around you is molecules flying at about 500 m/s, and how hot it is just says how fast they move. The Boltzmann constant, $k_B$ , turns degrees into energy.

Each molecule carries about $\tfrac32 k_B T$ of movement energy. Since 2019 $k_B$ is fixed exactly, and it defines the kelvin.

1 · Play first

Heat a box of gas and weigh its energy

Here is a flat box of atoms bouncing around. Warm the gas and every atom speeds up, so the average energy of each atom goes up with it.

Record the average energy at a few temperatures. Plot energy against temperature and the slope of the line is $k_B$ . Swap the gas and see what changes.

Temperature300 K (27 °C)

Gas

Heavier atoms move slower at the same temperature, but their average energy is the same. That is the clue: energy tracks temperature, whatever the atom.

Average energy per atom: 4.142 zJ
Energy ÷ temperature: 1.381 × 10⁻²³ J/K

Record two or more temperatures. The slope of energy against temperature is k, and it appears here.

Bars: how many of your atoms have each speed. Line: the Maxwell–Boltzmann curve at this temperature. 1 zJ (zeptojoule) is 10⁻²¹ joules. In a flat box each atom moves in two directions, so its average energy is exactly k × T.

What to notice

Double the temperature (in kelvin) and the average energy per atom doubles.
The atoms have a spread of speeds. Only their average is set by temperature.
Heavier atoms move slower, yet at the same temperature they carry the same average energy.

2 · The equation, piece by piece

The ideal gas law, written for single molecules instead of moles. Tap any piece.

\htmlClass{t0}{p}\,\htmlClass{t1}{V} = \htmlClass{t2}{N}\,\htmlClass{t3}{k_B}\,\htmlClass{t4}{T}

p

Pressure

The push on each square metre of wall.

Multiply $k_B$ by Avogadro's number and you get the gas constant $R = 8.314$ J/(mol K), the version chemists use.

3 · How we found it

1From squeezing air to absolute zero

Robert Boyle (1662) found squeezing air raises its pressure. Jacques Charles and Joseph Gay-Lussac found warming it does too. Extend the line backwards and the pressure hits zero at about −273 °C: absolute zero.

2Heat is motion

James Clerk Maxwell (1860) and Ludwig Boltzmann worked out how the speeds of gas molecules spread out, and linked temperature to their average energy. Boltzmann's entropy formula, $S = k \log W$ , is carved on his grave.

Max Planck was the first to write the constant as $k$ and gave it a value in 1900, in the same paper as $h$ .

3Fixing the kelvin

The kelvin used to be set by the triple point of water. Labs measured $k_B$ by timing sound in argon inside a copper sphere, since sound speed depends on $k_B T$ . In 2019 $k_B$ was frozen at their best value.

1662
Boyle's law links pressure and volume.
1802
Gay-Lussac publishes how gases expand with heat.
1848
Kelvin proposes an absolute temperature scale.
1860
Maxwell's speed distribution for gas molecules.
1877
Boltzmann links entropy to counting arrangements.
1900
Planck writes $k$ and gives its first value.
2019
$k_B$ is fixed exactly and the kelvin is redefined.

4 · How the value is pinned down

Since 2019 $k_B$ is exactly 1.380649 × 10⁻²³ J/K. The best measurements before that, mostly acoustic gas thermometers, agreed to better than one part per million.

Now one kelvin simply means a change in thermal energy of $k_B$ joules per molecule.

5 · What if it were different?

If k_B were bigger

The same temperature would mean far more molecular energy. Room-temperature air would tear molecules apart.

At absolute zero

Molecules reach their lowest possible energy. Quantum physics says they still keep a tiny jiggle.

Why it matters to you

Engine efficiency, weather forecasts and fridge design all use $k_B$ hidden inside the gas laws.

6 · Try it at home

Find absolute zero with a balloon

Gas shrinks as it cools, in a straight line. Measure a balloon warm and cold, extend the line and see where it hits zero size.

You need

A round balloon
A tape measure or string
A freezer and a warm room (and a thermometer if you have one)

Do this

Blow up the balloon to medium size and tie it. Measure round its widest part at room temperature.
Put it in the freezer for 30 minutes, then measure it again quickly.
Work out the volume each time from the circumference $C$ , using $V = C^3/(6\pi^2)$ .

The maths

Plot volume against temperature in °C, for example 20 °C and −18 °C. Draw a straight line through the two points and extend it to volume zero. You should land somewhere near −273 °C; within 50 degrees is good for a balloon.

Stay safe: Do not put your hand or face against freezer surfaces for long, and ask before using the freezer.

7 · Common mix-ups

MythCold is a thing that flows into objects.

ActuallyOnly heat flows. Cold just means slower jiggling.

MythAll the molecules in a gas move at the same speed.

ActuallyThey have a wide spread of speeds. Temperature sets the average.

MythAbsolute zero can be reached with a good enough freezer.

ActuallyIt can be approached ever closer, but never reached.

8 · References

CODATA value: Boltzmann constant, NIST, 2018
The International System of Units (SI Brochure, 9th edition), BIPM, 2019
A low-uncertainty measurement of the Boltzmann constant, M. de Podesta et al., Metrologia 50, 2013
Kelvin, Boltzmann constant, NIST, 2019

Next constantThe Avogadro constant: How many atoms are in a spoonful, and how did anyone count them?