Dimensional Formula of Boltzmann Constant: [ML²T⁻²K⁻¹]

The dimensional formula of Boltzmann constant is [M¹ L² T⁻² K⁻¹], which simply means energy divided by temperature. Its value is k = 1.381 × 10⁻²³ J/K, and it links the temperature of a gas to the energy of its individual molecules.

The key relation is the mean translational kinetic energy of one gas molecule, E = (3/2)kT. Energy carries [ML²T⁻²] and absolute temperature carries [K], so k must carry their ratio. The letter K inside the bracket is the temperature dimension; do not confuse it with the constant k itself, written in lower case and often as k_B.

Quick facts

Dimensional formula
[M¹ L² T⁻² K⁻¹]
Defining relation
$\bar{E} = \frac{3}{2} k_B T$
SI unit
joule per kelvin (J/K)
Value
1.381 × 10⁻²³ J/K
Link to R
$k_B = R / N_A$
Same bracket
Entropy, heat capacity of a body

Steps to the dimensional formula of Boltzmann constant

Mean kinetic energy of a molecule: $\bar{E} = \frac{3}{2}k_B T$ .
Rearrange: $k_B = \frac{2\bar{E}}{3T}$ . The 2/3 has no dimension.
Energy: $[E] = [ML^{2}T^{-2}]$ . Temperature: $[T_{abs}] = [K]$ .
So $[k_B] = [ML^{2}T^{-2}K^{-1}]$ .

An alternative route uses the ideal gas law written per molecule, $PV = Nk_BT$ . Pressure times volume is $[ML^{-1}T^{-2}][L^{3}] = [ML^{2}T^{-2}]$ , an energy. N is a count with no dimension. Dividing by temperature gives the same bracket. Both derivations rest on the dimensions of energy.

Why J/K and how big that is

Put kg, m, s and K in place of the dimensions and you get kg m² s⁻² K⁻¹. Since kg m² s⁻² is a joule, the unit is the joule per kelvin. The tiny value tells you how little energy one molecule holds.

Example: find kT at room temperature, 300 K, in joules and in electron volts.

$k_BT = (1.381 \times 10^{-23}\ \text{J/K})(300\ \text{K}) = 4.14 \times 10^{-21}\ \text{J}$ .

In eV: $\frac{4.14 \times 10^{-21}}{1.602 \times 10^{-19}} = 0.0259\ \text{eV}$ .

Mean translational energy: $\frac{3}{2} \times 4.14 \times 10^{-21} = 6.21 \times 10^{-21}\ \text{J}$ .

Use kelvin, never °C, inside kT. At 27 °C, T = 300 K.

Boltzmann constant versus the gas constant

The gas constant R appears in PV = nRT, where n is the amount of substance in moles. Because moles count particles in bulk, R picks up the amount dimension N: $[R] = [ML^{2}T^{-2}K^{-1}N^{-1}]$ . The Boltzmann constant is R shared out per particle, $k_B = \frac{R}{N_A}$ , and Avogadro's number $N_A$ has the bracket [N⁻¹]. Dividing removes the N.

Example: compute k using R, which is 8.314 J mol⁻¹ K⁻¹, together with N_A = 6.022 × 10²³ mol⁻¹.

$k_B = \frac{8.314}{6.022 \times 10^{23}} = 1.381 \times 10^{-23}\ \text{J/K}$ . The mol⁻¹ units cancel, matching the bracket.

Other quantities measured in joules per kelvin

Brackets of energy over temperature
Quantity	Formula	Bracket
Boltzmann constant	$k_B = R/N_A$	[M L² T⁻² K⁻¹]
Entropy	$\Delta S = Q/T$	[M L² T⁻² K⁻¹]
Heat capacity of a body	$C = Q/\Delta T$	[M L² T⁻² K⁻¹]
Gas constant	$PV/nT$	[M L² T⁻² K⁻¹ N⁻¹]
Specific heat	$Q/(m\Delta T)$	[L² T⁻² K⁻¹]

The statistical definition of entropy, $S = k_B \ln W$ , shows the match neatly: ln W is a pure number, so entropy inherits the bracket of k. Compare with thermal conductivity, another heat quantity whose bracket needs K.

Where students slip

Leaving out K. Without temperature the bracket is that of energy, which is wrong.
Adding N. That belongs to R, the per mole constant, not to k.
Writing T for temperature in the bracket. T is time. Use K or Θ.
Mixing up k and K. Lower case k is the constant; capital K is the temperature dimension and the kelvin.

Universal constants such as h, G, ε₀ and k sit in one block of the dimensional formulas table. Compare k with Planck's constant, which is energy times time instead of energy per temperature.

Frequently asked questions

What is the dimensional formula of Boltzmann constant?

It is [M¹ L² T⁻² K⁻¹]. The constant equals energy divided by absolute temperature, from E = (3/2)kT.

Which unit does the Boltzmann constant take?

Joule per kelvin, J/K. Written with SI base units: kg m² s⁻² K⁻¹.

Do Boltzmann constant and entropy have the same dimensions?

Yes. Entropy change is heat divided by temperature, which is energy over temperature, the same bracket as k.

How is the Boltzmann constant related to the gas constant?

k equals R divided by Avogadro's number. R works per mole and carries the amount dimension N⁻¹; k works per molecule and does not.

Why is the Boltzmann constant so small?

It gives the energy per kelvin for a single molecule. One molecule carries very little energy, so the number in joules is tiny.