Boltzmann Constant — Value and Dimensional Formula

The easiest way to derive the dimensional formula is from the famous kinetic energy formula for a monoatomic gas molecule:

E = (3/2) k T

Where:

• E = Average Kinetic Energy
• T = Absolute Temperature
• k = Boltzmann constant

Rearranging to solve for k: k = E / T (Ignoring the 3/2 since numbers are dimensionless).

To find the dimensions of k, we divide the dimensions of Energy by the dimensions of Temperature.

1. Dimensional formula of Energy (E): Energy = Work = Force × Distance = [M¹ L¹ T⁻²] × [L¹] = [M¹ L² T⁻²]

2. Dimensional formula of Temperature (T): Temperature is a fundamental quantity, represented by [K] (Kelvin) or sometimes [θ].

3. Putting it together: [k] = [E] / [T] [k] = [M¹ L² T⁻²] / [K¹] [k] = [M¹ L² T⁻² K⁻¹]

The dimensional formula for the Boltzmann constant is [M¹ L² T⁻² K⁻¹].

Based on the formula k = E / T, the SI unit is Energy divided by Temperature.

• SI Unit: Joules per Kelvin (J/K or J·K⁻¹)

The accepted numerical value of the Boltzmann constant is: k = 1.380649 × 10⁻²³ J/K

The Boltzmann constant is simply the Universal Gas Constant (R) divided by Avogadro's Number (N_A). k = R / N_A While 'R' applies to a whole 'mole' of gas, 'k' applies to just one single molecule of gas.