Question

Give the dimensional formula for Boltzmann’s constant.

Answer 1

Boltzmann's constant:

1. Boltzmann constant is a proportionality constant that connects the average kinetic energy of a system with its absolute temperature.
2. The exact value of the Boltzmann constant is $1.380649\times {10}^{-23}{m}^{2}kg{s}^{-2}{K}^{-1}$.
3. For example, for an ideal gas $\frac{m{v}^2}{2}=\frac{3}{2}KT$ , where $\frac{m{v}^2}{2}=\mathrm{average}\mathrm{kinetic}\mathrm{energy}\mathrm{of}\mathrm{molecule}$.
4. Boltzmann's constant, $k_B=\frac{2E_k}{3T}$.
5. The dimensional formula for energy is $\frac{m{v}^2}{2}=\left[M^1{L}^2{T}^{-2}\right]\mathrm{as}v=\left[L^1{T}^{-1}\right]$.
6. $k_B=\frac{2E_k}{3T}=\frac{\left[M^1{L}^2{T}^{-2}\right]}{K}=\left[M^1{L}^2{T}^{-2}{K}^{-1}\right]$
7. Hence, the dimensional formula of the Boltzmann constant is $\left[M^1{L}^2{T}^{-2}{K}^{-1}\right]$, Here, $M$ implies dimension of mass, $L$ implies length, $T$ implies time, $K$ implies temperature.