The dimensional formula for Boltzmann's constant is

The correct option is [ML²T⁻²θ⁻¹].

To find the dimensional formula of Boltzmann's constant ($k_B$), we can use the relation for the kinetic energy of a gas molecule:

$E=\frac{3}{2}{k}_{B}T$

Where:
- $E$ is the kinetic energy of the molecule
- $k_B$ is the Boltzmann's constant
- $T$ is the absolute temperature

Rearranging the equation to solve for $k_B$:

$k_B=\frac{2E}{3T}$

Since numerical fractions like $\frac{2}{3}$ are dimensionless, the dimensions of Boltzmann's constant are given by:

$\left[k_B\right]=\frac{\left[E\right]}{\left[T\right]}$

Next, we determine the dimensions of each quantity:
1. Energy ($E$) has the same dimensions as work (Force × Displacement):
$\left[E\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$

2. Temperature ($T$) is a fundamental quantity denoted by $\left[\theta \right]$ (or $\left[K\right]$):
$\left[T\right]=\left[\theta \right]$

Substituting these dimensions into the expression for $\left[k_B\right]$:

$\left[k_B\right]=\frac{\left[{\mathrm{ML}}^2{T}^{-2}\right]}{\left[\theta \right]}=\left[{\mathrm{ML}}^2{T}^{-2}{\theta }^{-1}\right]$

Therefore, the dimensional formula for Boltzmann's constant is $\left[{\mathrm{ML}}^2{T}^{-2}{\theta }^{-1}\right]$.