Dimensional formula for Boltzmann's constant is $\ldots \ldots \ldots \ldots \ldots$ (a) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (d) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{1} \mathrm{~K}^{-2}\)

Boltzmann's constant (k) is related to the gas constant (R) and Avogadro's number (N_A) through the following equation: \[k = \frac{R}{N_A}\] \

The dimensional formula for the gas constant (R) can be found by analyzing the ideal gas equation \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, and T is the temperature. The formula for the gas constant is: \[R = \frac{PV}{nT}\] Pressure (P) has dimensions of \(\mathrm{M}^1\mathrm{L}^{-1}\mathrm{T}^{-2}\), volume (V) has dimensions of \(\mathrm{L}^3\), temperature (T) has dimensions of \mathrm{K}^1, and the number of moles (n) has the dimension of \mathrm{mol}^1. By substituting these dimensions, we find the dimensional formula for R: \[R = \frac{\mathrm{M}^1\mathrm{L}^{-1}\mathrm{T}^{-2} \cdot \mathrm{L}^3}{\mathrm{mol}^1\mathrm{K}^1} = \frac{\mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}}{\mathrm{mol}^1\mathrm{K}^1}\] \

Avogadro's number (N_A) is a dimensionless constant because it is essentially the number of particles (atoms or molecules) in one mole. Therefore, its dimensional formula is: \[N_A = \mathrm{mol}^{-1}\] \

Now we can find the dimensional formula for Boltzmann's constant (k) by dividing the dimensional formula for R by the dimensional formula for N_A: \[k = \frac{\frac{\mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}}{\mathrm{mol}^1\mathrm{K}^1}}{\mathrm{mol}^{-1}} = \mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}\mathrm{K}^{-1}\] Comparing our result to the given options, we can see that the correct answer is: (c) \(\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\)