Explanation of Dimensional Analysis

In order to match the quantities from LIST-I with their respective dimensions in LIST-II, we need to analyze their dimensional formulas:

(A) Boltzmann Constant [k]:

The equation relating pressure (P), volume (V), number of moles (N), and temperature (T) gives us:

$ [\mathrm{k}] = \frac{\mathrm{PV}}{\mathrm{NT}} = \frac{\mathrm{ML}^2 \mathrm{~T}^{-2}}{\mathrm{~K}} = \mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1} $

This corresponds to option III.

(B) Coefficient of Viscosity [η]:

The coefficient of viscosity can be defined using the relation:

$ [\eta] = \frac{\mathrm{F}}{6 \pi \mathrm{rv}} = \frac{\mathrm{MLT}^{-2}}{\mathrm{~L}^2 \mathrm{~T}^{-1}} = \mathrm{ML}^{-1} \mathrm{~T}^{-1} $

This corresponds to option IV.

(C) Planck's Constant [h]:

The relationship between energy (E) and frequency (f) gives the dimensional formula:

$ [\mathrm{h}] = \frac{\mathrm{E}}{\mathrm{f}} = \frac{\mathrm{ML}^2 \mathrm{~T}^{-2}}{\mathrm{~T}^{-1}} = \mathrm{ML}^2 \mathrm{~T}^{-1} $

This corresponds to option I.

(D) Thermal Conductivity [k]:

From the heat conduction equation, we get:

$ \mathrm{k} = \frac{\left(\mathrm{ML}^2 \mathrm{~T}^{-3}\right) \mathrm{L}}{\mathrm{~L}^2 \cdot \mathrm{~K}} = \mathrm{MLT}^{-3} \mathrm{~K}^{-1} $

This corresponds to option II.

To conclude, the correct matches are:

A - III

B - IV

C - I

D - II

This analysis ensures each physical quantity is aligned with its correct dimensional representation.