Question

Match List I with List II and select the correct answer using the codes given below the lists:

List I	List 2
Boltzmann Constant	$[M{L^2}{T^{ - 1}}]$
Coefficient of viscosity	$[M{L^{ - 1}}{T^{ - 1}}]$
Plank Constant	$[ML{T^{ - 3}}{K^{ - 1}}]$
Thermal Conductivity	$[M{L^2}{T^{ - 2}}{K^{ - 1}}]$

Answer 1

Hint:In order to answer this question we should first get some idea about the dimension. The powers to which the fundamental units are elevated to obtain one unit of a physical quantity are called the dimensions of that quantity.

Complete step by step answer:
Dimensional analysis is the process of determining the dimensions of physical quantities in order to check their relationships. All quantities in the world can be expressed as a function of the fundamental dimensions, which are independent of numerical multiples and constants.

Boltzmann Constant:
Boltzmann constant (k) $ = $Energy $ \times $${[Temperature]^{ - 1}}$ ….$(1)$
Since, Energy $ = $M$ \times $a$ \times $displacement$ = [{M^1} \times {M^0}{L^1}{T^{ - 2}} \times {L^1}]$
So, the dimensions of energy $ = [{M^1}{L^2}{T^{ - 2}}]$ ……$(2)$
Dimension formula of temperature $ = [{M^0}{L^0}{T^0}{K^1}]$ …..$(3)$
Using equation $(2)\,and\,(3)$ in equation $(1)$ we will get,
Boltzmann constant (k) $ = $Energy $ \times $${[Temperature]^{ - 1}}$
$k = [{M^1}{L^2}{T^{ - 2}}] \times {[{M^0}{L^0}{T^0}{K^1}]^{ - 1}} \\
\Rightarrow k= [{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}]$

Coefficient of Viscosity: So, coefficient of viscosity is given by,
$(\eta ) = F \times r \times {[A \times v]^{ - 1}}$ …..$(1)$
Where $F = $Tangential force, $r = $Distance between layers, $A = $Area , and $v = $velocity
Tangential Force $(F) = $Mass$ \times $Acceleration $ = $$M \times [L{T^{ - 2}}]$
Dimensional formula of force $ = {M^1}{L^1}{T^{ - 2}}$ …..$(2)$
Dimensional formula of Area and velocity $ = {L^2}$and ${L^1}{T^{ - 1}}$ …..$(3)$
Using Equation $(2)\,and\,(3)\,in\,equation\,(1)$ we will get,
Coefficient of viscosity$ = F \times r \times {[A \times v]^{ - 1}}$
$\eta = [ML{T^{ - 2}}] \times [L] \times {[{L^2}]^{ - 1}} \times {[{L^1}{T^{ - 1}}]^{ - 1}} \\
\Rightarrow \eta = [{M^1}{L^{ - 1}}{T^{ - 1}}]$

Planck's Constant: Planck's constant is the proportionality constant that connects the energy of a photon to the frequency of its corresponding electromagnetic wave.
The dimensional formula of energy $ = [{M^1}{L^2}{T^{ - 2}}]$
The dimensional formula of frequency $ = [{M^0}{L^0}{T^{ - 1}}]$
Planck’s Constant$ = \dfrac{{[{M^1}{L^2}{T^{ - 2}}]}}{{[{M^0}{L^0}{T^{ - 1}}]}}$
Planck’s Constant$ = [{M^1}{L^2}{T^{ - 1}}]$

Thermal Conductivity: Thermal conductivity is described as the flow of energy across a temperature gradient due to the random movement of molecules. In layman's terms, it's the measurement of a material's ability to conduct heat. The letter k stands for it.
$k = \dfrac{{QL}}{{A\Delta T}}$
$k$ is the thermal conductivity $(W{m^{ - 1}}{K^{ - 1}})$, $Q$ is the amount of heat transferred through the material $(J{s^{ - 1}})$, $A$ is the area of the body $({m^2})$ and $\Delta T$ is the temperature difference $(K)$
$k = \dfrac{{[M{L^2}{T^{ - 2}}][L]}}{{[{L^2}][K][T]}}$
$\Rightarrow k = \dfrac{{[M{L^3}{T^{ - 2 - 1}}]}}{{[K]}}$
$\Rightarrow k = \dfrac{{[ML{T^{ - 3}}]}}{{[K]}}$
$\Rightarrow k = [ML{T^{ - 3}}{K^{ - 1}}]$

Hence, the correct pairs are;

List I	List 2
Boltzmann Constant	$[{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}]$
Coefficient of viscosity	$[{M^1}{L^{ - 1}}{T^{ - 1}}]$
Plank Constant	$[{M^1}{L^2}{T^{ - 1}}]$
Thermal Conductivity	$[ML{T^{ - 3}}{K^{ - 1}}]$

Note: Let us see some more important points regarding dimensions. This approach is incapable of determining dimensional less quantities. This approach cannot calculate the proportionality constant. They can be discovered by experiment (or) theory. Trigonometric, logarithmic, and exponential functions are not relevant to this approach. This strategy will be difficult to use when physical quantities are reliant on more than three physical qualities.