Question 50: Units , Dimensions and Error Analysis

Question

$$ 	ext { Match the LIST-I with LIST-II } $$

LIST-I
LIST-II

A.
$$	ext { Boltzmann constant }$$
I
$$\mathrm{ML}^2 \mathrm{~T}^{-1}$$

B
$$	ext { Coefficient of viscosity }$$
II
$$\mathrm{MLT}^{-3} \mathrm{~K}^{-1}$$

C
$$	ext { Planck's constant }$$
III
$$\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}$$

D
$$	ext { Thermal conductivity }$$
IV
$$\mathrm{ML}^{-1} \mathrm{~T}^{-1}$$

Choose the correct answer from the options given below:

Accepted Answer

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 [ÃÂ·]:
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.

Answer

A - III, B - IV, C - I, D - II

Answer

A - III, B - IV, C - II, D - I

Answer

A - III, B - II, C - I, D - IV

Answer

A - II, B - III, C - IV, D - I

Units , Dimensions and Error Analysis

Choose the correct answer: