Question

The dimensions of universal gas constant are
\[
  {\text{A}}{\text{. }}\left[ {{L^2}{M^1}{T^{ - 2}}{K^{ - 1}}} \right] \\
  {\text{B}}{\text{. }}\left[ {{L^1}{M^2}{T^{ - 2}}{K^{ - 1}}} \right] \\
  {\text{C}}{\text{. }}\left[ {{L^1}{M^1}{T^{ - 2}}{K^{ - 1}}} \right] \\
  {\text{D}}{\text{. }}\left[ {{L^2}{M^2}{T^{ - 2}}{K^{ - 1}}} \right] \\
\]

Answer 1

Hint: We know the ideal gas equation in which we have a product of pressure and volume of an ideal gas equal to the product of number of moles, universal gas constant and the temperature of the gas. If we know the dimensions of the various quantities in the equation, we can obtain the dimensions of universal gas constant.

Complete step by step answer:
The ideal gas equation is given as
$
  PV = RT \\
  R = \dfrac{{PV}}{T} \\
$
Here P is the pressure of the gas, V is the volume of the gas, R is the gas constant and T is the temperature of the gas and we have taken 1 mole of the ideal gas. Now dimensions of the various quantities that we already know are as follows:
Dimensions of P $ = \left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$
Dimensions of V $ = \left[ {{L^3}} \right]$
Dimensions of T $ = \left[ K \right]$
Now we can obtain the dimensions of the gas constant in the following way:
Dimensions of R $ = \dfrac{{\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]\left[ {{L^3}} \right]}}{{\left[ K \right]}} = \left[ {{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}} \right]$

Hence, the correct answer is option A.

Additional Information
Dimensional formula: A dimensional formula of a physical quantity is an expression which describes the dependence of that quantity on the fundamental quantities.
All physical quantities can be expressed in terms of certain fundamental quantities. Following table contains the fundamental quantities and their units and dimensional notation respectively.

No.	Quantities	unit	Dimensional formula
1.	Length	metre (m)	$\left[ {{M^0}{L^1}{T^0}} \right]$
2.	Mass	kilogram (g)	$\left[ {{M^1}{L^0}{T^0}} \right]$
3.	Time	second (s)	$\left[ {{M^0}{L^0}{T^1}} \right]$
4.	Electric current	ampere (A)	$\left[ {{M^0}{L^0}{T^0}{A^1}} \right]$
5.	Temperature	kelvin [K]	$\left[ {{M^0}{L^0}{T^0}{K^1}} \right]$
6.	Amount of substance	mole [mol]	$\left[ {{M^0}{L^0}{T^0}mo{l^1}} \right]$
7.	Luminous intensity	candela [cd]	$\left[ {{M^0}{L^0}{T^0}C{d^1}} \right]$

Note:
In this question, we have considered 1 mole of the gas in the ideal gas equation which has affected the units of gas constant. If we also consider the moles of the gas then the dimensions are given as $\left[ {{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}mo{l^{ - 1}}} \right]$.