Question

The dimensions of universal gas constant $ R $ are
(A) $ {M^2}{L^2}{T^{ - 2}} $
(B) $ M{L^2}{T^{ - 2}}{\theta ^{ - 1}} $
(C) $ {M^2}{L^2}{T^{ - 2}}{\theta ^{ - 2}} $
(D) $ ML{T^{ - 2}}{\theta ^{ - 2}} $

Answer 1

Hint : To solve this question, we need to use the ideal gas equation. Then we have to separate the universal gas constant $ R $ in that equation and write it in the form of other quantities. Finally writing the dimensions of each of the quantities will give the final answer.

Formula used: The formula which is used in solving this question is given by,
 $ PV = nRT $ , where $ P $ is the pressure, $ V $ is the volume, $ n $ is the number of moles and $ T $ is the temperature.

Complete step by step answer
We know that the ideal gas equation is given by
 $ PV = nRT $
So the universal gas constant is given as
 $ R = \dfrac{{PV}}{{nT}} $
 $ \Rightarrow \left[ R \right] = \dfrac{{\left[ P \right]\left[ V \right]}}{{\left[ n \right]\left[ T \right]}} $ ………………...(1)
Now we write the dimensions of each of the quantities present in the above equation.
 $ \left[ P \right] = {M^1}{L^{ - 1}}{T^{ - 2}} $ , $ \left[ V \right] = {M^0}{L^3}{T^0} $ , $ \left[ T \right] = {M^0}{L^0}{T^0}{\theta ^1} $
As $ n $ is the number of moles, so it is dimensionless.
So from (1) the dimensions of the universal gas constant are given by
 $ \left[ R \right] = \dfrac{{\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]\left[ {{M^0}{L^3}{T^0}} \right]}}{{\left[ {{M^0}{L^0}{T^0}{\theta ^1}} \right]}} $
On simplifying, we finally get
 $ \left[ R \right] = M{L^2}{T^{ - 2}}{\theta ^{ - 1}} $
Hence the dimensions of the universal gas constant are $ M{L^2}{T^{ - 2}}{\theta ^{ - 1}} $ .
Hence, the correct answer is option B.

Additional Information
All the quantities of the physical world have their units and dimensions in terms of the combination of some fundamental or base quantities. These are the seven fundamental quantities in terms of which the dimensions of other physical quantities are expressed. These, along with their SI base units and dimensions are:
Mass – kg (kilogram) – $ \left[ M \right] $
Length – m (meter) – $ \left[ L \right] $
Time – s (second) – $ \left[ T \right] $
Electric Current – A (Ampere) – $ \left[ A \right] $
Temperature – K (Kelvin) – $ \left[ K \right] $
Luminous intensity – cd (candela) – $ \left[ {cd} \right] $
Amount of substance – mol (mole) – $ \left[ {mol} \right] $
(Square brackets [ ] round a quantity represents the dimensions of that quantity.)

Note
For solving such types of questions, it is required to obtain each physical quantity given in the problem in terms of the fundamental or base quantities. So we must know the physical formula which can relate the quantity with the fundamental quantities. It must be noted that there can be multiple such formulas and any one of them can be used for this purpose.