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Question: The dimension of universal gas constant R is: A. \({{M}^{2}}{{L}^{2}}{{T}^{-2}}\) B. \(M{{L}^{2...

The dimension of universal gas constant R is:
A. M2L2T−2{{M}^{2}}{{L}^{2}}{{T}^{-2}}
B. ML2T−2θ−1M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}
C. M2L2T−2θ−2{{M}^{2}}{{L}^{2}}{{T}^{-2}}{{\theta }^{-2}}
D. MLT−2θ−2ML{{T}^{-2}}{{\theta }^{-2}}

Explanation

Solution

A representation of derived quantities by the fundamental units is known as dimension. This is commonly named a dimensional formula. The dimensions are denoted by capital letters such as mass (M), length (L), time (T), etc.

Formula used: PV=nRTPV=nRT

Complete step by step solution:
The ideal gas equation gives a relation between the pressure, volume, number of molecules, temperature, and the universal gas constant. According to the general gas equation
PV=nRTPV=nRT
Where,
PP Is the pressure
VV Is the volume
nn is the number of molecules.
RR is the universal gas constant
TT Is the temperature
This equation is a state of a hypothetical situation. It was created just for the simplicity and study of gases.
By rearranging the above equation. The universal gas constant can be written as,
R=PVnT....(1)R=\dfrac{PV}{nT}\quad ....\left( 1 \right)
As the number of molecules is in moles it does not have any dimension.
An external force applied per unit area is known as pressure.
Mathematically,
P=FA ⇒P=m×aA \begin{aligned} & P=\dfrac{F}{A} \\\ & \Rightarrow P=\dfrac{m\times a}{A} \\\ \end{aligned}
Where,
mm Is the mass of the body
aa is acceleration with which it is coming towards the body
AA is the area on which force is applied
So the dimension of pressure can be given as,
P=[ML−1T−2]P=[M{{L}^{-1}}{{T}^{-2}}]
By considering dimensions of each quantity in equation (1),
R=[ML1T−2][L3][θ] ⇒R=[ML2T−2θ−1] \begin{aligned} & R=\dfrac{[M{{L}^{1}}{{T}^{-2}}][{{L}^{3}}]}{[\theta ]} \\\ & \Rightarrow R=[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}] \\\ \end{aligned}
So, the dimension of a universal gas constant is given by,
ML2T−2θ−1M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}

So, the correct answer is “Option B”.

Note: Ideal gas equation is a good approximation for the study of the behavior of many gases in certain conditions. But this equation has many limitations. This law does not comment as to whether a gas heats or cools during compression or expansion.