Question

The dimension of universal gas constant R is:
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: 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: \[PV=nRT\]

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
$PV=nRT$
Where,
$P$ Is the pressure
$V$ Is the volume
$n$ is the number of molecules.
$R$ is the universal gas constant
$T$ 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,
$R=\dfrac{PV}{nT}\quad ....\left( 1 \right)$
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,
$\begin{align}
  & P=\dfrac{F}{A} \\
 & \Rightarrow P=\dfrac{m\times a}{A} \\
\end{align}$
Where,
$m$ Is the mass of the body
$a$ is acceleration with which it is coming towards the body
$A$ is the area on which force is applied
So the dimension of pressure can be given as,
$P=[M{{L}^{-1}}{{T}^{-2}}]$
By considering dimensions of each quantity in equation (1),
$\begin{align}
  & R=\dfrac{[M{{L}^{1}}{{T}^{-2}}][{{L}^{3}}]}{[\theta ]} \\
 & \Rightarrow R=[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}] \\
\end{align}$
So, the dimension of a universal gas constant is given by,
$M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}$

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.