Question

The dimension of the 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 Use the ideal gas equation which gives a relation between the pressure, volume, temperature, number of moles, and the universal gas constant $R$ being the constant of proportionality for the equation. Use the dimensions of all the quantities in the equation and you will get the dimensions of the universal gas constant.

Complete Step by step solution
The ideal gas equation is given by
$PV=nRT$
Here, $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $T$ is the temperature, and $R$ is the universal gas constant, which is the constant of proportionality in this equation.
Therefore, we can write the equation of the gas constant $R$ as
$R=\dfrac{PV}{nT}$
Let us take the number of moles to be $1$ . Also, the number of moles is a dimensionless quantity.
Hence,
$R=\dfrac{PV}{T}$
The pressure is defined as force per unit area. And force is defined as mass multiplied by its acceleration.
Therefore, the dimensions of pressure will be $ML{{T}^{-2}}{{L}^{-2}}$ .
Volume is the cube of length. So, the dimensions of volume will be ${{L}^{3}}$.
Temperature is a fundamental quantity so its unit is $\theta $ .
Therefore, substituting these dimensions to the equation for the universal gas constant, we get the dimensions of $R$ as
Dimension of $R=\dfrac{ML{{T}^{-2}}{{L}^{-2}}{{L}^{3}}}{\theta }$
Simplifying, we get
Dimension of $R=M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}$

Therefore, option (B) is the correct answer.

Note
We have used dimensional analysis in this question. Dimensional analysis is the analysis of different physical quantities and writing their dimensions in terms of their base quantities. Dimensional analysis can be useful to check the correctness of an equation. It also helps to reduce the number of variables in an equation.