Question

What are the dimensions of gas constant?
A. $\left[ ML{{T}^{-2}}{{K}^{-1}} \right]$
B. $\left[ {{M}^{0}}L{{T}^{-2}}{{K}^{-1}} \right]$
C. $\left[ M{{L}^{2}}{{T}^{-2}}{{K}^{-1}}mo{{l}^{-1}} \right]$
D. $\left[ {{M}^{0}}{{L}^{2}}{{T}^{-2}}{{K}^{-1}} \right]$

Answer 1

Hint: Convert the derived physical quantities into fundamental physical quantities. First try to find the dimensional formula of the components of the other quantities in which we can express the gas constant. So, we can find the dimensional formula of gas constant from the dimension of the other quantities.

Complete step by step answer:
All the derived physical quantities can be expressed in terms of the fundamental quantities. The derived units are dependent on the 7 fundamental quantities. Fundamental units are mutually independent of each other.
Dimension of a physical quantity is the power to which the fundamental quantities are raised to express that physical quantity.
Ideal gas law is expressed as,
$PV=nRT$
Where, P is the pressure, V is the volume, n is the number of moles, T is the temperature and R is the gas constant.
So, we can express the gas constant as,
$R=\dfrac{PV}{nT}$
Now. dimension of volume, V = $\left[ {{M}^{0}}{{L}^{3}}{{T}^{0}} \right]$
And, dimension of pressure = $\dfrac{F}{A}=\dfrac{ma}{A}=\dfrac{\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right]}{\left[ {{M}^{0}}{{L}^{2}}{{T}^{0}} \right]}=\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-2}} \right]$
Here, m is the mass and a is the acceleration.
Dimension of n = $\left[ mo{{l}^{1}} \right]$
Dimension of temperature, T = $\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}}{{K}^{1}} \right]$
So, Now, we can find the dimension of the gas constant by putting the values of the above four quantities.
So, dimension of gas constant will be = $\dfrac{\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-2}} \right]\left[ {{M}^{0}}{{L}^{3}}{{T}^{0}} \right]}{\left[ mo{{l}^{1}} \right]\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}}{{K}^{1}} \right]}=\left[ {{M}^{1}}{{L}^{2}}{{T}^{-2}}{{K}^{-1}}mo{{l}^{-1}} \right]$
So, the correct option is (C)

Additional information:
The quantity R is called the gas constant with its universal value $R=8.314Jmo{{l}^{-1}}{{K}^{-1}}$

Note: Don’t try to remember the dimensional formula. You may get confused. Always express the derived quantities in terms of the fundamental quantities and you will get the dimension of quantities.