Question

At low pressure, Van der Waals equation is reduced to $ \left[ {{\text{P}} + \dfrac{{\text{a}}}{{{{\text{V}}^{\text{2}}}}}} \right]{\text{V}} = {\text{RT}} $ . The compressibility factor can be given as:
(A) $ 1 - \dfrac{{\text{a}}}{{{\text{RTV}}}} $
(B) $ 1 - \dfrac{{{\text{RTV}}}}{{\text{a}}} $
(C) $ 1 + \dfrac{{\text{a}}}{{{\text{RTV}}}} $
(D) $ 1 + \dfrac{{{\text{RTV}}}}{{\text{a}}} $

Answer 1

Hint: To answer this question, you must recall the Van der Waal equation, also known as the real gas equation. The compressibility factor is an important thermodynamic quantity that is used in modifying the ideal gas law to make it valid for real gases.
Formula used: The compressibility factor is given by, $ {\text{z}} = \dfrac{{{\text{PV}}}}{{{\text{RT}}}} $

Complete step by step solution
We know that the real gas equation is given by $ \left( {{\text{P}} + \dfrac{{{\text{a}}{{\text{n}}^{\text{2}}}}}{{{{\text{V}}^{\text{2}}}}}} \right)\left( {{\text{V}} - {\text{nb}}} \right) = {\text{nRT}} $
In the question, we are given that the real gas equation is reduced to a form $ \left[ {{\text{P}} + \dfrac{{\text{a}}}{{{{\text{V}}^{\text{2}}}}}} \right]{\text{V}} = {\text{RT}} $
Rearranging this equation to find the compressibility factor $ \left( z \right) $ , we get,
 $ \Rightarrow {\text{PV + }}\dfrac{{\text{a}}}{{\text{V}}} = {\text{RT}} $
So, $ {\text{z}} = \dfrac{{{\text{PV}}}}{{{\text{RT}}}} = \left( {1 - \dfrac{{\text{a}}}{{{\text{RTV}}}}} \right) $
Thus, the correct answer is A.

Note
An ideal gas is one that follows the postulates of the kinetic molecular theory of gas. The postulates are given as:
A gas is composed of a large number of very tiny spherical particles (atoms or molecules, and the particles are present far from each other. All the particles are identical to each other
The volume of the particles is negligible as compared to the total volume occupied by the gas
The particles of the gas are moving constantly. This constant motion also leads to collisions amongst the particles and also of the particle with the walls of the container. All these collisions are assumed to be completely elastic. The pressure exerted by the gas is due to these collisions with the walls of the container.
There are no forces of attraction or interaction between the particles of a gas.
Although, in a real gas, the intermolecular forces as well as the volume occupied by the gas particles are significant enough. Thus, the real gas equation was given so as to compensate for the factors missing in the ideal gas equation