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Vanderwall’s equation of State- Thermodynamics

The Joule’s law ( dU/dP) = 0 is true for a perfect gas only as it is based on assumption that, no intermolecular attractions existed in gas. PV = RT is an ideal gas equation also based on the assumption that molecules are infinitesimally small in size and collisions are perfectly elastic. But this equation is insufficient to describe the behavior of real gases because experiments over a wide range of temperature shows that actually gas deviates from perfect gas equation. 
It indicates that these deviations are due to mutual attraction between the molecules, and due to finite size of molecules. 


(i) Corrections in Intermolecular Attractions :
When a molecule strikes the wall of the container, it is pulled back by other molecule and hence observed pressure of the gas at the walls of the container is less than the actual pressure. 
This decrease in pressure is proportional to :-
(a) the number of attracting molecules per unit volume. 
(b) the number of attracted molecules striking per unit area of the wall per second. 
These both factors depends upon density of the gas. Therefore  decrease in pressure :
       

(ii) Correction of finite size of molecules :

As the molecules  have finite size. Hence they occupy space. Hence the actual space or their movement in the equation can be given as :
     

             
Thus, the Vanderwall’s equation or real gas equation is transformed as :-

This above equation is called Vanderwall’s equation of state. 

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