Real Gas Equation

When a real gas has a pressure lower than the expected value from ideal gas (because of the intermolecular force of attraction) , we have -


$$P\_{\text{ideal}}= P\_{\text{real}} + \frac{an^2}{V^2}$$ where a is the Van der Waal's constant for intermolecular forces of attraction between a gas.


Similarly, a real gas has a lower volume than the calculated volume in ideal gas. Here, when we say volume , we mean the empty space around molecules or atoms.
So, for volume, we have -
$$ V\_{\text{real}}=V\_{\text{ideal}}-nb$$
where b is the Van der Waal's constant for the volume of a molecule or atom.


Having said this, when we plug in these values into the ideal gas equation to get the real gas equation.


We put $\rightarrow$


$$ \left(P+\frac{an^2}{V^2}\right)(V-nb)=nRT $$


But shouldn't this be wrong? Because for pressure, we put $P\_{\text{ideal}}$ but for Volume we put $V\_{\text{real}}$. Shouldn't we put both $P\_{\text{ideal}}$, $V\_{\text{ideal}}$ so as to satisfy ideal gas equation? 


Why do we not do so? 


Please give a theoretical explanation along with a mathematical one if possible.




You have the volumes wrong. The ideal gas law is: $$\mathrm{ P\_{ideal}V\_{ideal} = nRT}$$


So: $$ V\_{\text{ideal}}=V\_{\text{real}}-nb$$ when compared to Van der Waal's equation. Thus the "ideal" volume is less than the "real" volume because the molecules take up space. At least that is the rationale.


I think the "truth" is closer to the fact that the extra terms allow more constants to be fit to a particular gas. There are a number of ["real gas"](https://en.wikipedia.org/wiki/Real_gas) equations basede on various assumptions about how a "real" gas would behave. Which equation works best for a particular gas is a matter of the temperature and pressure range. So you have to try each and just see which one works the best. 


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