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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.

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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" 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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    $\begingroup$ So you're saying that Ideal gases have lesser volume than real gases? The real gases have considerably large sized molecules and thus the net volume for them to move is lessened. Define "Volume" of a gas. Is it the space left empty ? or the space occupied by the particles? $\endgroup$ – Quark2 Mar 16 '16 at 2:52
  • $\begingroup$ I was regurgitating the assumptions behind Van der Waal's equation. An ideal gas is supposed to contain gas molecules which are points in space with no volume, not molecules of finite volume. So an "ideal gas" is supposed to occupy 22.4 liters at STP. But the "real gas" molecules occupy some of that volume, so the "empty" space is less than 22.4 liters. Look at a quick dimensional analysis. In the correction the factor $n$ is the number of moles of gas and $b$ is the volume per mole of the molecules. $\endgroup$ – MaxW Mar 16 '16 at 3:29
  • $\begingroup$ 22.4 liters is 22.4 liters. But if you put a mole of real gas let's say the molecules occupy 0.2 liter of the volume. Then the empty space is only 22.2 liters, but it should be 22.4. So you let the pressure drop and the gas expands to 22.6 liters. Now we have 0.2 liters for the molecules and 22.4 liters of empty space. $\endgroup$ – MaxW Mar 29 '16 at 8:04

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