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In the empirical demonstration of the Van der Waals equation at Khan Academy, they made a demonstration of the real volume, for example, it was said that it has to be larger than the ideal volume, since the size of the molecules is not disregarded, hence it would be:
$$ V_{\text{ideal}} = V_{\text{real}} + nb $$
Where, $ b $ was said to be related to the size of the molecule.
But why don't I also include intermolecular interaction? Depending on it, the volume may even be smaller.
Intermolecular attraction can reduce the volume, in the same way that increasing the pressure can lower the volume (Boyle's law).
One way to see the effect is to take the derivative of the volume, as given by the van der Waals (vdW) equation, with respect to the attractive parameter $a.$
The vdW equation is
$$\left(p+\frac{a}{\bar{V}^2}\right)(\bar{V}-b)=RT\tag{1}$$
Ignore for simplicity the excluded volume term $b,$ so that
$$p\bar{V}=RT-\frac{a}{\bar{V}}\tag{2}$$
Then the derivative of the volume wrt $a$ is
$$\left(\frac{\partial V}{\partial a} \right)_{p,T} = \frac{\bar{V}}{a-p\bar{V}^2}=-(RT)^{-1}\tag{3}$$
In essence what this result says is that the volume decreases as you increase the parameter $a.$
