Derive the van der Waals equation.

(N/A) The ideal gas equation $PV = nRT$ assumes that gas molecules are point masses with no intermolecular forces. Real gases deviate from this behavior due to molecular interactions and finite molecular volume.
$1$. Correction in Pressure: Real gas molecules experience intermolecular attractive forces. When a molecule approaches the wall of the container,it is pulled back by other molecules,reducing the impact force. The observed pressure $p$ is less than the ideal pressure $p_{ideal}$. The correction term is proportional to the square of the density,$\frac{n^2}{V^2}$. Thus,$p_{ideal} = p + \frac{an^2}{V^2}$,where $a$ is the van der Waals constant representing the magnitude of attractive forces.
$2$. Correction in Volume: Real gas molecules occupy a finite volume. The effective volume available for movement is $V - nb$,where $b$ is the excluded volume per mole,representing the volume occupied by the molecules themselves.
Substituting these into the ideal gas equation $p_{ideal} V_{ideal} = nRT$:
$(p + \frac{an^2}{V^2})(V - nb) = nRT$.