Deriving the van der Waals equation of state involves modifying the ideal gas law to account for the finite size of gas molecules and the attractive forces between them. Here’s a step-by-step derivation:

1. Ideal Gas Law

   The ideal gas law is given by:

   PV = nRT

   Where:

   • P is the pressure,
   • V is the volume,
   • n is the number of moles,
   • R is the ideal gas constant,
   • T is the temperature.
2. Correction for Molecular Volume

   In an ideal gas, molecules are considered point masses. However, real gas molecules occupy a finite volume, reducing the space in which they can move. The effective volume is reduced by nb, where b is the volume excluded per mole of gas. Therefore, the corrected volume V' is:

   V' = V - nb
3. Correction for Intermolecular Forces

   Ideal gas molecules do not interact. In reality, gas molecules attract each other, reducing the pressure exerted on the container walls. This reduction in pressure is proportional to the square of the concentration (n/V). The corrected pressure P' is:

   P' = P + a(n/V)^2

   Where a is a constant that depends on the strength of the attractive forces between the molecules.

4. Van der Waals Equation

   Substitute the corrected pressure P' and corrected volume V' into the ideal gas law:

   P'V' = nRT

   Which gives:

   (P + a(n/V)^2)(V - nb) = nRT

   This is the van der Waals equation of state.

In summary, the van der Waals equation accounts for the volume occupied by gas molecules and the attractive forces between them, providing a more accurate description of real gases compared to the ideal gas law.

More information can be found on:

• LibreTexts - The van der Waals Equation of State
• FSU Chemistry - van der Waals Equation