Question

Write Van der Waals equation for n moles of gas.

Answer 1

Hint: Van der Waals equation is a thermodynamic equation of state which is based on the theory that fluids are composed of particles with non-zero volumes, and subject to an interparticle attractive force.

Complete step by step solution:

Van der Waals equation of state for real gases is the modified form of ideal gas law. According to ideal gas law, PV = nRT where P is the pressure, V is the volume, n is the number of moles, T is the temperature and R is the universal gas constant.
The Van der Waals equation takes into consideration the molecular size and molecular interaction forces i.e. both the attractive and repulsive forces.
Volume correction by Van der Waals equation:
As the particles have a definite volume, the volume available for their movement is not the entire container volume but less.
Volume of an ideal gas is an overestimation and has to be reduced for real gases.
\[Volume\,of\,gas\,({{V}_{r}})=\dfrac{Volume\,of\,the\,container}{ideal\,gas\,({{V}_{i}})-Correction\,factor\,(b)}\]
Volume correction for each particle is not the volume of the particle but four times of it, e each particle has a sphere of influence 4 times its volume.
\[Volume\,correction\,for\,n\,particles\,(nb)=\dfrac{4n\times 4}{3\pi {{r}^{3}}}\]
Thus, volume of real gas = \[{{V}_{i}}-nb\]
Pressure correction of Van der Waals equation:
For inside particles, the interactions cancel each other. But the particles which are present on the surface and near the walls of the container do not have particles above the surface and on the walls. So, there will be net pulling of the bulk molecules towards the bulk that is away from the surface. The molecules experiencing a net interaction away from the walls will hit the walls with less force and pressure. Thus, real gases will exhibit lower pressure than shown by ideal gases.
Pressure of real gas = \[P-\dfrac{a{{n}^{2}}}{{{v}^{2}}}\]
Substituting the pressure and volume correction in the ideal gas equation
\[\left( P-\dfrac{a{{n}^{2}}}{{{v}^{2}}} \right)\left( {{V}_{i}}-nb \right)=nRT\]
where, ‘a’ and ‘b’ are Van der Waals constants and contain positive values. The constants are the characteristic of the individual gas.

Note: As the correction factor becomes negligible, pressure and volume of the real gases will be equal to that of ideal gases. Moreover, all real gases behave like ideal gases at low pressures and high temperatures.