Question

Write Vander Waals equation and explain the terms.

Answer 1

Hint: Vander Waal equation is a thermodynamic equation of state based on the theory that liquids are composed of particles with non-zero volumes but have interparticle attractive force. It is for real gases.

Complete step-by-step answer:
Vander Waals equation of state for real gases is obtained from the modification of the ideal gas equation or law. According to an ideal gas equation, \[PV = nRT\] where P is the pressure, V is the volume, T is the temperature, R is the universal gas constant and n is the number of moles of an ideal gas.
The Vander Waals equation considers the molecular interaction forces i.e. both the attractive and repulsive forces and the molecular size. There comes a volume correction in the Vander Waal equation. As the particles have a definite volume, the volume available for the movement of molecules is not the entire volume of the container but less than that.
Volume of an ideal gas is considered as an overestimation and therefore it is reduced in case of 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 in actual it is four times of it i.e. each particle has a sphere of influence 4 times its volume.
Volume correction for n particles (nb)= \[\dfrac{{4n \times 4}}{{3\pi {r^3}}}\]
Therefore, volume of real gases = \[{V_i}\] -nb
Now coming to pressure correction of Vander Waals equation:
For particles that are inside, the interactions cancel each other but the particles which are on the surface or near the wall of the container experience net pulling of the bulk molecules towards the bulk i.e. away from the surface. The molecules experiencing net interaction away from the walls will hit the walls with less pressure. That is why, real gases exhibit lower pressure than shown by the ideal gas.
Pressure of real gas = \[P - \dfrac{{a{n^2}}}{{V_i^2}}\]
On substituting these pressure and volume correction in ideal gas law equation, we get
 \[\left( {P - \dfrac{{a{n^2}}}{{V_i^2}}} \right)({V_i} - nb) = nRT\]
Where ‘a’ and ‘b’ are Van Der Waals constants and have positive values. They are the characteristics of an individual gas.

Note: When in case the correction factor of pressure and volume becomes negligible, the pressure and volume of real gas becomes equal to ideal gas. All real gases start to behave as ideal gases at low pressure and high temperature.