Write the Van der Waals equation for a real gas. Explain the correction term for pressure and volume.

Explanation

The vander equation for a real gas is
`(P+(an^(2))/(V^(2)))(V-nb)=nRT`
Pressure correction : the pressure of a gas is directly proportional to the force created by the bombardment of molecules on the walls of the container. The speed of a molecule moving towards the wall of the container is reduced by the attractive Force exerted by is neighbours. Hence , the measured gas pressure is lower than the ideal pressure of the gas. Hence, van der waals introduced a correction term to this effect .

Van der Waal' s found out the forces of attraction experienced by a molecule near the wall are directly proportional to the square of the the density of the gas
`P'" a "r^(2)`
`rho=n/V`
When n is the number of moles of gas and V is the volume of the container
`impliesP'alpha(n^(2))/(V^(2))`
`P'=(an^(2))/(V^(2))`
Where a is proportionality. Constant and depends on the natural of gas
Therefore ,
`P_("ideal")=P+(an^(2))/(V^(2))`
Volume correction: As very individual molecule of a gas occupies a certain volume. The actual volume is less than the volume of the container, V Van der waals introduced a correction factor V ' to this effect . Let us calculate the correction term by. Considering gas molecules as spheres.

V = excluded volume
Excluded volume for two molecules
`4/3pi(2r)^(3)`
`=8(4/3pir^(3))=8v_(m)`
Where v is a volume of a single molecule Excluded volume for single molecule
`(8V_(m))/(2)=4V_(m)`
Excluded volume for n molecule= `n(4V_(m))=nb` Where b is van der waals constant
which is equal to `4V_(m)`
`impliesV=nb`
`V_("ideal")=V-nb`
Replacing the corrected pressure and volume in the ideal gas equation of state for real gases as below.
`(P+(an^(2))/(V^(2)))(V-nb)=nRT`
The constants a and b are van der waals constants and their values vary with the natural of the gas. It is an appointment formula for the non- ideal gas.