Van der Waals Equation

In the year 1873 Johannes Diderik van der Waals derived the Van der Waals equation. This equation is actually a revised version of ideal gas law. According to ideal gas law, the gases consist of point masses that undergo perfectly elastic collision. This law was not able to clearly explain the behaviour of real gases. So, this equation was revised and it now helps in defining the physical state of real gases. Van Der Waals equation takes into account the molecular size and molecular interaction forces like attractive and repulsive forces. This equation relates with a relationship between pressure, volume, temperature and amount of real gases. Therefore, for ‘ n ’ moles of a real gas the equation can be written as:

Units of Van der Waals Equation

• ‘a’ has a unit of atm lit² mol⁻².
• ‘b’ has a unit of litre mol⁻¹.

Van der Waals Equation Derivation

This equation is derived on the basis of correcting pressure and volume of the ideal gases given via Kinetic Theory of gases, and is also based upon the potential of  the particles.

Van der Waals equation of State as seen real gases

The kinetic theory of ideal gases presumes the gaseous particles like:

• Point masses that have no volume.
• Having no interactions is believed to be independent.
• They are able to go through perfectly elastic collisions.

Previously Van der Waals presumed the gaseous particles:

• Very hard sphere like
• Having definite volume and cannot be compressed after a certain limit
• Two particles when they are in close proximity to each other interact, and creates an exclusive spherical volume around them.

Volume Correction as seen in Van der Waals Equation

Since the particles have a definite volume so the volume available for their movement does not represent the entire container volume but less. Thus, the volume in ideal gas is overvalued and it needs to be reduced for real gases. ‘ VR ’ represents the volume of real gas which is equals to ‘ VI ’ which is the volume of an ideal gas or the volume of container, minus the correction factor (b):
V _R = V _I – b
Van der Waal observed that the two hard sphere particles could come as close as to each other, and would not allow any other particle to enter in that volume. The radius of these two particles is found to be ‘ 2r ‘. Thus the volume correction for ‘ n ‘ number of particles can be represented as:
b = 4 N_A . 4/3 π.r³
Here, N_A represents the Avogadro’s number and r is the radius of the particles.

Pressure Correction as seen in Van der Waals Equation

Gaseous particles are able to interact but the particles present inside, among them the interaction gets cancelled. Whereas the particles that are present on the surface or near the walls of the container do not contain particles above the surface and on the walls. This results in net interactions in the bulk molecules towards the bulk which is located away from the walls and the surface. The molecules that experience the  interactions away from the wall, hit the wall with little force and pressure. Therefore in real gases the particles have lower pressure than those shown by ideal gases. The equation for correction in pressure can be written as:
P_i = P_r + an² / V²
After substituting the pressure and volume corrections in ideal gas equation, gives us the Van der Waals Equation for real gases :
(P + an² / V²)(V – nb) = nRT
The constants a and b represent the characteristics of the individual gases. If both the gases are found to be ideal or if they behave ideally then the value of both the  constants will be zero.

Relation between Ideal Gas and Van der Waals Equation

The ideal gas equation is written as PV = nRT and Van der Waals Equation can be written as (P + an2 / V2)(V – nb) = nRT . At constant temperature a decrease in pressure and an increase in volume can be seen. So at low pressure volume will be larger, thus the correction factor in pressure becomes very small.
Volume of the gas will become larger as compared to the volume of the molecules (i.e. n, b) therefore the volume correction will also become very small.
Since the correction factor becomes small the pressure and volume of real gases becomes equal to that of the ideal gases. Also at low pressure and high temperatures all real gases behave like that of ideal gases.

Advantages of Van der Waals Equation

• This equation helps in predicting the behaviour of gases better than those of ideal gas equations.
• Not only this equation is valid for gases but also for all fluids.
• This equation helps in calculating the critical conditions of liquefaction and to derive an equation of Principle of Corresponding States.

Disadvantages of Van der Waals Equation

• Only above critical temperatures this equation can give more accurate results of real gases.
• The results can also be accepted below the critical temperatures.
• Below the critical temperatures this equation completely failed in the transition phase of gas to the liquid.

Conclusion

Our conclusion is that the Van der Waals Equation of State gives a critical point since the equation has a unique solution. In the equation V is very large, so (a/V2 ) becomes very small therefore it can be neglected. b can also be neglected in response to V. Thus, Van der Waals Equation now becomes PV = RT this is the main reason why at low pressure real gases behave like ideal gases.