Van der Waals equation was derived by Johannes Diderik van der Waals in the year 1873. The equation is basically a modified version of the Ideal Gas Law which states that gasses consist of point masses that undergo perfectly elastic collisions. However, this law fails to explain the behavior of real gasses. Therefore, the Van der Waals equation was devised and it helps us define the physical state of a real gas.

More significantly, the Van der Waals equation takes into consideration the molecular size and molecular interaction forces (attractive and repulsive forces). Sometimes, it is also referred to as the Van der Waals equation of state.

Van der Waals equation is an equation relating the relationship between the pressure, volume, temperature, and amount of real gasses. For a real gas containing ‘n’ moles, the equation is written as;

Where, P, V, T, n are the pressure, volume, temperature and moles of the gas. ‘a’ and ‘b’ constants specific to each gas.

Van der Waals Equation Derivation

Van der Waals equation derivation is based on correcting the pressure and volume of the ideal gasses given by the Kinetic Theory of Gases. Another derivation is also used that is based on the potentials of the particles. Nonetheless, both derivations help us establish the same relationship.

Van der Waals Equation of State for Real Gases – Derivation

Kinetic theory of ideal gasses assumes the gaseous particles as –

• Point masses without any volume,
• Independent having no interactions and
• Undergo perfectly elastic collisions.

In practice, Van der Waals assumed that, gaseous particles –

1. Are hard spheres.
2. Have definite volume and hence cannot be compressed beyond a limit.
3. Two particles at close range interact and have an exclusive spherical volume around them.

Volume Correction in 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 in the ideal gas is hence an overestimation and has to be reduced for real gases.

Volume of the real gas VR = Volume of the container/ideal gas (VI) – Correction factor(b)

VR = VI – b.

If the particles are independent, then,

Total volume of the particle = number of particle x volume of one particle

=

But, the particles are not independent, they do interact.

Van der Waal considered two hard-sphere particles can come as close as to touch each other and they will not allow any other particle to enter in that volume as shown in the diagram.

The two sphere model, has a total radius of ‘2r’ (r is the radius of the sphere particle) and Volume of

=8×=8×volume of single particle.

Then, each of the two particles has a sphere of influence of 4 times the volume of the particle.

Volume correction for each particle is not volume of the particle but four times of it

=b=

Volume correction for ‘n’ particles

=nb=n

Volume (V) of the real gas = Vi – nb

Pressure Correction in Van der Waals Equation

Gaseous particles do interact. For inside particles, the interactions cancel each other. But, the particles 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 interactions or pulling of the bulk molecules towards the bulk that is away from the walls and surface. The molecules experiencing a net interaction away from the walls will hit the walls with less force and pressure. Hence, in real gases, the particles exhibit lower pressure than shown by ideal gases.

The reduction in pressure α square of the particle density in the bulk α (particle density/volume)2

ɑ

where, a is the proportionality constant.

Pressure of the real gas,

Pi= Pr +

Substituting the pressure and volume correction in the ideal gas equation, we get the Van der Waals equation for real gases as;

(Pr+)(Vi₋nb)=nRT

or in general,

(P+)(V₋nb)=nRT

Here, ‘a’ and ‘b’ are Van der Waals constants and contain positive values. The constants are the characteristic of the individual gas. When gas is ideal or that it behaves ideally then both the constant will be zero. Generally, a constant help in the correction of the intermolecular forces while the b constant helps in making adjustments for the volume occupied by the gas particles.

Merits and Demerits of Van Der Waals Equation of State:

Merits:

It can predict the behavior of gas much better and accurately than the ideal gas equation.

It is also applicable to fluids in spite of gasses.

The arrangement is made in a manner of cubic equation in volume.

The cubic equation can give three volumes which can be used for calculating the volume at and below the critical temperatures.

Demerits:

It can only get accurate answers for real gasses which are above the critical temperature.

Below critical temperature results also get accepted.

In the transition phase of gas, the equation is a failure.

Recommended Articles:

Potential Energy: Introduction, Definition, Types, Unit, And Derivation
Derivation Of Prism Formula
Derivation Of Schrodinger Wave Equation
Derivation of the Terminal Velocity
Derivation of Torsion Equation