# Van Der Waals Equation Derivation

In 1873, Johannes Diderik Van Der Waals derived the Van Der Waals equation. The equation is considered as the updated version of the ideal gas law that states that there are some point masses present in gases that undergo perfectly elastic collisions. However, the real gas law is incapable of explaining the behaviour of real gases. Due to this reason, the Van Der Waals equation was derived to define the physical state of a real gas.

More importantly, the Van Der Waals equation tends to take into consideration the molecular size and the molecular interaction forces which can be attractive or repulsive forces. Sometimes it is also known as Van Der Waals equation of state. In this article, students will learn how to derive the Van Der Waals equation of state.

### What is Van Der Waals Equation?

Van Der Waals equation is an equation that is used to relate the relationship existing between the pressure, volume, temperature, and amount of real gases. For a real gas containing ‘n’ moles, the real gas equation derivation is as follows.

[ P + (an2  / V2)] [ V - nb] = nRT

Where,

P is the pressure,

V is the volume,

T is the temperature,

n is the number of moles of gases,

‘a’ and ‘b’ are the constants that are specific to each gas.

The equation can be written as:-

1. Cube power of volume:

V3 - [b + (RT / P)] V2 + ( a / P) V - ab / P = 0

1. Reduced equation (Law of corresponding states) in terms of critical constants:

( 𝜋 + 3 / φ2) ( 3φ - 1 ) = 8τ

Where:

𝜋 = P / Pc

φ = V/Vc

τ = T/Tc

The units of Van Der Waals constants are:

For unit ‘a’ = atm lit2 mol-2

For unit ‘b’ = litre mol-1

### Derivation of Van Der Waals Equation For Real Gases

It is easy to derive Van Der Waals equation for real gases but only if the right steps are followed. Any mistakes committed in the process to deduce Van Der Waals equation of state can be crucial and affect the whole process.

Let us discuss the process to derive the Van Der Waals gas equation.

In the case of a real gas when students are using Van Der Waals equation, the volume of a real gas is considered as (Vm - b), where b can be considered as the volume occupied by per mole.

Therefore, when the ideal gas law gets substituted with V = Vm  - b, it is given as :

P(Vm  - b) = nRT

Due to the presence of intermolecular attraction P was modified as follows.

(P + a / V2) ( Vm  - b)  = RT

( P + an2 / V2) ( V - nb) = nRT

Where,

• Vm: molar volume of the gas

• R: universal gas constant

• T: temperature

• P: pressure

• V: volume

Thus, it is possible to reduce Van Der Waals equation to ideal gas law as PVm = RT.

### Van Der Waals Derivation For One Mole of Gas

To derive the Van Der Waals equation or to deduce the Van Der Waal equation of state for one mole of gas can be turned into an easy process if the right steps are followed.

The steps for derivation of the real gas equation for one of gas are as follows.

p = RT / V = (RT / v) p = RT / Vm  - b

C = Na - Vm   (proportionality between particle surface and number density)

a'C2 = a' ( Na / Vm)2 = a / Vm2

p = RT / (Vm - b) - a /  Vm2  =>  [ p + (a /  Vm2)] [ Vm - b] = RT

[ p + ( n2a / V2) ] [ V - nb] = nRT

( substituting nVm  = V )

### Derivation of Real Gas Equation When Applied to Compressible Fluids

Van Der Waals equation derivation, when applied at compressible fluids, can be understood if the concepts are clear.

Compressible fluids like polymers have fluctuating specific volume and this can be expressed as follows.

(p + A)(V - B) = CT

Where,

P: pressure

V: Specific volume

T: Temperature

A, B, C: Parameters

All this above information is used to derive the Van Der Waals equation of state.

### Merits

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

• It is also applicable to fluids in spite of gases.

• 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 gases which are above the critical temperature.

• Below critical temperature results also get accepted.

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