**Derivation of Vander Waals equation of State**

**or**

**Equation of State for Real (Imperfect) Gases**

Many attempts have been made in order to get an equation of state viz. applicable to real gases.

A number of equations of state have been suggested to describe the P-V-T relationship in case of real gases.

**The ****Van-der-Waals equation**** is the earliest and best known of them.**

**In 1873**, Van der Walls proposed an equation of state for real gas (imperfect gas), viz. Known as vanderwaals equation.

**He modified the ideal gas equation (PV=nRT) by suggesting the following points:**

**• Gas molecules were not mass points but behave like rigid spheres. **

**• There exists intermolecular forces of attraction between them.**

**In order to introduce these points the following corrections were made in the ideal gas equation :**

**1. Correction for the volume of gas molecule**

** In case of ideal gas **, we consider ( assume) the molecules of gas as

**mass points**that’s why we

**neglect**the volume of gas molecules in comparison to the volume of container(V).

** But in case of real gases**, the gas molecules are considered as

**rigid spheres**that’s why

*we can not neglect the volume of gas molecules.*So

**,**

__in order to get the ideal volume which is compressible__**van der Walls**suggested that a correction term

*nb***should be**

**substracted**

**from the total volume V.**

**∴ Corrected Volume= V – nb**

__Calculation of____ ____b__

__Calculation of__

__b__

**Consider 2 gas molecules as unpenetrable and incompressible spheres, each having radius r.**

**Thus the ****cente****r of these two spheres*** **can not approach each other ***more closely than distance*** ***2r***.*

** For this pair of molecules**, therefore,

**a sphere of radius 2r , having volume =**

**\( \frac{4}{3} \) \(𝝅(2r)^3\)**

This is known as

**excluded Volume**

**for the pair of molecules.**

**∴ Excluded**

**Volume per molecule**=

**half of \( \frac{4}{3} \) \(𝝅(2r)^3\)**

=

**\( \frac{1}{2} \)**×

**\( \frac{4}{3} \) \(𝝅(2r)^3\)**

**As the** **Actual volume of one gas molecule having radius r = ****\( \frac{4}{3} \) \(𝝅(r)^3\)**

**⇒****Excluded Volume per molecule= 4 × ****Actual volume of one gas molecule**

** Co-volume (b)**: The excluded Volume

**is known as the**

__per mole__**co-volume.**

**Excluded Volume per mole = N**

_{A }× Excluded Volume per molecule=

**N**

_{A }× 4 ×**\( \frac{4}{3} \) \(𝝅(r)^3\)**

**⇒b=**

**N**

_{A }× 4 ×**\( \frac{4}{3} \) \(𝝅(r)^3\)**

where **N _{A}** = Avagadro number

Thus the **compressibe volume per mole of gas** = **total volume – ****Excluded Volume per mole**

= **V – b**

**If the volume V of gas contains n moles then the excluded Volume** = **nb**

∴** ****Ideal volume which is compressible**** = ****V – nb**

*In this way Van der waal made correction in volume term by replacing the volume **V** by ideal compressible volume** V-nb *

**2. Correction due to Intermolecular Forces of Attraction **

__In case of ideal ga__** s equation**, it was

**assumed**that there are

**no intermolecular forces of attraction**

**between the molecules.**

**Actually it was not so**.

**×××**

**Because pressure will affected due to these forces of attraction between the molecules.**

**(**

**As Pressure = force/Area)**

**• When a molecule approaches the walls of the container, it**

**experiences attractive forces from the bulk of molecules behind it.**

**•**

**Thus**

__In case of real gases__**the molecules strike the wall with a**

**lower velocity**

**and hence exert a**

**lower**

**pressure**

**than they exert in case of ideal gases.**

**• Thus P**

_{ideal}> P_{real}(always)**So van der Waal take into account the effect of intermolecular forces of attraction in VWL equation.**

Thus In orderto get the ideal pressurevanderwaal made correction in the pressure term byaddingacorrection factor p’to the pressure of the gas.

p’is also calledinternal pressureof the gas

∴Corrected pressure = P + p’

__Calculation ____of ____correction factor____ p’__

__Calculation__

__of__

__correction factor__

__p’__

__ __

**The total inward attractive pull on the molecules which accounts for p’ is directly proportional to ****ρ ^{2}**

∵ There are two types of molecules A-type and B-type as shown in fig.

A-type is that single molecule which is about to strike the wall and B-type molecules are those which strike the wall at any instant.

Thus force of attraction exerted on both of these depends upon the number of molecules per unit volume in the bulk of the gas.

i.e., depends directly upon the density of the gas

**i.e., p’ ****∝ ****ρ ^{2}**

**As we know, density**

**∝**

**\( \frac{1}{V} \)**

**if there are n moles of gas occupying volume V then**

**ρ ∝**

**\( \frac{n}{V} \)**

∴

**p’**

**∝**

**\( \frac{n^2}{V^2} \)**

**p’=**

**\( \frac{an^2}{V^2} \)**

**Thus,Ideal pressure or ****Corrected pressure **

**= P + p’**

**= ****P+ ****\( \frac{an^2}{V^2} \)**

**Hence van der waals equation for n mole of real gas becomes:**

**\( \fbox{\( \left( {P + \frac{{an^2 }}{{V^2 }}} \right)\left( {V – nb} \right) = nRT \)} \)**

**where a & b are van der Walls constants.**

__Units of Vanderwaal constants a and b__

__Units of Vanderwaal constants a and b__

As p’= **\( \frac{an^2}{V^2} \)**

a= p’ **\( \frac{V^2}{n^2} \)**

__Unit of a__**= ****atm L²mol ^{-2}**

**As regards b, it is the incompressible volume per mole of a gas**

**∴ ****Unit of b = ****litre/mole**

__Significance of Vanderwaal constants a and b__

__Significance of Vanderwaal constants a and b__

**Significance of a :**

It is the measure of the magnitude of the intermolecular forces of attraction within the gas.

**a ∝ intermolecular forces of attraction**

i.e., greater the value of a, the greater is the intermolecular forces of attraction among the gas molecules.

**⇒**more internal pressure generate

**⇒**more is the value of p’

• **Value of a** for **H(0.024)** and **He(0.246)** are **very small ****because van der Walls forces are very weak in these gases.**

• **Value of a** for **NH _{3}(4.17),**

**CO**

_{2}(3.59), SO_{2}(6.71)**are very high**.

**Thus van der Walls forces in these gases are very strong.**

•

**As the value of a increases, intermolecular forces of attraction increases; and the liquification tendency also increases,**i.e.,

**Greater the value of a, greater is the ease with which a gas can be liquified**

**Significance of b :**

**b represents the Excluded Volume per mole of the gas. It is also called co-volume**

**It is the measure of effective size of the gas molecules.**

**b ****∝ size of gas molecules **

**i.e., more is the size of gas molecule, more will be the value of b, i.e., more volume will have to be excluded**