Van der Waals Equation Derivation

According to the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, T is the temperature, and R is the universal gas constant. The Van der Waals equation is also known as the Van der Waals equation of state for real gases which do not follow ideal gas law. The Van der Waals Equation derivation is explained below.

Derivation of Van der Waals equation

For a real gas, using the Van der Waals equation, the volume of a real gas is given as (Vm – b), where b is the volume occupied per mole.

Therefore, ideal gas law when substituted with V = Vm – b is given as:

\(\begin{array}{l}P(V_{m}-b)=nRT\end{array} \)

Because of intermolecular attraction, P was modified as below.

\(\begin{array}{l}(P+\frac{a}{V^{2}_{m}})(V_{m}-b)=RT\end{array} \)
\(\begin{array}{l}(P+\frac{an^{2}}{V^{2}})(V-nb)=nRT\end{array} \)

Where,

Vm: molar volume of the gas

R: universal gas constant

T: temperature

P: pressure

V: volume

Thus, Van der Waals equation can be reduced to ideal gas law as PVm = RT.

Recommended Videos

Van der Waals Equation Derivation for one mole of gas

Following is the derivation of Van der Waals equation for one mole of gas that is composed of non-interacting point particles which satisfies ideal gas law:

\(\begin{array}{l}p=\frac{RT}{V_{m}}=\frac{RT}{v}\end{array} \)
\(\begin{array}{l}p=\frac{RT}{V_{m}-b}\end{array} \)
\(\begin{array}{l}C=\frac{N_{a}}{V_{m}}\end{array} \)
(proportionality between particle surface and number density)

\(\begin{array}{l}{a}’C^{2}={a}'(\frac{N_{A}}{V_{M}})^{2}=\frac{a}{V_{m}^{2}}\end{array} \)
\(\begin{array}{l}p=\frac{RT}{V_{m}-b}-\frac{a}{V^{2}_{m}}\Rightarrow (p+\frac{a}{V^{2}_{m}})(V_{m}-b)=RT\end{array} \)
\(\begin{array}{l}(p+\frac{n^{2}a}{V^{2}})(V-nb)=nRT\end{array} \)
(substituting nVm = V)

Van der Waals equation applied to compressible fluids

Compressible fluids like polymers have varying specific volume which can be written as follows:

\(\begin{array}{l}(p+A)(V-B)=CT\end{array} \)

Where,

p: pressure

V: specific volume

T: temperature

A,B,C: parameters

This was the derivation of Van der Waals equation. Stay tuned with BYJU’S and learn various other Physics related topics.

Related Physics Articles:

Impact Of Temperature Kinetic Energy
Energy: The Property Of Objects Thermodynamics

 

Test your knowledge on Van der Waals equation derivation
Quiz Image

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button
Check your score and answers at the end of the quiz

Congrats!

Visit BYJU’S for all Physics related queries and study materials

Your result is as below

0 out of 0 arewrong
0 out of 0 are correct
0 out of 0 are Unattempted
PHYSICS Related Links
Ampere Circuital Law Bjt Configurations
12th Physics Circular Motion Pdf Refraction Of Light Through A Glass Slab
Uses Of Electricity In Daily Life Heating Effect Of Electric Current Class 7
Difference Between Comet And Meteor Scalar Definition
Strain Definition Physics Unpolarized Light

Comments

Leave a Comment

Your Mobile number and Email id will not be published.

*

*

close
close

Play

&

Win