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.

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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.

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