Van der Waals equation is also known as Van der Waals equation of state for real gases which do not follow ideal gas law. According to 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 derivation is explained below.
Derivation of Van der Waals equation
For real gas, using Van der Waals equation, the volume of a real gas is given as (Vm – b), where b is volume occupied by per mole.
Therefore, ideal gas law when substituted with V = Vm – b is given as:
\(P(V_{m}-b)=nRT\)Because of intermolecular attraction P was modified as below
\((P+\frac{a}{V^{2}_{m}})(V_{m}-b)=RT\) \((P+\frac{an^{2}}{V^{2}})(V-nb)=nRT\)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.
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:
\(p=\frac{RT}{V_{m}}=\frac{RT}{v}\) \(p=\frac{RT}{V_{m}-b}\) \(C=\frac{N_{a}}{V_{m}}\) (proportionality between particle surface and number density) \({a}’C^{2}={a}'(\frac{N_{A}}{V_{M}})^{2}=\frac{a}{V_{m}^{2}}\) \(p=\frac{RT}{V_{m}-b}-\frac{a}{V^{2}_{m}}\Rightarrow (p+\frac{a}{V^{2}_{m}})(V_{m}-b)=RT\) \((p+\frac{n^{2}a}{V^{2}})(V-nb)=nRT\) (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:
\((p+A)(V-B)=CT\)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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