Ideal Gas Equation and Van der Waals Equation

The Ideal Gas law or Ideal Gas Equation are the equations which represent the ideal gas. In 1834, Benoit Paul Emile Clapeyron was the first physicist and engineer who stated this gas law or Ideal Gas equation as the conjugation of Charle’s law, Gay-Lussac’s law, Boyle’s law and Avogadro’s law and can be represented wholly as;

PV=nRT

where, P represents pressure of the gas, V represents the volume of the gas, T represents the absolute temperature, R represents the constant of proportionality or the ideal gas constant and n represents the number of moles of the gas.

The shared electron pair that exists in between the two atoms which are covalently bonded is called bond pair. The other valence shell electron pairs of the bonded atoms which are not taking part in the process of bond formation are called lone pairs of electrons. The bond formed by sharing between the electron pairs and the electrons of other atoms is known as a shared pair of electrons.

In 1873, Van Der Waals made necessary corrections corresponding to the faulty assumptions of kinetic molecular theory in the ideal gas equation, PV=nRT and modified it in the form of a new equation known as Van Der Waals Equation of state and for one mole of gas the equation is represented as;

[P+ (a/V²)] (V – b) = nRT

where, a and b are the van der Waals constants.

Derivation of Ideal Gas Equation

Charle’s law, Boyle’s law and Avogadro’s law can be combined together to obtain a very important equation known as the ideal gas equation. This equation correlates pressure, volume, temperature and n i.e. number of moles of a gas. The equation can be derived as follows;

According to Boyle’s law, the volume of a given mass of a gas is inversely proportional to its pressure provided temperature is kept constant, i.e., 

V ∝ 1/P (at constant T and n)

According to Charle’s law, the volume of a given mass of a gas is directly proportional to its absolute temperature if pressure remains constant, i.e.,

V ∝ T (at constant P and n)

According to Avogadro’s law, the volume of a gas is directly proportional to its number or moles of both the pressure and temperature are constant, i.e.,

V ∝ n (at constant T and P)

On combining all the laws, we get;

V ∝ (1/P)Tn

Or

PV ∝ nT

Or

PV = nRT

where, R is the universal gas constant and has energy per degree per mole or work done per degree per mole as it’s SI unit.

Corrections and Derivations of Van Der Waals Equation

The corrections made by van der Waals and the derivation of the new Van Der Waals Equation are as follows;

1. Volume Correction: The molecules of a gas are of finite size and possess a definite volume. Their volume cannot be neglected in comparison to the total volume of gas. In ideal gas equation PV=nRT, the volume V of the gas is taken exactly equal to the volume of the container. This is not true for real gas. The actual volume of a gas should be less than the volume of the container in which it has been placed because the molecules of finite size and of finite volume are moving in the same space. The actual volume of a real gas can thus be obtained by excluding the volume occupied by the molecules. Van Der Waals’ introduced a correction factor b (for one mole of a gas) known as excluded volume to account for the space diminished due to finite size and finite volume of molecules. This excluded volume depends upon the nature of the gas. If V is the observed volume of one mole of a real gas, then the actual volume or corrected volume of a real gas is given by;

Corrected volume of the gas = V-b (for one mole of a gas)

2. Pressure Correction: Although the molecules of a real gas are very small, yet they exert appreciable attraction on one another. The presence of attractive forces affects the pressure exerted by the gas. The pressure of a gas is the force exerted by the molecules per unit area of the walls of the container, the observed pressure of the gas would be less than the ideal pressure i.e., the pressure that a gas should have exerted in the absence of attractive forces. In order to account for this decrease in pressure due to the presence of attractive forces, a correction term p is added to the observed pressure. Hence,

The Corrected Pressure = P+p

where P is the observed pressure of the gas.

For one mole of a gas, the value of p is given by,

p = a/V2

where V is the volume of the gas and a is a constant which depends upon the nature of the gas. Thus,

Corrected Pressure = P+a/V2 (for 1 mole of gas)

Substituting bothe the equations in PV=RT

[P + (a/V2)] (V – b) = RT

This is the Van Der Waals Equation of state for one mole of a gas.

For n moles of a gas, it is;

[P + (a/V2)] (V – b) = nRT

Conclusion

Ideal gas equation is the combination of several equations or laws like Charle’s law, Boyle’s law and Avogadro’s law and for n number of moles with R as the universal gas constant can be represented by; PV=nRT. In order to explain the behaviour of real gases, Van Der Waals’ Equation is derived in 1873 by Van Der Waals’ by modifying the ideal gas equation.