# Van der waal’s equation

Van der waal’s equation

Thought the equation PV=RT was arrived at first experimentally and then theoretically, yet it fails to explain the behaviour of real gases. van der Waals attributed the deviation of real gases from gas equation to the following faulty assumptions of the kinetic theory:

(i)   The actual volume of the gas molecules is negligible as compared with the total volume of the gas.

(ii)  The gas molecules do not exert any appreciable attraction on each other.

Both these assumptions are not true at high pressure and low temperature. At high pressure the volume is reduced to a great extent and the actual volume of the molecules cannot be supposed as negligible under such conditions. Moreover, the molecules come closer and the attractive force between them should also be taken into consideration.

Van der Waals introduced the necessary corrections as follows:

(a)    Volume correction: At higher pressure, the volume is much reduced and at this state the volume of gas molecules no longer remains negligible in comparison with the total volume V occupied by the gas. Thus the free space available for motion of molecules is reduced. If v be the volume of the molecules at rest, effective volume when they are in motion must be greater. It has been calculated to be approximately four times the actual volume of the molecules.

I. e.

4v and is generally denoted by b. Therefore the actual volume available in which the molecules are free to move will be

= Total volume — Effective volume.

i.e., correct volume(V – b)

(b)   Pressure correction: The pressure of a gas is due to the hits of the molecules on the walls of the containing vessel. The attractive force between the molecules comes into play when the molecules are brought close together by compressing the gas. A molecule in the body of the gas is attracted in all the directions when forces acting in opposite directions cancel out, but a molecule, the boundary of the gas is subjected to an inward pull due to unbalanced molecular attraction.

In this way some of the energy of the molecule about to strike the wall of the vessel is used up in overcoming this inward pull. Therefore, it will not strike the opposite wall with the same force. The observed pressure consequently will be less than the ideal pressure. Therefore the ideal pressure, Pi will be equal to observed pressure P plus a pressure correction, Pa depending upon the attractive forces,

I.e. Pi = P + Pa

Now, force of attraction $\alpha$ number of molecules in the interior which are attracting (i.e. , density)

Total force of attraction $\propto (\text{density})^2 \propto \dfrac{1}{(\text{Volume})^2}$

Or, pressure due to force of attraction $\propto \dfrac{1}{(\text{Volume})^2}$

i.e., $Pa = \dfrac{a}{V^2}$

(where V is the volume of the gas and ‘a’ is the coefficient of attraction)

$\therefore \text{ideal pressure } = P + \dfrac{a}{V^2}$

Introducing, both these corrections, the equation becomes

$\left(P + \dfrac{a}{V^2} \right) (V- b) = RT \cdots (i) \\[3mm] \left(P + \dfrac{an^2}{V^2} \right) (V- nb) = nRT \cdots (ii)$

It is known as van der Waals equation. This equation explains the behaviour of real gases with great accuracy and also the deviations of gas laws from ideal behaviour.

The units of ‘a’ and ‘b’ are expressed in the units of pressure and volume as the unit of ‘a’ is  atm $L^2 mol^{-2}$ that of ‘b’ is $L mol^{-1}$ and

Significance of van der Waals Constants

(i) The value of ‘a’ is higher for easily liquefiable gases (as $SO_2,NH_3 ,H_2S,CO_2$ etc.) while lower for permanent gases (as $H_2, N_2, O_2$ , He etc.). This is due to the fact that liquefiable gas has greater intermolecular forces of attraction, therefore the value of ‘a’ is said to be a measure of the intermolecular forces of attraction. Order of liquification of some gases:

$\underset{a}{\text{Gas}} \underset{\to}{to} \underset{6.71}{SO_2} > \underset{6.5}{Cl_2} > \underset{4.17}{NH_3} > \underset{3.58}{CO_2} > \underset{1.39}{N_2} > \underset{1.36}{O_2} > \underset{0.24}{H_2} > \underset{0.24}{Ne} > \underset{0.03}{He}$

(ii) Since the constant ‘b’ is the effective volume of the gas molecules hence it indicates that the gas molecules are incompressible.

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