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in States of Matter - Gas and Liquid by (48.1k points)

How real gases are different from ideal gases? Derive van der Waal’s equation by pressure and volume modifications.

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Van der Waal’s Equation:

On the basis of modification of pressure and volume in kinetic theory of gases, van der Waals corrected ideal gas equation and gave modified form of ideal gas equation. This equation is known as van der Waal’s equation. The corrections made in this equation are following :

1. Correction in Pressure: The real gases deviate from ideal gas behaviour because of interaction of molecules. At high pressure, molecules do not strive the walls of the container with full impact these are dragged back by other molecules due to molecular attractive force. This affects the pressure exerted by the molecules on the walls of the container. Thus the pressure exerted by the gas is lower than the pressure exerted by ideal gas.

Hence,
P = Pi – p
Pi = P + p …(1)
Here p is the pressure correction.

The correction in pressure ‘p’ depends upon following two factors :

(i) Number of molecules of gas which exert force on striking molecules.

(ii) Number of molecules which strike the wall of container per unit area per second.
The decreasing in pressure p is directly proportional to square of density.

2. Volume Correction : van der Waals considered the molecules of real gases as inpentrable spheres having definite volume ‘b’ for ‘n’ number of moles, the restricted volume is ‘nb’. At high pressure, when the molecules are almost in contact, repulsive interactions become significant. Hence, the volume occupied by molecules also became significant because instead of moving in volume V, these are now restricted to volume (V – nb). It can be represented as follows :
Volume of ideal gas (Vi) = Volume of real gas – Volume of molecules of gas
Vi = V – b
Here b is exclude volume of 1 mol of real gas. for n moles Vi=V – nb
Excluded volume (b) is 4 times of volume of molecules of real gas. So for 1 mol of ideal gas,
PiVi = RT …(1)
for n moles
PiVi = nRT …(2)
where, Vi = ideal volume and Pi = ideal pressure.
Substituting the value of corrected pressure and corrected volume in eq. (2).

The equation (3) is van der Waal’s equation for n moles of gas, For n = 1 mol, the eq. (3) becomes

Distinction between Ideal gas and Real Gas
The differences between Ideal gas and Real Gas are given as below:

Ideal Gas Real Gas
1. It follows gas law at all temperature and pressure It follows gas law only at low pressure and high temperature
2. Volume of the molecule of gas is negligibly small in comparision to the space occupied by gas. 2. The volume of the molecules of a gas is not negligible in comparisoin to the volume occupied by gas
3. In all conditions of temperature and pressure, the attraction force between molecule is negligible At high pressure and low temperature, the attraction force between the molecules of gas is not negligible
4. Ideal gas obeys ideal gas equation
PV = nRT
Real gas obeys Van der Waal's equation

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