The relationship between critical constants of the gases and their Van der Waal constants is as follows:
(i) Vc = 3b
(ii) Pc = a/27b2
(iii) Tc = 8a/27Rb
(iv) The critical compressibility factor Zc is given by,
These relations have been derived from the calculations based on Van der Waal's equation.
Boyle's Law
When the volume of the gas is reduced, the molecules move with the same speed but in lesser space. As a result, the frequency of collisions with the walls of the container increases and a pressure increase is noticed. This is the statement of Boyle's law and this can be deduced from the kinetic gas equation,
PV = (1/3) m Nu2
PV = (2/13) (1/2) mNu2, PV = (2/3) x N(1/2mu2)
At a fixed temperature the average kinetic energy 1/2 mu2
and the number of molecules N remain constant. Thus
PV = (2/3) x constant = constant
P a 1/v which is Boyle’s law
Charle's Law
When a gas is heated the molecules move faster increasing the pressure. But to maintain the pressure constant, the force of collision is compensated with an increase in volume. So, at constant pressure the volume of the gas increases with temperature. By kinetic gas equation we have,
PV = (1/3) mNu2
PV = (2/3) x N{(1/2)mu2} where p, (2/3) and N are constant
V = constant (1/2) mu2)
Since Kinetic energy {(1/2) mu2} is proportional to T,
substituting this we have,
V a T or (V/T) = constant
This is the statement of Charles' law.
Dalton's Law
When more than one type of molecule is present (at constant T) then, the total pressure exerted by the mixture should be equal to the sum of their individual pressures when present alone. The pressure exerted by each gas is
P1 = (2/3) (e1N1) and P2 = (2/3)( e2N2)
where e1 and e2 are the average molecular kinetic energies of the two gases containing N1 and N2 molecules per unit volume. Since the molecules behave independent of each other, e1 and e2 = e , hence
P total = (2/3) [e(N1+N2)] = (2/3)( e1N1) + (2/3)( e2N2),
= p1 + p2
The total pressure is equal to the sum of the pressures, which each gas would exert if allowed to occupy the entire space. This is Dalton's law of partial pressures.
Graham's Law of Diffusion
From kinetic theory,
PV = (1/3) m nu2
U2 = (3P/P) or mrms =
mrms = 1.085 mav
mav = 
mav a
Since the rate of diffusion of a gas is determined by the molecular speed,
Rate of diffusion (r) = constant x aav = constant = constant x 
The rate of diffusion of a gas depends inversely on the square root of its density. This is Graham's law of diffusion.
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