Deduce (a) Graham's law and (b) Daltons law from Kinetic gas equation.

Explanation

A. Graham.s law: According to kinetic gas equation,
`PV = 1/3 mnc^(2)`
(or) `PV = 1/3 mc^(2)` (or) `c^(2) = (3PV)/m` But `V/m = 1/d`
`therefore c^(2) = (3P)/d` (or) `c = sqrt((3P)/d)`
At constant, pressure, c =-constant ` xx 1/sqrt(d)` (or) `c prop 1/sqrt(d)`
In the case of gases r.m.s. velocity (c) is directly-proportional to rate of diffusion. (r)
`therefore r prop 1/sqrt(d)` (At constant T and P)
i.e.. At constant temperature and pressure the rate of diffusion of a gas is inversely proportional to the square root of its density. This is Graham.s law,
Dalton.s Jaw of partial pressure : Consider a given mass of gas
(1) in a container of volume V.
Let number of molecules in the gas = `n_(1)`
Mass of each molecule = `m_(1)`, RMS velocity `= c_(1)`
Then according to kinetic gas equation.
Pressure `(P_(1)) = 1/3 (m_(1)n_(1)c_(1)^(2))/V`
Now- replace gas .(1) by gas (2)
Let number of molecules in the gas = `n_2`
Mass of each molecule = `m_2`
RMS velocity = `c_2`
then according to kinetic gas equation
Pressure `P_(2) = 1/3 (m_(2)n_(2)c_(2)^(2))/V`
Suppose, the two gases are mixed in the same container: ■Let the total pressure of the gas be P.
Then, `P = 1/3 (m_(1)n_(1)c_(1)^(2))/V + 1/3 (m_(2)n_(2)c_(2)^(2))/V`.
`therefore P = P_(1) + P_(2)`
This-is Dalton.s law of partial pressures, (i.e.) At constant temperature, the total pressure exerted by a mixture of gases which do not react chemically with each other is equal to the sum of partial pressures of the-individual gases. This is Dalton.s law of partial pressures.