Gas laws on the basis of kinetic theory

Boyle’s law

At a constant temperature the product of the pressure and volume for a given gas is always constant.
From kinetic theory of gas

$P=\frac{1}{3}\frac{mn}{V}\;\overline{c^{2}}$

$or\;\;\;\;\;PV=\frac{2}{3}\left(\frac{1}{2}\frac{mn}{V}\;\overline{c^{2}}\right)n$

If T remains constant, then average kinetic energy of molecule remains constant.
In a given volume the number of molecule of gas remains constant.

PV = constant

At constant pressure the volume of any gas is directly proportional to its absolute temperature.
From kinetic theory of gas

$P=\frac{1}{3}\frac{mn}{V}\;\overline{c^{2}}$

V ∝ T

At constant volume, the pressure of a gas is directly proportional to the absolute temperature.
Perfect gas equation

PV = RT

or P = (R / V) T

At constant temperature and pressure, equal volume of different gases have equal number of molecules.
Let m₁, n₁ and c₁ are mass per molecule, number of molecules per c.c. and r.m.s. velocity of one gas respectively.
And m₂, n₂ and c₂ are mass per molecule, number of molecules per c.c. and r.m.s. velocity of second gas respectively.
If two gases exert the same pressure P, then since P = (1/3)mnc²

$\therefore\;\;\;\;\;P=\frac{1}{3}m_{1}n_{1}c_{1}^{2}=\frac{1}{3}m_{2}n_{2}c_{2}^{2}$

$or\;\;\;\;\;m_{1}n_{1}c_{1}^{2}=m_{2}n_{2}c_{2}^{2}$

$\therefore\;\;\;\;\;\frac{1}{2}m_{1}c_{1}^{2}=\frac{1}{2}m_{2}c_{2}^{2}$

$or\;\;\;\;\;n_{1}=n_{2}$

It means equal volume of two gases at the same temperature and pressure have the same number of molecules, which is Avogadro’s hypothesis.

If a large number of gases available in a container do not chemically reacted, then the total pressure of the gas will be equal to the partial pressure of individual gases.
Let there are number of gases

$P=\frac{1}{3}\rho _{1}\overline{c_{1}^{2}}+\frac{1}{3}\rho _{2}\overline{c_{2}^{2}}+\frac{1}{3}\rho _{3}\overline{c_{3}^{2}}+...$

$or\;\;\;\;\;P=P_{1}+P_{2}+P_{3}+...$

Here P₁, P₂, P₃, … are the pressures, which each gas would exert it along occupies the whole volume.
This is called Dalton’s law of partial pressure.

According to it the rate of diffusion of a gas is inversely proportional to the square root of its density.

According to Maxwell the average velocity of molecules is directly proportional to the root mean square velocity of them.
But c_rms ∝ 1 / √ ρ …
Thus the average velocity of molecules is inversely proportional to the square root of density of gas.
Since the rate of diffusion of any gas depends on the average velocity of molecules.
Therefore the rate of diffusion of any gas is inversely proportional to the square root of the density of gas.
This is called Graham’s law of diffusion.

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