Kinetic Molecular Theory Of Gases

Kinetic Molecular Theory Of Gases

The various gas laws which we have studied so far were based an experimental facts.

The theory that provides an explanation for the various experimental observations about a gas is based on the Kinetic Molecular model. It is assumed that all gases are made up of identical molecules, which are in a state of constant random motion.

Postulates of the Model

The following are the postulates of the model:

$\bullet$A gas consists of a large number of identical molecules of mass m. The dimensions of these molecules are very small compared to the space between them. Hence the molecules are treated as point masses.

$\bullet$There are practically no attractive forces between the molecules. The molecules therefore more independently.

$\bullet$The molecules are in a state of ceaseless and random motion, colliding with each other and with the walls of the container. The direction of their motions changes only on collision. These collisions are known as elastic collisions in which the energy and momenta of the molecules are concerned. In non-elastic collisions these quantities are not conserved.

$\bullet$The pressure of a gas is the result of collision of molecules with the wall of the container.

$\bullet$The average kinetic energy of the colliding molecules is directly proportional to its temperature.

Velocities of gas molecules

Average Velocity

As per kinetic theory of gases, each molecule is moving with altogether different velocity. Let 'n' molecules be present in a given mass of gas, each one moving with velocity v₁, v₂, v₃… v_n. The average velocity or U_aV = average of all such velocity terms.

Average velocity = $\frac{u_1+u_2+u_2+\cdots u_n}{n}$ = $\frac{n_1{u}_1+n_2{u}_2+n_3{u}_3}{n_1+n_2+n_3\cdots }$

U_av = $=\sqrt{\frac{8RT}{\pi M}}=\sqrt{\frac{8PV}{\pi M}}=\sqrt{\frac{8P}{\pi b}}$

Root Mean Square Velocity

Maxwell proposed the term U_rms as the square root of means of square of all such velocities.

$U_{rms}^2$= $\frac{u_1^2+u_2^2+u_3^2+\cdots }{n}$ =$\frac{n_1{u}_1^2+n_2{u}_2^2+n_3{u}_3^2+}{n_1+n_2+n_3}$

Also U_rms $=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3PV}{M}}=\sqrt{\frac{3P}{d}}$

Most probable velocity

It is the velocity which is possessed by maximum no. of molecules.

V_mp = $=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2PV}{M}}=\sqrt{\frac{2P}{d}}$

Furthermore U_MP : U_AV : U_rms : :$\sqrt{\frac{2RT}{M}}$:$\sqrt{\frac{8RT}{\pi M}}$: $\sqrt{\frac{3RT}{M}}$

$\sqrt{2}$: $\sqrt{\frac{8}{\pi }}$ :$\sqrt{3}$

1 : 1.128 : 1.224

Also U_av = U_rms$\times$0.9213

Kinetic Energy of Gas

As per kinetic equation PV = $\frac{1}{3}mn{u}_{rms}^2$

For 1 mole m´ n = Molecular Mass (M)

$\therefore$ PV = $\frac{1}{3}M{u}_{rms}^2=\frac{2}{3}\times \frac{1}{2}M{u}^2=\frac{2}{3}\times K.E./mole$ or$\frac{K.E.}{mole}=\frac{3}{2}RT$.

Also $\frac{K.E.}{molecule}=\frac{3}{2}\frac{RT}{N}$ = $\frac{3}{2}$kT

Where k is the Boltzmann constant $\left(k=\frac{R}{N}\right)$

Illustration 1. Calculate rms speed of O₂ at 273 K and 1$\times$10⁵Pa pressure. The density of O₂ under these conditions is 1.42 kg m^–3.

Solution: Data are given in SI units

$C=\sqrt{\frac{3P}{d}}=\sqrt{\frac{3\times {10}^5}{1.42}}$= 459.63 m sec^–1

Kinetic Gas Equation

The kinetic gas equation has been derived on the basis of postulates of the kinetic theory of gases. The equation is:

PV = $\frac{1}{3}nN{u}^2$

Where, P = Pressure

V = Volume

m = Mass of gas molecules

N = No. of molecules in Volume V

u = Root mean square velocity (rms)

Kinetic Energy of Gas Molecules

The kinetic energy of gas molecules can be easily calculated on the basis of kinetic gas equation.

Average K.E. of one molecule = $\frac{1}{2}$mu²

K.E. of N molecules = $\frac{1}{2}$mNu² = $\frac{3}{2}\times \frac{1}{3}mN{u}^2$

= $\frac{3}{2}PV=\frac{3}{2}RT$ [$\because PV=\frac{1}{3}mN{u}^2$and PV = RT fare 1 mole of gas]

$\therefore$ R is constant

$\therefore$ KE of a gas $\propto$ T

Illustration 2. Calculate K.E. of 4g of N₂ at – 13°C.

Solution: E_K = $\frac{3}{2}nRT$ = $\frac{3}{2}\times \frac{4}{28}\times 8.314\times 260$

= 463.21 J mol^–1