Kinetic Molecular Theory of Gases

Table of Content

Postulates of Kinetic Molecular Theory
The Kinetic Equation
Velocities of Gas Molecules
Kinetic Energy of Gas
Related Resources

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 Kinetic Molecular Theory

Maxwell proposed the following postulates under the heading of kinetic theory of gases.

Each gas is made up of a large number of small (tiny) particles known as molecules.
The volume of a molecule is so small that it may be neglected in comparison to total volume of gas.
The molecules are never in stationary state but they are believed to be in chaotic (random) motion. They travel in straight line in all possible directions with altogether different but constant velocities. The direction of motion is changed by the collision with container or with other molecules.
The molecules are perfectly elastic and bear no change in energy during collision.
The effect of gravity on molecular motion is negligible.
The temperature of gas is a measure of its kinetic energy.
The pressure of the gas is thus due to the continuous collision of molecules on the walls of container. Pressure P = Force/Area.

Refer to the following video for kinetic molecular theories of gases

The Kinetic Equation

Maxwell also derived an equation on the basis of above assumption as

$pV = \frac{1}{3}mnu^2$

Where P = Pressure of gas

V = Volume of gas

m = mass of one molecule of gas

n = no. of molecules of gas

u = root mean square velocity of molecules

For 1 mole n = N is Avogadro number

m $\times$ N = Molecular mass M

$pV = \frac{1}{3}Mu^2$

$\Rightarrow u^2=\frac{3pV}{M}$

For 1 mole of an Ideal gas, pV = RT

so,

$u^2=\frac{3RT}{M}$

$\Rightarrow u^2=\sqrt{}\frac{3PV}{M}=\sqrt{}\frac{3RT}{M}$

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 =

$\frac{u_1+u_2+u_3+u_4......+u_n}{n}=\frac{n_1u_1+n_2u_2+n_3u_3+...+n_nu_n}{n_1+n_2+n_3+.........+n_4}$

$U_{av}=\sqrt{\frac{8RT}{\pi M}} =\sqrt{\frac{8pV}{\pi M}}$

Root Mean Square Velocity:-

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

$U^2_{rms}\frac{u_1^2+u_2^2+u_3^2+u_4^2......+u_n^2}{n}=\frac{n_1u_1^2+n_2u_2^2+n_3u_3^2+...+n_nu_n^2}{n_1+n_2+n_3+.........+n_4}$

also

$U_{rms}=\sqrt{\frac{3RT}{M}} =\sqrt{\frac{3pV}{M}}$

Most probable velocity:-

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

$U_{mp}=\sqrt{\frac{2RT}{M}} =\sqrt{\frac{2pV}{M}}$

Furthermore

$u_{mp}: u_{av}:u_{rms}:::\sqrt{\frac{2RT}{M}}:\sqrt{\frac{8RT}{\pi M}}:\sqrt{\frac{3RT}{M}} = \sqrt{2}:\sqrt{\frac{8}{\pi }}:1.224$

$\Rightarrow u_{mp}: u_{av}:u_{rms}:::1:1.128:1.224$

Kinetic Energy of Gas

As per kinetic equation

$pV = \frac{1}{3}mnu^2$

For 1 mole m × n = Molecular Mass (M)

$pV = \frac{1}{3}Mu^2=\frac{2}{3}\times \frac{1}{2}Mu^2=\frac{2}{3}\times \frac{2}{3}\times K.E./mole=\frac{3}{2}RT$

Also

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

Where k is the Boltzmann constant (k = R / N)

Solved Examples

Question 1:

Calculate rms speed of O₂ at 273 K and 1 × 10⁵Pa pressure. The density of O₂under these conditions is 1.42 kg m^–3.

Solution:

Data are given in SI units

C = √3P / d = √3 × 10⁵ / 1.42 = 459.63 m sec^–1

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Question 2:

At what temperature will the r.m.s. velocity of oxygen be one and half times of its value at N.T.P.?

Solution:

1/2 mc² = 3/2 KT

Suppose the temperature required is T¢ when the velocity will be 3/2 C

3/2C / C = √T'/T or, T' = 9/4 × 273 = 614.25°K

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Question 3:

Calculate K.E. of 4g of N₂ at – 13°C.

Solution:

E_K = 3/2 nRT = 3/2 × 4/28 × 8.314 × 260 = 463.21 J mol^–1

Question 1: Which of the following statements of kinetic molecular theory is incorrect?

a. The effect of gravity reduced the molecular motion .

b. The temperature of gas is a measure of its kinetic energy. KE µ absolute temperature.

c. The pressure of the gas is thus due to the continuous collision of molecules on the walls of container

d. The volume of a molecule is so small that it may be neglected in comparison to total volume of gas.

Question 2: In the equation $pV = \frac{1}{3}mnu^2$

u² represents

a. average velocity of molecules.

b. root mean square velocity of molecules.

c. most probable velocity of molecules.

d. final velocity of molecules.

Question 3: Kinetic energy of gas is given by

a. 3/2 nRT

b.1/2 nRT

c. 3/2 nKT

d. 1/2 nRT

Question 4: The velocity which is possessed by maximum no. of molecules is called

a. average velocity.

b. root mean square velocity.

c. most probable velocity.

d. escape velocity.

Q.1	Q.2	Q.3	Q.4
a	b	a	c

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