## Introduction

**Gases** have negligible force of molecular
interaction. They have no shape and size and can be contained in vessels of any shape and size. So, **gases**
expands indefinitely and uniformly to fill the available space by
considering that gases are made up of atomic particles, many
scientists like **Boyle** and **Newton** tried to explain the behaviour of **gases**.

But the real theory was developed in the nineteenth century by **Maxwell** and **Boltzmann**. In this chapter we shall study some of the features of kinetic theory.

## Molecular Theory of Matter

According to the law of **definite proportions**, a given compound always contains same element in the same proportion, irrespective of the source. According to the laws of **multiple proportions** if two elements can combine to form more than one compound, the masses of one element that combine with a fixed mass of other element, are in the same ratio.

**John Dalton**, about 200 years ago, proposed the atomic theory. According to this theory

The smallest constituents of an element are atoms.

Atoms of one element are identical but differ from those of other elements.

A small number of atoms of each element combine to form a molecule of a compound.

## Behaviour of Gases

In gases molecules are far from each other and due to this the interatomic forces between the molecules is negligible except, when two molecules collide. Hence, the properties of gases are easier to understand than those of solids and liquids.

Gases satisfy a simple relation between pressure, temperature and volume at low pressure and high temperature, this relation is given by equation

PV = nRT

Where all symbols have their usual meaning.

**Read also:** Oscillations Class 11 Physics Notes Chapter 14

### (i). Avogadro’s Hypothesis

According to this hypothesis, "At a fixed temperature and pressure the number of molecules per unit volume is same for all gases".

The number of molecules in 22.4 litres of any gas is 6.02 × 10^{23}. This is known as Avogadro number and is denoted by N_{A}. The mass of 22.4 litres of any gas at S.T.P. (standard temperature 273 K and pressure 1 atm) is equal to its molecular weight which is equal to one mole.

### (ii). Boyle’s law

According to this law, keeping temperature constant, the pressure of a given mass of a gas varies inversely with volume. If n and T are fixed in ideal gas equation then, PV is constant.

`P\propto\frac{1}{V}`

PV = constant

**Read also:** Hydrocarbons Class 11 Notes Chemistry Chapter 13

### (iii). Charles’ law

According to this law, the volume (V) of a given mass of a gas is directly proportional to the temperature of the gas, provided pressure of the gas remains constant.

`V\propto T`

`\frac{V}{T}=`constant

### (iv). Gay Lussac’s Law

According to this law, the pressure P of a given mass of a gas is directly proportional to its absolute temperature T, provided the volume V of the gas remains constant.

`P\propto T`

`\frac{P}{T}=`constant

### (v). Dalton’s law of partial pressure

According to this law, the total pressure of a mixture of non-interacting ideal gases is the sum of partial pressures.

P = P_{1} + P_{2} + P_{3} + .....

### (vi). Graham’s Law of Diffusion

It states that rate of diffusion of a gas is inversely proportional to the square root of the density of the gas.

`r\propto\sqrt{\frac{1}{\rho}}`

Hence, denser the gas, the slower is the rate of diffusion.

**Read also:** Conceptual Questions for Class 11 Physics Chapter 13 Kinetic Theory

## Kinetic Theory of an Ideal Gas

**Kinetic theory** of gases is based on the molecules picture of matter. According to which

A given amount of gas is a mixture of very large number of identical molecules of the order of

**Avogadro’s number**.The molecules are moving randomly in all directions.

At ordinary temperature and pressure, the size of the molecules is very small as compared to the distances between them. Thus, the interaction between them is negligible.

The molecules do not exert any force of

**attraction**or**repulsion**on each other, except during collisions.The collisions of molecules against each other or with the walls of the container are perfectly elastic. Such that the momentum and the kinetic energy of the molecules are conserved during collisions, though their velocities change.

## Law of Equipartition of Energy

For a dynamic system in thermal equilibrium, the energy of the system is equally distributed amongst the various degrees of freedom and the energy associated with each degree of freedom per molecule is 1/2 kT, where k is Boltzman constant.

`<\frac{1}{2}mv^2> =\frac{1}{2}kT`

## Degrees of Freedom

Degrees of freedom of a system is defined as the total number of co-ordinates or independent quantities required to describe the position and configuration of the system completely.

Mono-, di-, and polyatomic (N) molecules have, 3,5 or (3N-K) number of degrees of freedom where K is the number of constraints.

## Specific Heat Capacity

Specific heat capacity is defined as the amount of heat energy required to raise the temperature of a gas by one degree celsius.

### (i). Monoatomic Gas

The molecule of a monatomic gas has only three translational degrees of freedom. Thus, the average energy of a molecule at temperature T is (3/2)kB T . The total internal energy of a mole of such a gas is

`U=\frac{3}{2}k\times T\times N_{A}=\frac{3}{2}RT`

The molar specific heat at constant volume, C_{v}, is

`C_{v}=\frac{dU}{dT}=\frac{3}{2}RT`

Molar specific heat for ideal gas at constant pressure C_{P} is given by .

`C_{p}=R+C_{v}`

`C_{p}=R+\frac{3}{2}RT`

`C_{p}=\frac{5}{2}RT`

`\gamma=\frac{C_p}{C_v}=\frac{5}{3}`

## (ii). Diatomic Gas

A diatomic molecule treated as a rigid rotator, like a dumbbell, has 5 degrees of freedom: 3 translational and 2 rotational. Using the law of equipartition of energy, the total internal energy of a mole of such a gas is

`U=\frac{5}{2}k\times T\times N_{A}=\frac{5}{2}RT`

The molar specific heats are then given by

`C_{v}=\frac{dU}{dT}=\frac{5}{2}R`

Molar specific heat for ideal gas at constant pressure C_{P} is given by .

`C_{p}=R+C_{v}`

`C_{p}=R+\frac{7}{2}RT`

`C_{p}=\frac{9}{2}RT`

`\gamma=\frac{C_p}{C_v}=\frac{9}{7}`

## (iii). Polyatomic Gas

In general, a polyatomic molecule has 3 translational, 3 rotational degrees of freedom and f vibrational modes. According to the law of equipartition of energy, the total internal energy for one mole of polyatomic gas can be calculated as

`U=(\frac{3}{2}kT+\frac{3}{2}kT+fkT)N_{A}`

`U=(3+f)RT`

`C_{v}=\frac{dU}{dT}=(3+f)R`

Molar specific heat for ideal gas at constant pressure C_{P} is given by .

`C_{p}=R+C_{v}`

`C_{p}=R+(3+f)R`

`C_{p}=(4+f)R`

`\gamma=\frac{C_p}{C_v}=\frac{(4+f)}{(3+f)}`

## Mean Free Path

Mean free path of a molecule in a gas is the average distance travelled by the molecule between two successive collisions.

If λ_{1}, λ_{2}, λ_{3},.... λ_{N} be the free paths tavelled by the molecule in N successiv colllision, then mean free path is given by

`λ=\frac{λ_{1}+λ_{2}+λ_{3}+...+λ_{N}}{N}`

`λ=\frac{1}{sqrt{2}n\pi d^2}`

## Summary

**Ideal gas :**An ideal gas or a perfect gas is that gas which strictly obeys the gas laws. The size of molecules of an ideal gas is zero. There is no force of attraction or repulsion among the molecules of an ideal gas.**Absolute zero :**Absolute zero of temperature may be defined as that temperature at which the root mean square velocity of the gas molecules reduces to zero.**Mean speed or Average speed**(`\vec{v}`) : It is the average speed with which the molecules of a gas move.**Root mean square speed :**It is defined as the square root of the mean of the squares of the random velocities of the individual molecules of a gas.**Degree of freedom :**The number of degrees of freedom of a dynamical system is defined as the total number of co-ordinates or independent quantities required to describe completely the position and configuration of the system.**Mean free path :**It is defined as the average distance traveled by a molecule between two successive collisions.**Boyle’s law :**According to it, for a given mass of an ideal gas at constant temperature, the volume of a gas is inversely proportional to its pressure, i.e., V ∝ 1/P if mass of gas and T = constant.**Charles’s law :**According to it, for a given mass of an ideal gas at constant pressure, volume of a gas is directly proportional to its absolute temperature i.e., V ∝ T, if m and P are constant.**Gay-Lussac’s law :**According to it, for a given mass of an ideal gas at constant volume, pressure of a gas is directly proportional to its absolute temperature, i.e., P ∝ T, if m and v are constant.**Avagadro’s law :**According to it, at same temperature and pressure, equal volumes of all the gases contains equal number of molecules, i.e., N_{1}= N_{2}, if P, V and T are same.**Graham’s law :**According to it, at constant pressure and temperature, the rate of diffusion of a gas is inversely proportional to the square root of its density i.e., Rate of diffusion ∝ 1/√p , if P and T = constant.**Dalton’s law :**According to it, the pressure exerted by a gaseous mixture is equal to the sum of partial pressure of each component present in the mixture. P = P_{1}+ P_{2}+ P_{3}+ ……Heat gets transferred due to temperature difference between the system and the surroundings.

No engine can have efficiency equal to 1 or no refrigerator can have co-efficient of performance equal to infinity.

Carnot engine is an ideal engine.