## Kerala Plus One Physics Notes Chapter 13 Kinetic Theory

Introduction:

The behaviour of gases can be explained by the kinetic theory. This is possible as the inter-atomic force can be neglected for gases. The kinetic theory of gases is related to the three basic features of gases: volume, pressure and temperature. It was developed by Maxwell, Boltzmann and others in the nineteenth century.

Molecular nature of matter:

John Dalton proposed the scientific Atomic Theory. According to him, atoms are the smallest constituents of elements. All atoms of one element are identical, but atoms of different elements are different. As elements are often in the form of molecules. So this theory can also be called the molecular theory of matter. Every atom consists of protons, neutrons and electrons. Though the average properties like pressure, volume, temperature of a gas is constant. The atoms of the gas are colliding with one another and changing their speed during collision.

Behaviour of gases:

Properties of gases are easier to understand than those of solids and liquids. This is mainly because in a gas, molecules are far from each other and their mutual interactions are negligible except when two molecules collide.

At low pressures and high temperatures the gas follow the relation, PV = RT

where P – pressure, V – volume, T – temperature, R is a constant.

R= Nk_{B }N is the number of molecules of the gas, k_{B }is Boltzmann constant. k_{B} = 1.38 x 10^{-23}

JK^{-1 }PV = Nk_{B}T or PV = nRT. where n is the number of moles.

Avagadro’s Hypothesis:

Equal volumes of all gases under identical conditions of pressure and temperature would contain equal number of molecules.

“The number of moles in 22.4 litres of any gas is 6.02×10^{23}. This is known as Avogadro number and is denoted by N_{A}. N_{a}= 6.02 × 10^{23}

The mass of 22.4 litres of any gas is equal to its molecular weight in grams at S.T.P (standard temperature 273 K and pressure 1 atm). This amount of substance is called a mole. The ideal gas equation can be written as PV = nRT

where np is the number of moles and R = N_{A}k_{B} is a universal constant.

R = 8.314 J mol^{-1} MK^{-1}.

Ideal gas or Perfect gas:

A gas which obeys the ideal gas equation: PV = nRT, at all temperatures and pressures is called an ideal gas or perfect gas.

Note:

Its main characteristics are

- The size of the molecules is negligibly small.
- There is no force attraction or repulsion amongest its molecules.

Deviation from ideal behaviour of a real gas:

From the perfect gas equation, is a constant .The graph versus P must be straight line parallel to p axis as shown in figure.

But when we plot for real gases at different temperatures T_{1} , T_{2} , T_{3} we see deviation from this.

We noticed that all curves for real gases approach the ideal gas behaviour at low pressures and high temperatures.

Boyle’s Law :

Keeping temperature constant, pressure of a given mass of gas is inversely proportional to its volume. This is known as Boyle’s law.

Pα 1/v

PV = constant

∴ P_{1}V_{1}= P_{2} V_{2 }Graph shows comparison between experimental P-V curves and the theoretical curves predicted by Boyle’s law at three different temperatures T_{1} , T_{2} , T_{3} .

Once again, we noticed that the agreement of real gas behaviour with ideal gas

behaviour is good at low pressures and high temperatures.

Charles’law:

Keeping pressure constant, volume of the gas is directly proportional to the temperature of the gas. This is known as Charles’ law.

VαT

= constant

Given graph shows comparison between j experimental V-T curve and the theoretical curve predicted by Charles’ law at three different temperatures T_{1}, T_{2} , T_{3}.

From this, we observe that the agreement of real gas behaviour with ideal gas behaviour is good at low pressure and high temperature.

CAQ

Question 1.

Why do the gases at tow temperature and high pressure show large deviations from ideal behaviour?

Answer:

At low temperature and high pressure, the inter molecular attractions become appreciable. Moreover, the volume occupied by the gas molecules cannot be neglected in comparison to the volume of the gas. Hence the real gases show large deviations from ideal gas behaviour.

Dalton’s law of Partial Pressures:

‘The total pressure of a mixture of ideal gases is the sum of partial pressures exerted by the individual gases in the mixture”. This is famous the Dalton’s law of partial pressures

i.e., P = P_{1} + P_{2} +……………+P_{n}

Kinetic theory of an ideal gas:

- A gas consists of a very large number of atoms or molecules, which are perfect elastic spheres.
- Gas molecules randomly move all the time and collide with each other and with walls of container.
- Size of molecules is negligible compared to average distance between molecules.
- The molecules do not exert any force of attraction or repulsion on each other, except during collisions.
- The collisions of the molecules with themselves and with the walls of the vessel are perfectly elastic. As such, the momentum and the kinetic energy of the molecules are conserved during collission.
- The time of collision is of two molecule is negligible as compared to time interval between two successive collisions.
- Mean (or average) speed of molecules of a gas is defined as the arithmetic mean of the speeds of gas molecules.

where m is mass of 1 molecule of given gas and 7 = temperature of gas. - Root mean square of gas molecule is defined as the square root of the mean of the squares of the speeds of gas molecules.

CAQ

Question 2.

Calculate the r.m.s. velocity of air molecules at S.T.P. Given density of air at S.T.P. is 1.296 kg m^{-3 }Answer:

Here P= 1 atm = 1.013 × 10^{5} Nm^{-2}

ρ = 1.296 kgm^{-3 } Root mean square velocity of air molecules at S.T.P,

Pressure of an ideal gas:

According to kinetic theory of the gas, molecules collide with each other and with the walls of the container during their random motion.

When they collide with the walls of the container they exert force and thus pres¬sure. This pressure can be expressed as

But nm = ρ , the density of gas

where n = number of gas molecules per unit volume, m = mass of each molecule, ρ = density of the gas.

Kinetic interpretation of temperature:

The total average kinetic energy of all the molecules of a gas is proportional to its

absolute temperature (T). Thus, the temperature of a gas is a measure of the average kinetic energy U of the molecules of the gas.

According to this interpretation of temperature, the average kinetic energy U is zero at

T = 0, i.e., the motion of molecules stops at absolute zero.

Hence hydrogen gas will leak more rapidly because of its smaller molecular mass.

CAQ

Question 3.

When a gas is heated, its temperature increases. Explain it on the basis of kinetic theory of gases

Answer:

When a gas is heated, the root mean square velocity of its molecules increases As , so the temperature of the gas increases.

Degrees of freedom:

Degrees of freedom is the number of independent ways by which a molecule can possess kinetic energy of translation, rotation and vibration.

CAQ

Question 4.

A box contains equal number of molecules of hydrogen and oxygen. If there is a fine hole in the box, which gas will leak rapidly? Why?

Answer:

As

Hence hydrogen gas will leak more rapidly because of its smaller molecular mass.

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_{s} be the free paths travelled by the molecule in N successive collisions.

Then mean free path is given by,

d = molecular diameter, n = number of molecules per unit volume, T = absolute temperature, p = pressure

Law of equipartition energy:

It states that in any dynamical system in thermal equilibrium, the energy is equally distributed amongst its various degrees of freedom and the energy associated with each degree of freedom per molecule is ^{1}/2k_{B}T, where k_{B} is Boltzmann’s constant and T is the absolute temperature of the system.

Predicated value of specific heat capacities of gases:

Solids:

Consider a solid of N atoms, each vibrating about its mean position. According to law of equipartition of energy, average energy associated with an atom due to its oscillation in one dimensional,

Water:

Water molecule has three atoms two hydrogen and one oxygen.