Maharashtra Board Class 12 Physics Chapter 3 Kinetic Theory of Gases and Radiation PDF Download

3. Kinetic Theory of Gases and Radiation

Can you recall?

1. What are different states of matter?

2. How do you distinguish between solid, liquid and gaseous states?

3. What are gas laws?

4. What is absolute zero temperature?

5. What is Avogadro number? What is a mole?

6. How do you get ideal gas equation from the gas laws?

7. How is ideal gas different from real gases?

8. What is elastic collision of particles?

9. What is Dalton's law of partial pressures?

3.1 Introduction

You have been introduced to the three common states of matter namely solid, liquid and gas. You have also studied the gas laws: Boyle's law, Charles' law, and Gay-Lussac's law. The ideal gas equation can be obtained from the three gas laws.

The volume V of a gas is inversely proportional to the pressure P, temperature being held constant. Separately, volume V and pressure P are directly proportional to temperature. In a nut shell:

Boyle's law: \(V \propto \frac{1}{P}\) at constant T --- (3.1)

Charles' law \(V \propto T\) at constant P --- (3.2)

Gay-Lussac's law: \(P \propto T\) at constant V --- (3.3)

All the three laws apply to fixed mass m of an enclosed gas.

Combining the three laws into a single relation for a fixed mass of gas yields ideal gas equation. Thus:

\(PV \propto T\), or \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\)

Expressing the fixed mass of gas in the above three laws in terms of number of moles n of gas:

\(PV \propto nT\), or \(PV = nRT\)

where number of moles \(n = \frac{\text{mass of the gas (m)}}{\text{molar mass (M)}}\) = \(\frac{N}{N_A}\)

(Molar mass is the mass of 1 mole of gas)

Here, proportionality constant R is the universal gas constant, having the same value 8.314 J mol\(^{-1}\) K\(^{-1}\), for all the gases. N is the number of molecules in the gas and N\(_A\) is the Avogadro number and is the number of molecules in one mole of gas.

Alternatively:

\(PV = Nk_BT\) --- (3.4)

where k\(_B\) is the Boltzmann constant. R and k\(_B\) are related by the following relation:

\(R = N_A k_B\) --- (3.5)

The laws of Boyle, Charles, and Gay-Lussac are strictly valid for real gases, only if the pressure of the gas is not too high and the temperature is not close to the liquefaction temperature of the gas.

A gas obeying the equation of state \(PV = nRT\) at all pressures, and temperatures is an ideal gas.

Equation of State: For a gas, its state is specified by a number of physical quantities such as pressure P, temperature T, volume V, internal energy E, etc. Hence, the equation relating these quantities is known as the equation of state.

Teacher's Note

An ideal gas is like a perfect student who always follows rules. In real life, gases sometimes break these rules when they are very cold or under very high pressure. Just like air in your room behaves like an ideal gas, but air in a compressed cylinder does not.

Exam Trick

Remember: PV = nRT is the magic formula for ideal gases. Write it down first in every gas problem. It is like the 2 × table in math - you must know it by heart.

Points to Remember

An ideal gas has molecules with no size and no forces between them.
Real gases behave like ideal gases only at low pressure and high temperature.
The gas constant R has the same value for all gases.
PV = nRT connects pressure, volume, number of moles, and temperature.
Boyle's, Charles', and Gay-Lussac's laws are three parts of the ideal gas equation.

3.2 Behaviour of A Gas

A stone thrown upwards in air reaches a certain height and falls back to the ground. Its motion can be described well with the help of Newton's laws of motion. A gas enclosed in a container is characterized by its pressure, volume and, temperature. This is the macroscopic description of the gas. You know that the particles of the gas (molecules) are in constant motion. Unlike in the case of motion of the stone, it is very difficult to understand the behaviour of a gas in terms of motion of a single particle (molecule). The number of particles in the gas is itself so large (a 10\(^{23}\) particles per m\(^3\)) that any attempt to relate the macroscopic parameters P, V, T and E with the motion of individual particles would be futile.

Hence, certain assumptions are made regarding the particles (molecules) of a gas, averages of physical quantities over the large number of particles involved are obtained and these averages are finally related to the macroscopic parameters of the gas. This is the approach of kinetic theory of gases.

Teacher's Note

Think of a gas like a crowd of people in a busy market. We cannot study each person, so we study the crowd as a whole. The same idea is used to study gases.

Exam Trick

Remember: We cannot track each molecule, so we use averages. It is like finding the average height of students in your class instead of measuring each one.

Points to Remember

A gas has about 10\(^{23}\) molecules in every cubic meter.
We use average values to study gases, not individual molecules.
Macroscopic means we see the gas as a whole, not its tiny parts.
Kinetic theory connects the motion of molecules to gas properties like pressure and temperature.
The stone's motion is easy to describe, but gas molecules are too many to track one by one.

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