Kinetic Theory: Comprehensive NEET Physics Notes

1. Molecular Nature of Matter

1.1 Introduction

Boyle discovered his law in 1661, and Newton and others tried to explain the behavior of gases by considering them as made of tiny atomic particles. The kinetic theory, established by Maxwell, Boltzmann, and others in the 19th century, explains the behavior of gases based on the idea of rapidly moving atoms or molecules. This theory provides a molecular interpretation of gas pressure and temperature, is consistent with gas laws and Avogadro’s hypothesis, and explains specific heat capacities of many gases.

1.2 Historical Development of Atomic Theory

Early Speculation: Ancient philosophers like Kanada in India and Democritus in Greece speculated that matter is composed of indivisible constituents called atoms.
Dalton's Atomic Theory: Dalton's atomic theory explained the laws of definite and multiple proportions. According to him, atoms of one element are identical but differ from those of other elements. Small numbers of atoms combine to form molecules of compounds.
Avogadro's Law: Avogadro hypothesized that equal volumes of all gases at the same temperature and pressure have the same number of molecules, explaining Gay Lussac's law of combining volumes.

1.3 Atomic Hypothesis

According to Richard Feynman, the discovery that matter is made up of atoms is a significant one. Atoms are little particles that move in perpetual motion, attracting each other when a little distance apart and repelling upon being squeezed together.

Did You Know?

The size of an atom is about an angstrom $1 0^{- 10}$ . In solids, atoms are spaced about a few angstroms apart.

2. Behavior of Gases

2.1 Ideal Gas Equation

The behavior of gases can be described by the ideal gas equation:

$P V = k T$

where $P$ is the pressure, $V$ is the volume, and $T$ is the temperature. For a sample of gas, $k$ is proportional to the number of molecules $N$ in the sample. We can write $k = N k_{B}$ , where $k_{B}$ is the Boltzmann constant.

2.2 Avogadro's Hypothesis

Avogadro's hypothesis states that the number of molecules per unit volume is the same for all gases at a fixed temperature and pressure. The number of molecules in 22.4 liters of any gas is $6.02 \times 1 0^{23}$ , known as Avogadro's number, $N_{A}$

2.3 Real and Ideal Gases

A gas that perfectly satisfies the ideal gas equation at all pressures and temperatures is defined as an ideal gas. No real gas is truly ideal, but real gases approximate ideal behavior at low pressures and high temperatures.

NEET Problem-Solving Strategy:

Use the ideal gas equation P $P V = n RT$ to solve problems involving pressure, volume, and temperature changes in gases. Remember to use absolute temperature (Kelvin).

Common Misconception:

Students often confuse the number of molecules with the number of moles. Remember, one mole of any gas contains Avogadro's number of molecules, regardless of the type of gas.

3. Kinetic Theory of an Ideal Gas

3.1 Molecular Motion and Collisions

The kinetic theory of gases is based on the molecular picture of matter, where gas consists of a large number of molecules in incessant random motion. Molecules collide elastically with each other and with the walls of the container.

3.2 Pressure of an Ideal Gas

The pressure $P$ of an ideal gas is related to the average kinetic energy of the molecules. Considering a cubic container with side $l$ :

$P = \frac{1}{3} nm ⟨ v^{2} ⟩$

where $n$ is the number density of molecules, $m$ is the mass of a molecule, and $l an g l e v^{2} ⟩$ is the mean squared speed of the molecules.

3.3 Kinetic Interpretation of Temperature

Temperature $T$ of a gas is a measure of the average kinetic energy of its molecules: $\frac{1}{2} m ⟨ v^{2} ⟩ = \frac{3}{2} k_{B} T$

This equation shows that temperature is directly proportional to the average kinetic energy of the molecules.

Real-life Application:

The kinetic theory explains why a balloon expands when heated: increasing temperature increases the average kinetic energy of the gas molecules, causing them to collide more frequently and forcefully with the balloon's walls, increasing its volume.

NEET Tip:

Always use absolute temperature (Kelvin) when applying the kinetic theory equations. Convert Celsius to Kelvin by adding 273.15.

4. Law of Equipartition of Energy

4.1 Degrees of Freedom

A molecule free to move in space needs three coordinates to specify its location. If it is constrained to move in a plane, it needs two coordinates, and if constrained to move along a line, it needs just one coordinate. Motion of a body from one point to another is called translation. Thus, a molecule free to move in space has three translational degrees of freedom.

4.2 Equipartition of Energy

According to the law of equipartition of energy, each degree of freedom contributes a term to the total energy. Each translational and rotational degree of freedom contributes a term that contains the square of some variable of motion.

For a diatomic molecule treated as a rigid rotator:

$E = \frac{1}{2} m v_{x}^{2} + \frac{1}{2} m v_{y}^{2} + \frac{1}{2} m v_{z}^{2} + \frac{1}{2} I_{1} ω_{1}^{2} + \frac{1}{2} I_{2} ω_{2}^{2}$

where $o m e g a_{1}$ and $o m e g a_{2}$ are the angular speeds about the axes of rotation, and $I_{1}$ and $I_{2}$ are the corresponding moments of inertia.

4.3 Application to Specific Heat

For monatomic gases, the molar specific heat at constant volume $C_{v}$ is:

$C_{v} = \frac{3}{2} R$

For diatomic gases treated as rigid rotators:

$C_{v} = \frac{5}{2} R$

NEET Exam Strategy:

Understand the concept of degrees of freedom and apply the law of equipartition of energy to calculate the specific heats of different types of gases. Remember that vibrational modes contribute twice as much to the energy due to both kinetic and potential components.

Quick Recap

Kinetic theory explains the behavior of gases using the idea of rapidly moving molecules.
Dalton's atomic theory and Avogadro's hypothesis laid the foundation for understanding the molecular nature of gases.
The ideal gas equation $P V = n RT$ describes the relationship between pressure, volume, and temperature.
Temperature is a measure of the average kinetic energy of gas molecules.
The law of equipartition of energy distributes energy equally among all degrees of freedom.

Practice Questions

Calculate the pressure of an ideal gas: A 2-liter container holds 1 mole of an ideal gas at 300 K. Calculate the pressure inside the container.
Determine the volume change: A gas occupies 10 liters at a pressure of 2 atm and a temperature of 300 K. If the pressure is reduced to 1 atm and the temperature is increased to 400 K, what will be the new volume?
Find the root mean square speed: Calculate the root mean square speed of oxygen molecules at 300 K. (Molecular mass of $O_{2}$ = 32 u)
Compare kinetic energies: Compare the average kinetic energy of hydrogen and oxygen molecules at the same temperature.
Analyze gas mixture: A mixture contains 3 moles of hydrogen and 2 moles of nitrogen at 300 K. Calculate the total pressure of the mixture in a 10-liter container.

Solutions:

Using $P V = n RT$ , where $R = 8.314 J/mol K$ :
$P = \frac{n RT}{V} = \frac{1 \times 8.314 \times 300}{2} = 1247.1 Pa$
Using $P V = n RT$ :
$\frac{P _{1} V _{1}}{T _{1}} = \frac{P _{2} V _{2}}{T _{2}}$