Kinetic Theory: Comprehensive NEET Physics Notes

1. Introduction to Kinetic Theory

1.1 Molecular Nature of Matter

• Atomic Hypothesis: Matter consists of tiny atomic particles in perpetual motion, attracting each other at small distances and repelling upon close contact. This fundamental idea explains various phenomena, including gas behavior, pressure, and temperature relations.

Did You Know?

The idea that matter is made up of atoms was proposed long ago in ancient India by the philosopher Kanada.

2. Behavior of Gases

2.1 Ideal Gas Law

• Formula: $PV=\mu RT=k_{B}NT$
• Explanation:
• • $P$: Pressure of the gas
  • $V$: Volume of the gas
  • $T$: Absolute temperature
  • $\mu$: Number of moles
  • $N$: Number of molecules
  • $R$: Universal gas constant, $R=8.314{\text{J mol}}^{-1}{\text{K}}^{-1}$
  • $k_B$: Boltzmann constant, $k_B=1.38\times 10^{-23}{\text{J K}}^{-1}$

NEET Problem-Solving Strategy:

To solve problems involving the ideal gas law, always ensure that the temperature is in Kelvin, and pressures and volumes are consistent in their units.

Example Problem:

• Problem: Calculate the pressure exerted by 2 moles of an ideal gas occupying a volume of 22.4 liters at 273 K.
• Solution:
• • Given: $\mu =2\text{mol}$, $V=22.4\text{L}$, $T=273\text{K}$, $R=8.314{\text{J mol}}^{-1}{\text{K}}^{-1}$
  • Using the formula $PV=\mu RT$,
  • $P=\frac{\mu RT}{V}=\frac{2\times 8.314\times 273}{22.4}\text{atm}$
  • $P=2\text{atm}$

2.2 Kinetic Interpretation of Temperature

• Formula: $\frac{1}{2}m{v}^2=\frac{3}{2}{k}_{B}T$
• Explanation:
• • This equation relates the average kinetic energy of a gas molecule to the absolute temperature of the gas.
  • $m$: Mass of the molecule
  • $v$: Speed of the molecule
  • $T$: Absolute temperature

Common Misconception:

Students often confuse the average speed of molecules with the root mean square speed. The root mean square speed gives a better representation of the molecule's energy.

Example Problem:

• Problem: Find the root mean square speed of nitrogen molecules at 300 K. Given, $M=28{\text{g mol}}^{-1}$.
• Solution:
• • Convert molecular mass to kg: $M=28\times 10^{-3}{\text{kg mol}}^{-1}$
  • $v_{rms}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3\times 8.314\times 300}{28\times 10^{-3}}}\text{m/s}$
  • $v_{rms}=517\text{m/s}$

2.3 Pressure of an Ideal Gas

• Formula: $P=\frac{1}{3}nm\overline{v^2}$
• Explanation:
• • $p$: Pressure exerted by the gas
  • $n$: Number density (number of molecules per unit volume)
  • $m$: Mass of one molecule
  • $\overline{v^2}$: Mean square speed of the molecules

NEET Tip:

In NEET problems, focus on simplifying the expressions by considering the symmetry in the problem or using averages when dealing with large numbers of molecules.

Quick Recap

• Ideal Gas Law: $PV=\mu RT$
• Kinetic Energy and Temperature: $\frac{1}{2}m{v}^2=\frac{3}{2}{k}_{B}T$
• Pressure in Kinetic Theory:$P=\frac{1}{3}nm\overline{v^2}$

Concept Connection

• Link to Chemistry: The Ideal Gas Law is frequently used in Chemistry to calculate molar volumes and relate to chemical reactions involving gases.

Practice Questions

1. Question: Calculate the number of molecules in a gas occupying 10 liters at 300 K and 1 atm pressure. $R=8.314{\text{J mol}}^{-1}{\text{K}}^{-1}$.
2. • Solution: Use the Ideal Gas Law to find $N$.
2. Question: What is the root mean square speed of oxygen molecules at 500 K? Given $M=32{\text{g mol}}^{-1}$.
3. • Solution: Apply $v_{rms}=\sqrt{\frac{3RT}{M}}$.
3. Question: Derive the relation between pressure and volume for a given mass of gas when the temperature is constant (Boyle's Law).
4. Question: A gas is compressed to half its initial volume. How does this affect the pressure if the temperature remains constant?
5. Question: Compare the average kinetic energy of two gases at the same temperature, one being hydrogen and the other being oxygen.

These notes cover the fundamental aspects of Kinetic Theory as it relates to the behavior of gases. The explanations, examples, and practice questions are tailored to enhance your understanding and problem-solving skills for NEET.