JEE Physics: Gas and Kinetic Theory Complete Guide

The study of gases and kinetic theory is an important part of the JEE Physics syllabus. Understanding gas laws, molecular theory, and the statistical behavior of gas molecules forms the basis of many JEE problems. This detailed guide covers all the essentials of gas laws, kinetic theory, Maxwell-Boltzmann distribution, specific heat capacities, and related formulas and concepts to help you prepare effectively.

1. Introduction to Gases and Gas Laws

Gases are fluids that expand to fill their containers. The behavior of gases is described by empirical gas laws which relate pressure, volume, temperature, and amount of gas.

1.1 Boyle’s Law (Isothermal Process)

At constant temperature, the volume of a given mass of gas is inversely proportional to pressure:

\( \displaystyle P V = \text{constant} \quad \text{or} \quad P_1 V_1 = P_2 V_2 \)

1.2 Charles’s Law (Isobaric Process)

At constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature:

\( \displaystyle \frac{V}{T} = \text{constant} \quad \text{or} \quad \frac{V_1}{T_1} = \frac{V_2}{T_2} \)

1.3 Gay-Lussac’s Law (Isovolumetric Process)

At constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature:

\( \displaystyle \frac{P}{T} = \text{constant} \quad \text{or} \quad \frac{P_1}{T_1} = \frac{P_2}{T_2} \)

1.4 Avogadro’s Law

Equal volumes of gases at the same temperature and pressure contain equal number of molecules.

1.5 Ideal Gas Equation

Combining the above gas laws leads to the ideal gas equation:

\( \displaystyle PV = nRT \)

Where \( P \) = pressure, \( V \) = volume, \( n \) = number of moles, \( R = 8.314 \, \text{J/mol K} \), \( T \) = absolute temperature.

2. Kinetic Theory of Gases

Kinetic theory explains macroscopic properties of gases in terms of molecular motion and collisions.

2.1 Postulates of Kinetic Theory

Gas consists of a large number of identical molecules moving randomly in all directions.
The volume of molecules is negligible compared to the volume of the container.
Molecules undergo perfectly elastic collisions with each other and container walls.
There are no intermolecular forces except during collisions.
The average kinetic energy of molecules is proportional to the absolute temperature.

2.2 Pressure Exerted by Gas Molecules

Pressure arises from molecules colliding elastically with container walls. For \( N \) molecules of mass \( m \) in volume \( V \), average molecular speed squared is \( \overline{v^2} \). The pressure is:

\( \displaystyle P = \frac{1}{3} \frac{N}{V} m \overline{v^2} \)

2.3 Average Kinetic Energy and Temperature

The average kinetic energy per molecule is:

\( \displaystyle E_{\text{avg}} = \frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T \)

where \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \) is Boltzmann constant.

2.4 Root Mean Square Speed

The root mean square speed \( v_{\text{rms}} \) is:

\( \displaystyle v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{M}} \)

Where \( M \) is molar mass in kg/mol.

2.5 Average and Most Probable Speeds

Most probable speed: \( v_p = \sqrt{\frac{2RT}{M}} \)
Average speed: \( v_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}} \)

3. Maxwell-Boltzmann Distribution

This distribution describes the fraction of molecules having speed between \( v \) and \( v + dv \).

\( \displaystyle f(v) dv = 4 \pi \left(\frac{m}{2 \pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2 k_B T}} dv \)

The distribution curve shifts to higher speeds with increasing temperature.

4. Specific Heats of Gases

Specific heat is the heat capacity per unit mass or mole. For gases:

4.1 Molar Specific Heat at Constant Volume

\( \displaystyle C_V = \left( \frac{\partial Q}{\partial T} \right)_V \)

4.2 Molar Specific Heat at Constant Pressure

\( \displaystyle C_P = \left( \frac{\partial Q}{\partial T} \right)_P \)

4.3 Relation Between \( C_P \) and \( C_V \)

\( \displaystyle C_P - C_V = R \)

4.4 Ratio of Specific Heats \( \gamma \)

\( \displaystyle \gamma = \frac{C_P}{C_V} \)

Typical values: \( \gamma = \frac{5}{3} \) for monoatomic gases, \( \frac{7}{5} \) for diatomic gases.

5. Degrees of Freedom and Equipartition of Energy

Each molecule has degrees of freedom representing independent ways it can store energy: translation, rotation, and vibration.

5.1 Equipartition Theorem

Each degree of freedom contributes \( \frac{1}{2} k_B T \) energy per molecule.

5.2 Application to Specific Heat

For \( f \) degrees of freedom,

\( \displaystyle C_V = \frac{f}{2} R \quad ; \quad C_P = C_V + R = \frac{f + 2}{2} R \)

6. Real Gases and Van der Waals Equation

Real gases deviate from ideal gas behavior due to molecular size and intermolecular forces.

6.1 Van der Waals Equation

\( \displaystyle \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \)

where \( a \) accounts for attractive forces, and \( b \) for finite molecular volume.

6.2 Critical Constants

Critical temperature \( T_c \), pressure \( P_c \), and volume \( V_c \) characterize gas-liquid transition.

7. Thermodynamic Processes of Gases

Isothermal: \( PV = \text{constant} \)
Adiabatic: \( PV^\gamma = \text{constant} \)
Isobaric: \( V/T = \text{constant} \)
Isochoric: \( P/T = \text{constant} \)

7.1 Work Done in Processes

Isothermal: \( W = nRT \ln \frac{V_f}{V_i} \)
Adiabatic: \( W = \frac{P_i V_i - P_f V_f}{\gamma - 1} \)

8. Mean Free Path

The mean free path \( \lambda \) is average distance traveled between collisions:

\( \displaystyle \lambda = \frac{1}{\sqrt{2} \pi d^2 \frac{N}{V}} \)

where \( d \) is molecular diameter and \( \frac{N}{V} \) number density.

9. Diffusion and Effusion

Diffusion: Movement of gas molecules from high to low concentration.

Effusion: Escape of gas molecules through a tiny hole without collisions.

9.1 Graham’s Law of Effusion

Rate of effusion \( r \) is inversely proportional to square root of molar mass:

\( \displaystyle \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \)

10. Important Formulas Summary

Concept	Formula	Remarks
Ideal Gas Equation	\( PV = nRT \)	Relates pressure, volume, temperature
Pressure from Kinetic Theory	\( P = \frac{1}{3} \frac{N}{V} m \overline{v^2} \)	Pressure due to molecular collisions
Average Kinetic Energy	\( E_{\text{avg}} = \frac{3}{2} k_B T \)	Energy per molecule proportional to temperature
Root Mean Square Speed	\( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \)	Speed measure in kinetic theory
Van der Waals Equation	\( \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \)	Real gas behavior
Mean Free Path	\( \lambda = \frac{1}{\sqrt{2} \pi d^2 \frac{N}{V}} \)	Average distance between collisions

11. Sample Problems and Solutions

Problem 1: Calculate RMS Speed of Oxygen Molecules at 300 K

Given molar mass of oxygen \( M = 32 \times 10^{-3} \, \text{kg/mol} \), find \( v_{\text{rms}} \).

Solution:

\( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3 \times 8.314 \times 300}{0.032}} \)

Calculating,

\( v_{\text{rms}} = \sqrt{233606.25} = 483.3 \, \text{m/s} \)

Problem 2: Work Done During Isothermal Expansion

One mole of an ideal gas expands isothermally at 300 K from volume 10 L to 20 L. Calculate work done.

Solution:

\( W = nRT \ln \frac{V_f}{V_i} = 1 \times 8.314 \times 300 \times \ln \frac{20}{10} \)

\( W = 2494.2 \times 0.693 = 1728.5 \, \text{J} \)

12. Tips to Score High in JEE Gas and Kinetic Theory Questions

Memorize and understand all gas laws and their derivations.
Practice problems on root mean square, average, and most probable speeds.
Understand the physical meaning of Maxwell-Boltzmann distribution.
Revise relations between specific heats and degrees of freedom regularly.
Focus on conceptual questions on real gas behavior and deviations.
Work on thermodynamic process calculations thoroughly.

13. Conclusion

Mastering gases and kinetic theory requires understanding core concepts, practicing formulas, and solving a variety of problems. This guide compiles everything you need to confidently handle this important JEE physics chapter. Regular revision and problem solving will enhance your command over the topic and boost your JEE score.

Keep practicing, stay curious, and success will follow. Good luck!