Class 11 Physics | Chapter: Kinetic Theory (अणुगति सिद्धांत)

Quick Revision Notes & Formula Sheet (MP Board)

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1. Fundamental Ideal Gas Laws (गैस के नियम)

• Boyle’s Law (बॉयल का नियम): At constant temperature ($T$), $P \propto \frac{1}{V} \implies P_1 V_1 = P_2 V_2$
• Charles’s Law (चार्ल्स का नियम): At constant pressure ($P$), $V \propto T \implies \frac{V_1}{T_1} = \frac{V_2}{T_2}$
• Gay-Lussac’s Law (गे-लुसाक का नियम): At constant volume ($V$), $P \propto T \implies \frac{P_1}{T_1} = \frac{P_2}{T_2}$
• Avogadro’s Law (आवोगाद्रो का नियम): Equal volumes of all gases under the same temperature and pressure contain an equal number of molecules ($N_1 = N_2$).

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2. Ideal Gas Equation (आदर्श गैस समीकरण)

An ideal gas strictly obeys all gas laws at all pressures and temperatures.

$$\mathbf{P V = n R T = N k_B T}$$

Where:

• $P$ = Pressure (दाब)
• $V$ = Volume (आयतन)
• $n$ = Number of moles (मॉल की संख्या) $= \frac{m}{M} = \frac{N}{N_A}$
• $R$ = Universal Gas Constant $= 8.314 \text{ J/(mol}\cdot\text{K)}$
• $N$ = Total number of molecules
• $N_A$ = Avogadro's Number $= 6.022 \times 10^{23} \text{ molecules/mol}$
• $k_B$ = Boltzmann Constant $= \frac{R}{N_A} = 1.38 \times 10^{-23} \text{ J/K}$

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3. Assumptions of Kinetic Theory of Gases (अणुगति सिद्धांत की परिकल्पनाएं)

1. A gas consists of extremely small particles called molecules.
2. Molecules are in a state of continuous, random motion in all directions.
3. The actual volume of molecules is negligible compared to the total volume of the gas.
4. There are no intermolecular attractive or repulsive forces between molecules.
5. Collisions between molecules and with the container walls are perfectly elastic (Kinetic Energy and Momentum are conserved).
6. The time spent during a collision is negligible compared to the time between two consecutive collisions.

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4. Pressure Exerted by an Ideal Gas (आदर्श गैस का दाब)

$$\mathbf{P = \frac{1}{3} \rho v_{rms}^2 = \frac{1}{3} \frac{N m}{V} v_{rms}^2}$$

Where:

• $\rho$ = Density of the gas $= \frac{m N}{V}$
• $m$ = Mass of a single molecule
• $v_{rms}$ = Root Mean Square speed of gas molecules

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5. Kinetic Interpretation of Temperature (ताप की गतिज व्याख्या)

• Average Kinetic Energy per Molecule: $$E_k = \frac{3}{2} k_B T$$

• Total Kinetic Energy of 1 Mole of Gas: $$E_{mole} = \frac{3}{2} R T$$

Key Takeaway: Temperature is a direct measure of the average translational kinetic energy of gas molecules. At $T = 0 \text{ K}$ (Absolute Zero), molecular motion completely stops ($v_{rms} = 0$).

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6. Speeds of Gas Molecules (गैस अणुओं की चाल)

1. Root Mean Square Speed ($v_{rms}$): $$v_{rms} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3 P}{\rho}}$$

2. Average Speed ($v_{avg}$): $$v_{avg} = \sqrt{\frac{8 R T}{\pi M}} = \sqrt{\frac{8 k_B T}{\pi m}}$$

3. Most Probable Speed ($v_{mp}$): $$v_{mp} = \sqrt{\frac{2 R T}{M}} = \sqrt{\frac{2 k_B T}{m}}$$

• Comparison / Ratio: $$v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1 : 1.128 : 1.225$$ $$\mathbf{v_{rms} > v_{avg} > v_{mp}}$$

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7. Degrees of Freedom ($f$) (स्वातंत्र्य कोटि)

The total number of independent coordinates or ways in which a system can possess energy.

Gas Atomicity	Degrees of Freedom ($f$)	Breakdown	Examples
Monoatomic (एकपरमाणुक)	3	3 Translational	$\text{He, Ne, Ar}$
Diatomic (Rigid) (द्विपरमाणुक)	5	3 Trans + 2 Rotational	$\text{O}_2, \text{N}_2, \text{H}_2$
Diatomic (Non-rigid/High T)	7	3 Trans + 2 Rot + 2 Vibrational	$\text{O}_2 \text{ at high temp}$
Polyatomic / Non-linear	6	3 Trans + 3 Rotational	$\text{H}_2\text{O}, \text{NH}_3, \text{CH}_4$

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8. Law of Equipartition of Energy (ऊर्जा के समविभाजन का नियम)

For any dynamical system in thermal equilibrium, the total energy is equally distributed among all its degrees of freedom.

• Energy associated with each degree of freedom per molecule $= \frac{1}{2} k_B T$
• Total Internal Energy of 1 mole of gas with '$f$' degrees of freedom: $$U = \frac{f}{2} R T$$

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9. Specific Heat Capacities of Gases (गैसों की विशिष्ट ऊष्मा)

• Molar Heat Capacity at Constant Volume ($C_v$): $$C_v = \frac{dU}{dT} = \frac{f}{2} R$$

• Molar Heat Capacity at Constant Pressure ($C_p$): $$C_p = C_v + R = \left(\frac{f}{2} + 1\right) R$$

• Mayer’s Relation (मेयर का संबंध): $$C_p - C_v = R$$

• Adiabatic Index / Ratio of Specific Heats ($\gamma$): $$\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$$

Summary Table for Specific Heats:

Type of Gas	$f$	$C_v$	$C_p$	$\gamma = C_p / C_v$
Monoatomic	3	$\frac{3}{2} R$	$\frac{5}{2} R$	$\frac{5}{3} \approx 1.67$
Diatomic (Rigid)	5	$\frac{5}{2} R$	$\frac{7}{2} R$	$\frac{7}{5} = 1.40$
Polyatomic (Non-linear)	6	$3 R$	$4 R$	$\frac{4}{3} \approx 1.33$

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10. Mean Free Path ($\lambda$) (माध्य मुक्त पथ)

The average distance traveled by a gas molecule between two successive collisions.

$$\mathbf{\lambda = \frac{1}{\sqrt{2} n \pi d^2}}$$

In terms of Pressure and Temperature: $$\mathbf{\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}}$$

Where:

• $n$ = Number density (number of molecules per unit volume $= N/V$)
• $d$ = Diameter of the gas molecule
• $P$ = Pressure of the gas
• $T$ = Absolute temperature

Factors affecting Mean Free Path:

1. $\lambda \propto \frac{1}{n}$ (Inversely proportional to molecular density)
2. $\lambda \propto \frac{1}{d^2}$ (Inversely proportional to square of molecular diameter)
3. $\lambda \propto T$ (Directly proportional to temperature at constant pressure)
4. $\lambda \propto \frac{1}{P}$ (Inversely proportional to pressure at constant temperature)

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Quick Revision Tips for Board Exams

• Derivation Alert: Practice the derivation of Pressure exerted by an ideal gas ($P = \frac{1}{3}\rho v_{rms}^2$) — extremely popular in MP Board long-answer questions.
• Definitions: Memorize definitions of Degrees of Freedom, Law of Equipartition of Energy, and Mean Free Path.
• Numerical Focus: Practice finding $v_{rms}$ at different temperatures and calculating ratios of $C_p/C_v$ for gas mixtures.