Kinetic Theory Class 11 Physics Notes

The ideal gas equation combines Boyle's law (P ∝ 1/V at constant T), Charles' law (V ∝ T at constant P) and Avogadro's law (V ∝ n at constant P and T). R = 8.31 J mol⁻¹ K⁻¹ is the universal gas constant, and k_B = R/N_A ≈ 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant.

• Boyle's law: PV = constant at constant temperature.
• Charles' law: V/T = constant at constant pressure.
• Avogadro's law: equal volumes of gases at the same P and T contain equal numbers of molecules.

MCQ Questions

1 Mark Questions

1. 1
   State the main assumptions of the kinetic theory of gases.
2. 2
   Write the ideal gas equation and define each symbol.
3. 3
   State Boyle's law.

2 Mark Questions

1. 1
   What is an ideal gas? Under what conditions does a real gas approach ideal behaviour?
2. 2
   State Boyle's law, Charles' law and Avogadro's law and show how they combine into the ideal gas equation.
3. 3
   Find the number of moles in 44 g of CO₂. (Molar mass = 44 g mol⁻¹.)

3 Mark Questions

1. 1
   State the postulates of the kinetic theory of gases and explain why the ideal gas equation holds at low pressure and high temperature.
2. 2
   A gas occupies 2 m³ at a pressure of 10⁵ Pa and a temperature of 300 K. Find the number of moles and the number of molecules. (R = 8.31 J mol⁻¹ K⁻¹, N_A = 6.02 × 10²³.)

Using the average of the square of the molecular speed v², the pressure is P = (1/3)ρv²_avg, where ρ = Nm/V is the density. This derivation connects the macroscopic quantity pressure with the microscopic motion of the molecules.

• Pressure depends on the mean-square speed of the molecules, not their individual speeds.
• The same formula holds in three dimensions because each of the three velocity components contributes equally on average.
• Doubling the mean-square speed doubles the pressure at constant volume.

MCQ Questions

1 Mark Questions

1. 1
   Explain how pressure arises in the kinetic theory of gases.
2. 2
   Write the expression P = (1/3)ρv²_avg and identify the symbols.
3. 3
   On which molecular quantity does gas pressure depend?

2 Mark Questions

1. 1
   Derive the relation P = (1/3)ρv²_avg from the kinetic theory model.
2. 2
   The density of a gas is 1.5 kg m⁻³ and its molecules have a mean-square speed of 10⁶ m² s⁻². Find the pressure it exerts.
3. 3
   Show that the pressure of a gas is directly proportional to its density at constant mean-square speed.

3 Mark Questions

1. 1
   Use the kinetic theory to derive an expression for the pressure exerted by an ideal gas in terms of the mass, number density and mean-square speed of the molecules.
2. 2
   A gas has a density of 2 kg m⁻³ and exerts a pressure of 2 × 10⁵ Pa. Find the root-mean-square speed of its molecules.

At a given temperature all gases, regardless of molecular mass, have the same average translational kinetic energy per molecule (3/2)k_B T. Lighter molecules move faster to have the same energy — at the same temperature, hydrogen molecules are much faster than oxygen molecules.

• Temperature is proportional to the mean translational kinetic energy, not the total energy.
• At absolute zero all translational motion ceases in the classical model.
• Two gases at the same temperature have equal average translational KE per molecule.

MCQ Questions

1 Mark Questions

1. 1
   State the kinetic interpretation of temperature.
2. 2
   Write the expression for the average kinetic energy of a gas molecule.
3. 3
   Two gases at the same temperature: what is equal for their molecules?

2 Mark Questions

1. 1
   Show that the average kinetic energy per molecule of a gas equals (3/2)k_B T.
2. 2
   Find the average translational kinetic energy of a molecule at 300 K. (k_B = 1.38 × 10⁻²³ J K⁻¹.)
3. 3
   Why is the average translational kinetic energy of all gases the same at a given temperature?

3 Mark Questions

1. 1
   Derive the kinetic interpretation of temperature, starting from P = (1/3)ρv²_avg and the ideal gas equation.
2. 2
   Compute the total translational kinetic energy of 2 moles of an ideal gas at 400 K. (R = 8.31 J mol⁻¹ K⁻¹.)

The RMS speed is proportional to the square root of temperature and inversely proportional to the square root of molar mass. At 300 K, the RMS speed of nitrogen molecules (M = 28 g/mol) is about 517 m/s. For a given gas, raising the temperature increases the RMS speed, and for a given temperature, lighter gases have higher RMS speeds.

• v_rms ∝ √T at fixed mass.
• v_rms ∝ 1/√M at fixed temperature.
• The RMS speed is slightly larger than the mean speed and the most probable speed.

MCQ Questions

1 Mark Questions

1. 1
   Define the root-mean-square speed of gas molecules.
2. 2
   Write the expression for v_rms in terms of the molar mass M.
3. 3
   How does v_rms depend on temperature?

2 Mark Questions

1. 1
   Find the RMS speed of oxygen molecules at 300 K. (M = 32 × 10⁻³ kg mol⁻¹, R = 8.31 J mol⁻¹ K⁻¹.)
2. 2
   Compare the RMS speeds of hydrogen and oxygen at the same temperature.
3. 3
   If the temperature of a gas is raised from 300 K to 1200 K, by what factor does its RMS speed increase?

3 Mark Questions

1. 1
   Derive an expression for the root-mean-square speed of gas molecules in terms of T and molar mass.
2. 2
   At what temperature will the RMS speed of oxygen molecules equal that at which hydrogen has an RMS speed of 2000 m/s at 300 K? Derive your answer.

For a monatomic gas C_V = 3R/2 and γ = 5/3. For a diatomic gas (translational + rotational) C_V = 5R/2 and γ = 7/5. At very high temperatures the vibrational degrees are also excited and C_V rises.

• Monatomic (He, Ar): f = 3, C_V = 3R/2, γ = 5/3.
• Diatomic (N₂, O₂) without vibration: f = 5, C_V = 5R/2, γ = 7/5.
• Each degree of freedom contributes (1/2)R per mole to C_V.

MCQ Questions

1 Mark Questions

1. 1
   Define the degrees of freedom of a gas molecule.
2. 2
   State the law of equipartition of energy.
3. 3
   Write the value of C_V for a monatomic gas.

2 Mark Questions

1. 1
   Find the number of degrees of freedom of (a) monatomic and (b) diatomic gas molecules.
2. 2
   State the law of equipartition of energy and hence write C_V and C_P for a monatomic gas.
3. 3
   For a diatomic gas, using f = 5, find C_V, C_P and γ.

3 Mark Questions

1. 1
   State the law of equipartition of energy and use it to obtain the specific heats and γ for a monatomic and a diatomic gas.
2. 2
   Compute the internal energy of 3 moles of a monatomic gas at 400 K. (R = 8.31 J mol⁻¹ K⁻¹.)

The three speeds are related by v_mp : v_mean : v_rms = √2 : √(8/π) : √3 ≈ 1 : 1.128 : 1.225. As temperature rises, the distribution broadens and shifts to higher speeds, and the most probable, mean and RMS speeds all increase.

• The most probable speed is the speed possessed by the largest number of molecules.
• The distribution spreads out with increasing temperature.
• At the same temperature, lighter gases have their distribution peak at higher speeds.

MCQ Questions

1 Mark Questions

1. 1
   What is the Maxwell speed distribution?
2. 2
   Define the most probable speed.
3. 3
   Write the expressions for the three characteristic speeds of a gas.

2 Mark Questions

1. 1
   Find the most probable speed of nitrogen molecules at 300 K. (M = 28 × 10⁻³ kg mol⁻¹, R = 8.31 J mol⁻¹ K⁻¹.)
2. 2
   Explain why the Maxwell distribution broadens as the temperature increases.
3. 3
   State the ordering v_mp < v_mean < v_rms and explain it briefly.

3 Mark Questions

1. 1
   Describe the Maxwell speed distribution and explain the physical meaning of the most probable, mean and RMS speeds.
2. 2
   Compute the most probable, mean and RMS speeds of argon atoms at 300 K. (M = 40 × 10⁻³ kg mol⁻¹, R = 8.31 J mol⁻¹ K⁻¹.)

Exercises (NCERT Q12.1–Q12.10)

• 12.1) Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.
• 12.2) Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP: 1 atmospheric pressure, 0 °C). Show that it is 22.4 litres.
• 12.3) Figure 12.8 shows plot of PV/T versus P for 1.00 × 10⁻³ kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify? (b) Which is true: T₁ > T₂ or T₁ < T₂? (c) What is the value of PV/T where the curves meet on the y-axis? (d) If we obtained similar plots for 1.00 × 10⁻³ kg of hydrogen, would we get the same value of PV/T at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of PV/T (for low pressure high temperature region of the plot)? (Molecular mass of H₂ = 2.02 u, of O₂ = 32.0 u, R = 8.31 J mol⁻¹ K⁻¹.)
• 12.4) An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder (R = 8.31 J mol⁻¹ K⁻¹, molecular mass of O₂ = 32 u).
• 12.5) An air bubble of volume 1.0 cm³ rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C?
• 12.6) Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m³ at a temperature of 27 °C and 1 atm pressure.
• 12.7) Estimate the average thermal energy of a helium atom at (i) room temperature (27 °C), (ii) the temperature on the surface of the Sun (6000 K), (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
• 12.8) Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not, in which case is v_rms the largest?
• 12.9) At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at −20 °C? (atomic mass of Ar = 39.9 u, of He = 4.0 u).
• 12.10) Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 °C. Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N₂ = 28.0 u).

Answers at a glance

1. 1
   Q12.1 → The fraction is about 3.8 × 10⁻⁴.
2. 2
   Q12.2 → V = RT/P = 0.0224 m³ = 22.4 litres at STP.
3. 3
   Q12.3 → (a) ideal-gas behaviour (b) T₁ > T₂ (c) PV/T = 0.26 J K⁻¹ (d) no; 6.3 × 10⁻⁵ kg of hydrogen gives the same value.
4. 4
   Q12.4 → About 0.14 kg of oxygen is taken out of the cylinder.
5. 5
   Q12.5 → The bubble grows to about 5.3 cm³.
6. 6
   Q12.6 → There are about 6.11 × 10²⁶ air molecules in the room.
7. 7
   Q12.7 → (i) 6.21 × 10⁻²¹ J (ii) 1.24 × 10⁻¹⁹ J (iii) 2.07 × 10⁻¹⁶ J.
8. 8
   Q12.8 → Yes, equal numbers of molecules; v_rms differs, and it is largest for neon, the lightest.
9. 9
   Q12.9 → Approximately 2.52 × 10³ K (2523 K).
10. 10
    Q12.10 → Mean free path = 1.11 × 10⁻⁷ m; collision frequency = 4.58 × 10⁹ s⁻¹; free time ≈ 500 × collision time.

MCQ Questions

1 Mark Questions

1. 1
   Define the mean free path of a gas molecule.
2. 2
   State the ideal gas equation in terms of Boltzmann constant k. (Ans: PV = NkT)
3. 3
   What is the relation between rms speed and the pressure/density of a gas? (Ans: vrms = √(3P/ρ))
4. 4
   State the law of equipartition of energy.

2 Mark Questions

1. 1
   Find the rms speed of oxygen molecules (M = 32 g/mol) at 300 K. (Ans: vrms = √(3RT/M) ≈ 484 m/s)
2. 2
   A cylinder contains 0.2 mol of an ideal gas at 27°C occupying 5 L. Find its pressure. (Ans: P = nRT/V ≈ 9.97 × 10⁴ Pa)
3. 3
   At what temperature are the rms speed of nitrogen molecules double its value at 300 K? (Ans: T′ = 4T = 1200 K)

3 Mark Questions

1. 1
   Using kinetic theory, derive the expression P = ⅓ρc² for the pressure exerted by an ideal gas, defining the symbols.
2. 2
   Explain the mean free path of a gas molecule and write the factors on which it depends. (Ans: λ = 1/(√2 n πd²))
3. 3
   Distinguish between rms, mean and most probable speeds of gas molecules and write their relations with temperature.

5 Mark Questions

1. 1
   Derive the expression for the pressure of an ideal gas using kinetic theory and hence obtain PV = ⅓mnc², and the relation between average kinetic energy of a molecule and the temperature. (Ans: KE = (3/2)kT)
2. 2
   Explain the degrees of freedom of gaseous molecules, apply the law of equipartition of energy to monatomic, diatomic and polyatomic gases, and obtain the ratio γ = Cp/Cv for a diatomic gas. (Ans: γ = 7/5)