How to Use This Page
1. Solve first, check later — Attempt every exercise question on your own before reading the solutions below. 2. Follow the sequence — Work through sections I to VI in order; later sections build on earlier ones. 3. Understand the method — Focus on the reasoning and formula selection, not just the final answer. 4. Practise numericals — Re-work the worked examples with different values until you can set them up without hints. 5. Review mistakes — Read the "Common Mistakes to Avoid" section before every exam.
Key Concepts (Quick Recap)
Gas Laws
| Law | Statement | Formula | Condition |
|---|---|---|---|
| Boyle's law | Volume inversely proportional to pressure | PV = constant (or P₁V₁ = P₂V₂) | T = constant |
| Charles' law | Volume directly proportional to temperature | V/T = constant (or V₁/T₁ = V₂/T₂) | P = constant |
| Gay-Lussac's law | Pressure directly proportional to temperature | P/T = constant (or P₁/T₁ = P₂/T₂) | V = constant |
| Ideal gas law | Combines all gas laws | PV = nRT | All conditions |
| Combined gas law | Relates P, V, T changes | P₁V₁/T₁ = P₂V₂/T₂ | — |
Constants and Values
| Constant | Symbol | Value | Unit |
|---|---|---|---|
| Universal gas constant | R | 8.314 | J/(mol·K) |
| Boltzmann constant | k_B | 1.38 × 10⁻²³ | J/K |
| Avogadro number | N_A | 6.022 × 10²³ | mol⁻¹ |
| Relation | R | = N_A × k_B | — |
| Standard temperature | — | 273.15 K (0°C) | K |
| Standard pressure | 1 atm | 1.013 × 10⁵ Pa | Pa |
Molecular Speeds
| Speed | Formula | Ratio to v_rms |
|---|---|---|
| RMS speed | v_rms = √(3RT/M) = √(3kT/m) | 1.000 |
| Average speed | v_avg = √(8RT/πM) = √(8kT/πm) | ≈ 0.920 |
| Most probable speed | v_mp = √(2RT/M) = √(2kT/m) | ≈ 0.816 |
Kinetic Theory Results
| Quantity | Formula |
|---|---|
| Pressure | P = (1/3)ρv_rms² = (1/3)(Nm/V)v_rms² |
| Average KE per molecule | (1/2)mv² = (3/2)kT |
| Average KE per mole | (3/2)RT |
| Translational KE | (3/2)kT per molecule |
| Mean free path | λ = 1/(√2 π d² n) |
| Degrees of freedom (monatomic) | f = 3 |
| Degrees of freedom (diatomic) | f = 5 (moderate T) |
| Equipartition theorem | Energy per degree of freedom = (1/2)kT |
Specific Heats and Gamma
| Gas type | f | Cᵥ | Cₚ | γ = Cₚ/Cᵥ |
|---|---|---|---|---|
| Monatomic | 3 | (3/2)R | (5/2)R | 5/3 ≈ 1.67 |
| Diatomic | 5 | (5/2)R | (7/2)R | 7/5 = 1.40 |
| Polyatomic | 6 | (6/2)R = 3R | (7/2)R + R = 4R | 4/3 ≈ 1.33 |
I. Choose the Best Answer (MCQs)
1. According to Boyle's law, if the pressure of a gas is doubled at constant temperature, the volume becomes:
- (a) Four times
- (b) Double
- (c) Half
- (d) Remains the same
2. The RMS speed of gas molecules is proportional to:
- (a) √T
- (b) T
- (c) T²
- (d) 1/T
3. The kinetic energy of 1 mole of an ideal gas at temperature T is:
- (a) (3/2)RT
- (b) (3/2)kT
- (c) (1/2)RT
- (d) RT
4. The number of degrees of freedom for a diatomic gas molecule at moderate temperature is:
- (a) 3
- (b) 5
- (c) 6
- (d) 7
5. The ratio of specific heats (γ) for a monatomic gas is:
- (a) 1.33
- (b) 1.40
- (c) 1.67
- (d) 2.00
6. If the absolute temperature of a gas is doubled, the RMS speed becomes:
- (a) Double
- (b) Half
- (c) √2 times
- (d) 4 times
7. Mean free path of gas molecules depends on:
- (a) Only the diameter of the molecules
- (b) Only the number density of the molecules
- (c) Both the diameter and number density
- (d) Neither the diameter nor number density
II. Fill in the Blanks
1. The value of the universal gas constant R is 8.314 J/(mol·K). 2. The relation between R and k_B is R = N_A × k_B. 3. At constant temperature, the pressure of a gas is inversely proportional to its volume (Boyle's law). 4. The RMS speed of molecules of a gas of molar mass M at temperature T is √(3RT/M). 5. The average kinetic energy per molecule of an ideal gas is (3/2)kT. 6. The ratio Cₚ/Cᵥ for a diatomic gas at moderate temperature is 7/5 or 1.4. 7. According to the equipartition theorem, each degree of freedom contributes (1/2)kT of energy per molecule. 8. At standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 litres (or 0.0224 m³). 9. The mean free path is the average distance a molecule travels between two successive collisions. 10. The pressure exerted by a gas is proportional to the mean square velocity of the molecules. 11. The degrees of freedom for a monatomic gas molecule is 3. 12. If the volume of a gas at constant pressure is doubled, its absolute temperature is doubled (Charles' law).
III. True or False (with corrections)
IV. Match the Following
| Column A | Column B |
|---|---|
| 1. Boyle's law | (e) PV = constant (at constant T) |
| 2. Charles' law | (f) V/T = constant (at constant P) |
| 3. Gay-Lussac's law | (a) P/T = constant (at constant V) |
| 4. RMS speed | (b) √(3RT/M) |
| 5. Equipartition theorem | (d) Energy per degree of freedom = (1/2)kT |
| 6. Mean free path | (c) 1/(√2 π d² n) |
V. Answer Briefly (2-Mark Questions)
VI. Answer in Detail (5-Mark Questions)
1. Derive the expression for pressure exerted by an ideal gas using kinetic theory and solve a numerical.
Consider N molecules of gas in a cubical container of side L. Each molecule has mass m and velocity components (v_x, v_y, v_z).
Step 1: Force by one molecule on one wall (x-direction)
A molecule hitting the wall perpendicular to the x-axis rebounds with velocity −v_x. The change in momentum per collision: Δp = mv_x − (−mv_x) = 2mv_x.
Time between successive collisions on the same wall = 2L/v_x.
Force by one molecule = rate of change of momentum = 2mv_x / (2L/v_x) = mv_x²/L.
Step 2: Total force and pressure
Summing over all N molecules:
Worked Example: Calculate the RMS speed of oxygen molecules at 300 K. Also find the pressure exerted by 2 moles of oxygen in a 0.05 m³ container at 300 K. (M = 32 × 10⁻³ kg/mol, R = 8.314 J/(mol·K))
2. Derive the gas laws from the kinetic theory equation and verify Charles' law numerically.
Starting from: P = (1/3)(Nm/V)v_rms² = (1/3)(N/V)mv_rms²
Since KE per molecule = (1/2)mv_rms² = (3/2)kT, we get mv_rms² = 3kT.
Substituting:
Boyle's law: At constant T, PV = nRT = constant. So P₁V₁ = P₂V₂. Charles' law: At constant P, V/T = nR/P = constant. So V₁/T₁ = V₂/T₂. Gay-Lussac's law: At constant V, P/T = nR/V = constant. So P₁/T₁ = P₂/T₂.
Worked Example: A gas occupies 2 L at 27°C and 1 atm. What volume will it occupy at 127°C at the same pressure?
3. Derive the relations Cᵥ = (f/2)R and Cₚ = Cᵥ + R using the equipartition theorem, and solve numericals.
By the equipartition theorem, each degree of freedom contributes (1/2)kT of energy per molecule. For f degrees of freedom:
Worked Example 1: Calculate Cᵥ, Cₚ, and γ for (a) a monatomic gas, (b) a diatomic gas at moderate temperature. (R = 8.314 J/(mol·K))
Worked Example 2: 4 moles of a diatomic ideal gas at 300 K are heated at constant volume to 400 K. Find Cᵥ, the heat supplied, and the change in internal energy.
Common Mistakes to Avoid
1. Using Celsius instead of Kelvin — All gas law calculations (PV = nRT, v_rms, Charles' law, etc.) require absolute temperature in Kelvin. Always convert: K = °C + 273.15. This is the single most common mistake.
2. Confusing molar mass M with molecular mass m — M is the mass of one mole in kg/mol (e.g., O₂: M = 32 × 10⁻³ kg/mol), while m is the mass of one molecule in kg (m = M/N_A). In v_rms = √(3RT/M), use M in kg/mol; in v_rms = √(3kT/m), use m in kg per molecule.
3. Mixing up v_rms, v_avg, and v_mp — These are three different speeds: v_rms = √(3RT/M), v_avg = √(8RT/πM), v_mp = √(2RT/M). The order is always v_rms > v_avg > v_mp. Use v_rms when relating to kinetic energy.
4. Using wrong degrees of freedom — Monatomic: f = 3 (translational only). Diatomic at moderate T: f = 5 (3 translational + 2 rotational). Diatomic at high T: f = 7 (add 2 vibrational). Using f = 3 for a diatomic gas will give wrong Cᵥ, Cₚ, and γ values.
5. Forgetting the √2 in mean free path — The correct formula is λ = 1/(√2 π d² n), not λ = 1/(π d² n). The √2 factor accounts for the relative motion of molecules (they approach each other from different directions).
6. Assuming ideal gas behaviour at all conditions — The ideal gas law is an approximation that works well at low pressures and high temperatures. Real gases deviate significantly at high pressures and low temperatures where intermolecular forces and molecular volume matter.
7. Wrong Mayer's relation substitution — Cₚ − Cᵥ = R (per mole) or cₚ − cᵥ = R/M (per unit mass, where M is molar mass). Do not use Cₚ − Cᵥ = R/M — that applies to specific heats per unit mass, not per mole.
8. Confusing number density (n) with number of moles (n) — In the kinetic theory equation P = (1/3)ρv_rms², the number density is usually written as n = N/V (molecules per m³). In the ideal gas law PV = nRT, n is the number of moles. Use different symbols or clearly define which n you are using.
Exam Checklist
- [ ] I can state and apply Boyle's, Charles', and Gay-Lussac's laws with correct conditions
- [ ] I can use the ideal gas equation PV = nRT and solve for any variable
- [ ] I can derive the kinetic theory pressure expression P = (1/3)ρv_rms²
- [ ] I can calculate RMS speed, average speed, and most probable speed and distinguish them
- [ ] I can apply the equipartition theorem to find U, Cᵥ, Cₚ, and γ for mono/diatomic gases
- [ ] I can use Mayer's relation Cₚ = Cᵥ + R and verify γ = 1 + 2/f
- [ ] I can define and calculate mean free path
- [ ] I can convert between Celsius and Kelvin and between molar mass and molecular mass
- [ ] I can solve combined gas law problems (P₁V₁/T₁ = P₂V₂/T₂)
- [ ] I can list the postulates of kinetic theory and explain deviations of real gases