Questions & explanations
1. Which of the following statements correctly relates a gas law to the ideal gas equation?
- A) Boyle's law holds when pressure and temperature are constant
- B) Charles's law holds when volume and temperature are constant
- C) Gay-Lussac's law holds when pressure and volume are constant
- D) Boyle's law holds when number of moles and temperature are constant
Answer: D) Boyle's law holds when number of moles and temperature are constant
Boyle's law states that pressure is inversely proportional to volume at constant temperature and constant number of moles, which is directly derived from PV = nRT. Option D correctly identifies the constant conditions. Option A is wrong because Boyle's law requires constant temperature, not pressure. Option B is wrong because Charles's law requires constant pressure, not volume. Option C is wrong because Gay-Lussac's law requires constant volume, not pressure.
2. Which of the following is a fundamental assumption of the kinetic theory of an ideal gas?
- A) Molecules exert attractive forces on each other
- B) Collisions between molecules are inelastic
- C) The volume of individual molecules is negligible compared to the volume of the gas
- D) Molecules are stationary and do not move
Answer: C) The volume of individual molecules is negligible compared to the volume of the gas
The kinetic theory assumes that gas molecules are point particles with negligible volume, no intermolecular forces, and that collisions are perfectly elastic. Option C is correct because the volume of molecules is considered negligible relative to the container volume. Options A, B, and D are incorrect as they contradict the assumptions: molecules do not attract each other, collisions are elastic, and molecules are in random motion.
3. A fixed amount of an ideal gas undergoes a change of state from (P1, V1, T1) to (P2, V2, T2). Which relation always holds for the gas?
- A) P1 V1 = P2 V2
- B) V1 / T1 = V2 / T2
- C) P1 V1 / T1 = P2 V2 / T2
- D) P1 / T1 = P2 / T2
Answer: C) P1 V1 / T1 = P2 V2 / T2
The ideal gas equation is PV = nRT. For a fixed amount of gas (n constant), the ratio PV/T remains constant. Therefore, P1V1/T1 = P2V2/T2 is always true. Options A, B, and D are special cases of the ideal gas equation that hold only when one variable is constant: A for constant temperature (Boyle's law), B for constant pressure (Charles's law), and D for constant volume (Gay-Lussac's law).
4. According to the law of equipartition of energy, the total energy per molecule of a diatomic gas at temperature T, considering only translational and rotational degrees of freedom (vibrations neglected), is:
- A) (3/2) kT
- B) (5/2) kT
- C) (7/2) kT
- D) (1/2) kT
Answer: B) (5/2) kT
For a diatomic molecule, there are 3 translational and 2 rotational degrees of freedom when vibrations are neglected. Each degree of freedom contributes (1/2)kT, so total energy = (3+2) × (1/2)kT = (5/2)kT. Option A corresponds to a monatomic gas, option C includes vibrational modes, and option D is the contribution of a single degree of freedom.
5. The root mean square speed of oxygen molecules (O₂) at 27°C is approximately: (Given molar mass of O₂ = 32 g/mol, R = 8.314 J/mol·K)
- A) 484 m/s
- B) 682 m/s
- C) 342 m/s
- D) 123 m/s
Answer: A) 484 m/s
v_rms = √(3RT/M). T = 27°C = 300 K, M = 32 g/mol = 0.032 kg/mol. v_rms = √(3 × 8.314 × 300 / 0.032) = √(7482.6 / 0.032) = √(233831.25) ≈ 483.6 m/s ≈ 484 m/s. Option A is correct. Options B, C, and D arise from common errors such as using M in g/mol without conversion, incorrect temperature scaling, or wrong formula.
6. For an ideal gas in equilibrium, which of the following correctly arranges the root mean square speed (v_rms), average speed (v_avg), and most probable speed (v_mp) in increasing order?
- A) v_mp < v_avg < v_rms
- B) v_rms < v_avg < v_mp
- C) v_avg < v_mp < v_rms
- D) v_mp < v_rms < v_avg
Answer: A) v_mp < v_avg < v_rms
For a Maxwellian distribution, the speeds are related as v_rms = √(3RT/M), v_avg = √(8RT/πM), and v_mp = √(2RT/M). Numerically, v_rms ≈ 1.732 √(RT/M), v_avg ≈ 1.596 √(RT/M), and v_mp ≈ 1.414 √(RT/M). Hence v_mp < v_avg < v_rms. Option A gives this correct order. Options B, C, and D represent incorrect sequences.
7. The average translational kinetic energy of a molecule of an ideal gas is directly proportional to which of the following?
- A) √T
- B) T
- C) T²
- D) 1/T
Answer: B) T
According to kinetic theory, the average translational kinetic energy per molecule is (3/2)kT, where k is Boltzmann's constant. This shows it is directly proportional to the absolute temperature T. Option A (√T), C (T²), and D (1/T) are incorrect proportionality relations.
8. The kinetic theory gives the pressure of an ideal gas as P = (1/3) ρ c², where ρ is the density and c is the root mean square speed. If the number density of molecules is n (number per unit volume) and each molecule has mass m, which expression correctly represents the pressure?
- A) P = (1/3) n m c²
- B) P = (1/3) n m c
- C) P = (1/3) n m / c²
- D) P = (1/3) n m² c²
Answer: A) P = (1/3) n m c²
Density ρ = mass per unit volume = (N m)/V = n m, where n = N/V. Substituting into P = (1/3) ρ c² gives P = (1/3) n m c². Hence option A is correct. Options B, C, and D have incorrect powers of c or m and do not match the derived expression.
9. At room temperature, the number of degrees of freedom of a diatomic gas molecule is:
- 3
- 5
- 6
- 7
Answer: 5
Diatomic gas molecules at room temperature have 3 translational and 2 rotational degrees of freedom, as vibrational modes are not excited. Hence total degrees of freedom = 5.
10. When the temperature of a gas is increased at constant pressure, the mean free path of its molecules:
- Increases
- Decreases
- Remains unchanged
- Becomes zero
Answer: Increases
Mean free path λ is given by λ = kT/(√2 π d²P). At constant pressure, λ ∝ T, so increasing temperature increases the mean free path.