Pressure of an Ideal Gas - UNSOLVED PRACTICE SET
Chapter: Kinetic Theory of Gases | Topic: Pressure of an Ideal Gas
PRESSURE OF AN IDEAL GAS - UNSOLVED PRACTICE SET
Topic: Pressure of an Ideal Gas
Multiple Choice Questions
Q1. According to kinetic theory, the pressure of an ideal gas is given by:
- P = (1/2)nmv²_rms
- P = (1/3)nmv²_rms
- P = (2/3)nmv²_rms
- P = nmv²_rms
Q2. The pressure of a gas is directly proportional to:
- The average speed of molecules
- The square of the average speed of molecules
- The mean speed of molecules
- The most probable speed of molecules
Q3. If the rms speed of gas molecules is doubled while keeping the number density constant, the pressure becomes:
- Half
- Same
- Double
- Four times
Q4. The pressure exerted by a gas is due to:
- The weight of the gas molecules
- The collisions of molecules with the walls of the container
- The intermolecular forces
- The gravitational pull on the gas
Q5. For an ideal gas, the relation between pressure P and density ρ is:
- P = (1/3)ρv²_rms
- P = ρv²_rms
- P = (2/3)ρv²_rms
- P = (1/2)ρv²_rms
Q6. When you press the nozzle of a perfume bottle, the spray comes out with force. This is because:
- The liquid is pushed by gravity
- The high pressure inside forces molecules out
- The bottle is elastic
- The perfume molecules are heavy
Short Answer Questions
Q7. Derive the expression P = (1/3)nmv²_rms, explaining the meaning of each symbol.
Q8. Show that the pressure of an ideal gas can also be written as P = (2/3)(E/V), where E is the total translational kinetic energy.
Q9. Explain why the pressure of a gas in a container does not depend on the shape of the container.
Q10. If the volume of a gas is halved at constant temperature, what happens to the pressure? Explain using kinetic theory.
Q11. Why does the pressure of a gas increase when heated at constant volume? Explain using the molecular picture.
Q12. A gas is transferred from a small container to a larger container at the same temperature. What happens to the pressure? Explain using kinetic theory.
Long Answer Questions
Q13. Starting from the basic assumptions of kinetic theory, derive the expression for pressure of an ideal gas: P = (1/3)nmv²_rms. Show all steps including:
Consideration of molecular motion in a cubical container
Calculation of momentum change per collision
Calculation of collision frequency
Summation over all molecules
Final expression and interpretation
Q14. Using the kinetic theory expression for pressure, derive the ideal gas equation PV = nRT. Show clearly how the microscopic picture of molecular motion connects to the macroscopic variables P, V, and T. Explain the physical meaning of each step.
Q15. A container is divided into two equal compartments by a partition. One compartment contains helium gas at pressure P and temperature T. The other compartment is evacuated. The partition is suddenly removed.
(i) What happens to the pressure of the gas immediately after the partition is removed?
(ii) What happens to the temperature?
(iii) Explain your answers using kinetic theory.
(iv) Is this process reversible? Why or why not?
Numerical / Application-Based Problems
Q16. A cubical container of side 0.1 m contains 10²⁴ molecules of nitrogen gas. The mass of each N₂ molecule is 4.65 × 10⁻²⁶ kg. The rms speed of the molecules is 500 m/s.
(i) Calculate the number density of the gas.
(ii) Calculate the pressure exerted by the gas on the walls.
(iii) Calculate the temperature of the gas using kinetic theory.
(iv) Verify your pressure calculation using the ideal gas equation.
(Given: k = 1.38 × 10⁻²³ J/K, N_A = 6.022 × 10²³ mol⁻¹)
Q17. The pressure inside a car tire is 2.5 × 10⁵ Pa at 27°C. The tire contains 0.5 moles of air. The molar mass of air is 29 g/mol.
(i) Calculate the volume of the tire.
(ii) Calculate the density of air inside the tire.
(iii) Calculate the rms speed of air molecules inside the tire.
(iv) If the car is driven fast and the tire temperature rises to 57°C, what is the new pressure? (Assume volume remains constant)
(Given: R = 8.31 J mol⁻¹ K⁻¹)
Q18. A star is composed of hydrogen gas at a temperature of 10⁷ K. The number density of hydrogen atoms is 10³⁰ m⁻³.
(i) Calculate the pressure inside the star using kinetic theory. (Mass of H atom = 1.67 × 10⁻²⁷ kg)
(ii) Calculate the rms speed of hydrogen atoms.
(iii) Compare this speed to the escape velocity from the star's surface if the star has the same mass and radius as the Sun. (M_sun = 2 × 10³⁰ kg, R_sun = 7 × 10⁸ m)
(iv) What does this comparison tell you about the stability of the star?
(Given: G = 6.67 × 10⁻¹¹ N m² kg⁻², k = 1.38 × 10⁻²³ J/K)