RMS Speed Mean Speed Most Probable Speed - UNSOLVED PRACTICE SET

Q7. Define rms speed, mean speed, and most probable speed of gas molecules. Write the expression for each.

Q8. Show that the ratio of rms speed to most probable speed is √(3/2) for a gas.

Q9. Explain why lighter gas molecules have higher rms speed than heavier molecules at the same temperature.

Q10. Draw a rough sketch of the Maxwell-Boltzmann speed distribution curve. Label the positions of v_mp, v_mean, and v_rms on the curve.

Q11. The rms speed of nitrogen molecules at 300 K is 515 m/s. Calculate the rms speed at 1200 K without using the full formula.

Q12. Why is the rms speed used in the kinetic theory expression for pressure rather than the mean speed or most probable speed?

Q13. Derive the expression for rms speed of gas molecules: v_rms = √(3RT/M) = √(3kT/m) = √(3P/ρ). Starting from the kinetic theory expression for pressure, show all steps clearly. Explain the physical significance of rms speed and why it is called "root mean square."

Q14. Describe the Maxwell-Boltzmann distribution of molecular speeds. Explain:

(i) Why the distribution is not symmetric

(ii) Why the peak shifts to higher speeds as temperature increases

(iii) Why the peak becomes broader and flatter at higher temperatures

(iv) The physical meaning of the area under the distribution curve

Q15. The rms speed of helium atoms at temperature T₁ is equal to the rms speed of oxygen molecules at temperature T₂.

(i) Find the ratio T₂/T₁. (Molar mass of He = 4 g/mol, O₂ = 32 g/mol)

(ii) If T₁ = 300 K, calculate T₂.

(iii) At what temperature will the rms speed of oxygen be equal to the rms speed of helium at 300 K?

(iv) Explain why such a high temperature is needed for oxygen.

Q16. Calculate the rms speed, mean speed, and most probable speed of nitrogen molecules (M = 28 g/mol) at 300 K.

(i) Find all three speeds.

(ii) Calculate the ratio v_rms : v_mean : v_mp.

(iii) Verify that v_rms² = (3/2)v_mp².

(iv) At what temperature will the rms speed be double its value at 300 K?

(Given: R = 8.31 J mol⁻¹ K⁻¹)

Q17. The escape velocity from Earth's surface is 11.2 km/s. The temperature of the upper atmosphere is about 1000 K.

(i) Calculate the rms speed of hydrogen molecules (M = 2 g/mol) at 1000 K.

(ii) Calculate the rms speed of oxygen molecules (M = 32 g/mol) at 1000 K.

(iii) Compare these speeds with the escape velocity. Which gas is more likely to escape from Earth's atmosphere?

(iv) Explain why Earth has lost most of its hydrogen but retained oxygen and nitrogen.

(Given: R = 8.31 J mol⁻¹ K⁻¹)

Q18. A mixture contains 2 moles of hydrogen (M = 2 g/mol) and 1 mole of oxygen (M = 32 g/mol) at 300 K.

(i) Calculate the rms speed of hydrogen molecules.

(ii) Calculate the rms speed of oxygen molecules.

(iii) Calculate the average kinetic energy per molecule for each gas.

(iv) Calculate the total kinetic energy of the mixture.

(v) Explain why both gases have the same average kinetic energy per molecule despite different speeds.

(Given: R = 8.31 J mol⁻¹ K⁻¹, N_A = 6.022 × 10²³ mol⁻¹)