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?
(i) Yes, the vessels contain equal number of respective molecules. Since all three vessels have equal capacity (volume) and the gases are at the same temperature and pressure, according to the ideal gas law (PV = nRT), the number of moles and hence the number of molecules must be the same in all three vessels.
(ii) No, the root mean square speed of molecules is not the same in the three cases.
(iii) The v_rms is largest for neon. This is because at the same temperature, lighter molecules have greater rms speed. Neon has the smallest molecular mass among the three gases (Ne 20 u, Cl₂ 71 u, UF₆ 352 u), so it has the largest v_rms.
Explanation
This question tests two key concepts from kinetic theory: (1) Avogadro's law - equal volumes of gases at same temperature and pressure contain equal number of molecules, and (2) the relationship between rms speed and molecular mass. From the context, the average kinetic energy per molecule = (3/2)kT depends only on temperature. Since v_rms = , at the same temperature, lighter molecules have greater rms speed. Among Ne, Cl₂, and UF₆, neon is the lightest, hence has the largest v_rms.
Solution Steps
Step 1: Apply ideal gas law - same P, V, T means same number of moles (n), hence same number of molecules.
Step 2: Recall that average kinetic energy = (3/2)kT = (1/2)mv_rms², so v_rms = .
Step 3: Since temperature is same, v_rms depends only on molecular mass - lighter molecules have higher v_rms.
Step 4: Compare molecular masses: Ne (20 u) < Cl₂ (71 u) < UF₆ (352 u). Therefore, v_rms is largest for neon.