Three vessels of equal capacity have gases at the same temperature and

1.	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 Vrms the largest?
Answer» Since all vessels are of same capacity and are at same pressure and temperature, P = P₁ = P₂= P₃ ……..(1) V = V₁ = V₂ = V₃ ……..(2) T = T₁= T₂ = T₃ ……..(3) where P_i V_iand T_i are pressure, volume and temperature of i^th vessel respectively. From ideal gas law PV = nRT ∴ P_i V_i= n_i RT_i i = 1,2, 3 From equation (1), (2) and (3). PV = n_i RT or n_i = \(\frac{PV}{RT}\) = constant for all the three vessels. ∴ All three vessels have same no of moles of respective gases. From Avagadro’s law, equal mols of gases contain equal no of molecules. The root mean square (RMS) speed of a molecule is given by V_rms = \(\sqrt \frac{3kg}{m}\) where kB = Boltzmann constant T = temperature m = molecular mass of the gas. As Vrms α 1/√m the rms speed of gas molecules in the three vessels are different. Since neon has the least molecular mass of the three gases, it has the highest rms speed.

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 Vrms the largest?

Answer»

Since all vessels are of same capacity and are at same pressure and temperature,

P = P₁ = P₂= P₃ ……..(1)

V = V₁ = V₂ = V₃ ……..(2)

T = T₁= T₂ = T₃ ……..(3)

where P_i V_iand T_i are pressure, volume and temperature of i^th vessel respectively.

From ideal gas law

PV = nRT

∴ P_i V_i= n_i RT_i i = 1,2, 3

From equation (1), (2) and (3).

PV = n_i RT

or n_i = \(\frac{PV}{RT}\) = constant for all the three vessels.

∴ All three vessels have same no of moles of respective gases. From Avagadro’s law, equal mols of gases contain equal no of molecules. The root mean square (RMS) speed of a molecule is given by

V_rms = \(\sqrt \frac{3kg}{m}\)

where kB = Boltzmann constant

T = temperature

m = molecular mass of the gas.

As Vrms α 1/√m the rms speed of gas molecules in the three vessels are different. Since neon has the least molecular mass of the three gases, it has the highest rms speed.

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