Ncert Exercises & Solutions

13.8 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 the 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?

Since the three vessels have the same capacity, they have the same volume.

Hence, each gas has the same pressure, volume, and temperature.

According to Avogadro’s law, the three vessels will contain an equal number of the respective molecules. This number is equal to Avogadro’s number, N = 6.023 ×

10^{23}

The root-mean-square speed (

v_{rms}

) of a gas of molecular mass m, and temperature T, is given by:

v_{rms} = \sqrt{\frac{3 kT}{m}}

Where k is Boltzmann constant.

For the given gases, k and T are constants.

Hence

v_{rms}

depends only on the mass of the atoms, i.e.,

v_{rms} \propto \sqrt{\frac{1}{m}}

Therefore, the root-mean-square speed of the molecules in the three cases is not the same as the gases have different molecular masses.

Among neon, chlorine, and uranium hexafluoride, the molecular mass of neon is the smallest. Hence, neon has the largest root-mean-square speed among the given gases.

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