Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monoatomic), 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 $v_{r m s}$ the largest?

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Solution

All 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, $N = 6.023 \times 10^{23}$ .

The root mean square speed $(V_{r m s})$ of a gas of mass $m$ , and temperature $T$ , is given by the relation:
$V_{r m s} = \sqrt{\frac{3 k T}{m}}$
where,
$k$ is Boltzmann constant

For the given gases, $k$ and $T$ are constants.
Hence, $V_{r m s}$ depends only on the mass of the atoms, i.e., $V_{r m s} \propto (1 / m)^{1 / 2}$ .

Therefore, the root mean square speed of the molecules in the three cases is not the same. Among neon, chlorine, and uranium hexafluoride, the mass of neon is the smallest. Hence, neon has the largest root mean square speed among the given gases.

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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?

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