#### Question

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?

#### Solution 1

Yes. All contain the same number of the respective molecules.

No. The root mean square speed of neon is 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 mass *m*, and temperature *T*, is given by the relation:

`v_"rms" = sqrt((3kT)/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" prop sqrt(1/m)`

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

#### Solution 2

Equal volumes of all the gases under similar conditions of pressure and temperature contains equal number of molecules (according to Avogadro’s hypothesis). Therefore, the number of molecules in each case is same.

The rms velocity of molecules is given by

`v_"rms" = sqrt((3kT)/m)`

Clearly `v_"rms" prop 1/sqrtm`

Since neon has minimum atomic mass m, its rms velocity is maximum