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?

(A) Yes,all vessels contain the same number of the respective molecules.
No,the root mean square speed of molecules is not the same in the three cases.
Since the three vessels have the same capacity,they have the same volume $(V)$.
Given that the temperature $(T)$ and pressure $(P)$ are also the same,according to the Ideal Gas Law $(PV = nRT)$,the number of moles $(n)$ in each vessel must be equal.
Since $n = N/N_A$,where $N_A$ is Avogadro's number,the number of molecules $(N)$ in each vessel is equal.
The root mean square speed $(v_{rms})$ of a gas molecule is given by the relation: $v_{rms} = \sqrt{\frac{3kT}{m}}$,where $k$ is the Boltzmann constant,$T$ is the temperature,and $m$ is the mass of one molecule.
Since $k$ and $T$ are constants,$v_{rms} \propto \frac{1}{\sqrt{m}}$.
As the mass of a neon atom is the smallest among neon,chlorine,and uranium hexafluoride,the root mean square speed $(v_{rms})$ is the largest for neon.