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# Three vessels of equal capacity have glasses at the same temperature and pressure. The first vessel contains neon (monoatomic), the second contains chlorine (diatomic), and the third one contains uranium hexafluoride(polyatomic). Do the vessels contain an equal number of respective molecules? Is the root mean square speed of the molecules the same in three cases? If not, in which case is ${V_{rms}}$ the largest?

Hint: Firstly we should have a basic idea that all three vessels (at the same temperature and pressure) have the same volume. So, in accordance with Avogadro's law, the three vessels will contain equal numbers of respective molecules. We will find a relation between root mean square velocity and the mass of the atom.

Firstly, we will have a quick glimpse of the root mean square speed. The root mean square speed is the measure of the speed of particles in a gas, defined as the square speed of the average speed -squared of the molecules in a gas.
Now we all know that all the three vessels have the same capacity so they have the same volume. Hence, each gas has the same pressure, temperature, and volume.
So 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}$.
We all know that the root mean square speed $\left( {{V_{rms}}} \right)$ of gas of mass m, and temperature T, is given by the relation
${V_{rms}} = \sqrt {\dfrac{{3kT}}{m}}$
where,
k is Boltzmann constant
We know that for the given gases, k and T are constants.
Hence, ${V_{rms}}$ depends only on the mass of the atoms.
I.e. from the above line we can find out that
${V_{rms}} \propto {\left( {\dfrac{1}{m}} \right)^{\dfrac{1}{2}}}$
Therefore, the basis mean square speed of the molecules within the three cases isn't an equivalent. We can easily notice that among neon, chlorine, and uranium hexafluoride, the mass of neon is the smallest. Hence, neon has the most important root mean square speed among the given gases.

From the above solution, we can easily state that among three gases neon has the largest root mean square speed.

Note:
We all should have a perfect knowledge of root mean square speed and must remember the relation of ${V_{rms}}$. We know that the root mean square of the velocities of all the molecules in a volume of gas is directly proportional to the temperature in the case of an ideal gas. Also, the root-mean-square speed measures the average speed of particles in a gas.