Question

Which of the following has maximum root mean square velocity at the same temperature?
(1) \[{\rm{S}}{{\rm{O}}_{\rm{2}}}\]
(2)\[{\rm{C}}{{\rm{O}}_{\rm{2}}}\]
(3)\[{{\rm{O}}_{\rm{2}}}\]
(4)\[{{\rm{H}}_{\rm{2}}}\]

Answer 1

Hint: Root mean square(rms) velocity defines square root of average velocity. The formula used for calculation of this velocity is \[{v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \] , where, R stands for gas constant, T stands for temperature, M stands for molar mass of the gas.

Complete Step by Step Solution:
In the given question, using the root rms velocity formula we can identify the gas having maximum rms speed.

rms velocity can be find out by the following formula,
 \[{v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \]
Here, given a set of gases, we have to identify the gas which possesses maximum rms speed.
As R is gas constant and given that all gases are at same temperature, molar masses of each gas decide its rms speed.

So, \[{v_{rms}} = \sqrt {\dfrac{1}{M}} \], that means, rms velocity is indirectly proportional to molar mass of gases. So, the gas that possesses smaller molar mass has the maximum rms velocity.
So, now we will calculate the molar mass of each gas one by one.
Molar mass of \[{\rm{S}}{{\rm{O}}_{\rm{2}}} = 32 + 2 \times 16 = 32 + 32 = 64\,{\rm{u}}\]
Molar mass of \[{\rm{C}}{{\rm{O}}_{\rm{2}}} = 12 + 2 \times 16 = 12 + 32 = 44\,{\rm{u}}\]
Molar mass of \[{{\rm{O}}_{\rm{2}}} = 16 \times 2 = 32\,{\rm{u}}\]
Molar mass of \[{{\rm{H}}_{\rm{2}}} = 2 \times 1 = 2\,{\rm{u}}\]

As we have seen, hydrogen gas possesses the lowest molar mass among all the gases. Therefore, hydrogen gas possesses the highest rms velocity.
Hence, option (4) is right.

Note: It is to be noted that average velocity is different from rms velocity. Average velocity defines the arithmetic mean calculation of velocities of different gaseous molecules at a particular temperature. The average velocity can be found by the formula \[{v_{av}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} \] . The rm velocity is always greater than average velocity.