Question

Compare the RMS speed of hydrogen, oxygen, and carbon dioxide at the same temperature.
a. \[{{\rm{V}}_{{{\rm{H}}_{\rm{2}}}}}{\rm{ < }}{{\rm{V}}_{{{\rm{O}}_{\rm{2}}}}}{\rm{ < }}{{\rm{V}}_{{\rm{C}}{{\rm{O}}_{_{\rm{2}}}}}}\]
b. \[{{\rm{V}}_{{{\rm{H}}_{\rm{2}}}}}{\rm{ > }}{{\rm{V}}_{{{\rm{O}}_{\rm{2}}}}}{\rm{ > }}{{\rm{V}}_{{\rm{C}}{{\rm{O}}_{_{\rm{2}}}}}}\]
c. \[{{\rm{V}}_{{{\rm{H}}_{\rm{2}}}}}{\rm{ > }}{{\rm{V}}_{{{\rm{O}}_{\rm{2}}}}}{\rm{ < }}{{\rm{V}}_{{\rm{C}}{{\rm{O}}_{_{\rm{2}}}}}}\]
d. \[{{\rm{V}}_{{{\rm{H}}_{\rm{2}}}}}{\rm{ < }}{{\rm{V}}_{{{\rm{O}}_{\rm{2}}}}}{\rm{ > }}{{\rm{V}}_{{\rm{C}}{{\rm{O}}_{_{\rm{2}}}}}}\]

Answer 1

Hint: Root mean square speed is the square root of the average of squares of the speed of particular gas molecules.
It is denoted by \[{{\rm{V}}_{{\rm{rms}}}}\].

Formula Used:
\[{{\rm{V}}_{{\rm{rms}}}}{\rm{ = }}\sqrt {\frac{{{\rm{3RT}}}}{{\rm{M}}}} \]; Where
M = Molar mass of the gas (Kg/mole)
R = Molar gas constant
T =Temperature

Complete Step by Step Solution:
In this question, we have to compare the root mean square speed of Hydrogen, Oxygen and Carbon dioxide at the same temperature.

If we see the formula for root mean square speed we see that the value present in the numerator is constant for all three gases. It is because the temperature is constant and 3 and R are similarly constant values.
So, we have to now compare the molar mass of the gas in Kg per mole.

We know that,
The molar mass of Oxygen=0.016 kg/mol
The molar mass of Hydrogen=0.001 kg/mol
The molar mass of Carbon dioxide=0.044kg/mol

The square root of the molar mass of Oxygen=0.126
The square root of the molar mass of Hydrogen=0.031
The square root of the molar mass of Carbon dioxide=0.209

The square root of molar mass is inversely proportional to the root mean square speed.
Carbon dioxide which has the highest square root of molar mass out of the three gases will have the lowest root mean square velocity.
Hydrogen which has the lowest square root of molar mass will have the highest root mean square speed.
The molar mass of Oxygen has less molar mass than Carbon dioxide but more than Hydrogen will have a lower root mean square speed than Hydrogen.

So, the correct order will be \[{{\rm{V}}_{{{\rm{H}}_{\rm{2}}}}}{\rm{ > }}{{\rm{V}}_{{{\rm{O}}_{\rm{2}}}}}{\rm{ > }}{{\rm{V}}_{{\rm{C}}{{\rm{O}}_{_{\rm{2}}}}}}\].

So, option B is correct.

Note: By seeing the formula we see that rms speed is directly proportional to the square root of temperature and inversely proportional to the square root of molar mass. Hence, at a given temperature lighter molecules like Hydrogen and Helium move more rapidly than the heavier molecules.