Question

Calculate the root mean square speed of an oxygen molecule at 288K in SI unit.

Answer 1

Hint: Root mean square can be found out using the formula \[{{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}\] where T is the temperature in kelvin, R is the universal gas constant and M is the molar mass.


Complete step by step solution:
Root mean square or rms is the square root of the average mean square of a set of numbers
However, root mean square speed is the square root of the arithmetic average of the velocities of the molecules of a gas, squared.
We know, kinetic energy of a molecule is dependent on temperature and the molecular weight similarly, root mean square speed also changes depending on the same factors. It depends on the molecular mass of the molecule and the surrounding temperature.
In simpler words, we can say that we can calculate root mean square speed taking the square of the velocities and then taking their square root.
Mathematically, root mean square speed is written as-
     \[{{v}_{rms}}=\sqrt{\dfrac{3RT}{M}}\]
Where, \[{{v}_{rms}}\]is also written as ‘c’ and is the root mean square velocity
‘R’ is the universal gas constant and its value in SI unit is 8.314 KPa $d{{m}^{3}}$/mol.K
‘T’ is the temperature in kelvin and
‘M’ is the molar mass of the gas in kilogram per mole
We have to find the root mean square speed if oxygen in SI unit at 288K
Molar mass of a molecule of oxygen = 32
Molar mass in kilogram/mole = $32\div 1000$kg/mol =0.032 kg/mol
Temperature= 288K
Therefore, c=\[{{v}_{rms}}=\sqrt{\dfrac{3\times 8.314J/molK\times 288K}{0.032kg/mol}}\]=473.791m/s

Therefore, root mean square speed of a molecule of oxygen at 288K is 473.791m/s

Note: The value of the universal gas constant is the same for every gas as it is dependent on the units used for the calculation of pressure, temperature and volume.
It is important to remember that the molar mass is to be used in kg/mol. To convert it, we can simply divide the molar mass by thousand and the resultant will be the molar mass in kg/mol.