Not all the air molecules surrounding us travel at the same speed. Some air molecules travel faster, some move at a moderate speed and few others will hardly move at all. Hence, instead of asking the speed of any particular gas molecule we ask the distribution of speed in gas at a particular temperature. James Maxwell and Ludwig Boltzmann came up with a theory to show how the speeds of the molecule are distributed for an ideal gas. In the next section, we shall be discussing the Maxwell Boltzmann distribution formula in detail.
Maxwell Boltzmann Distribution Equation
The average kinetic energy of the gas molecules is given by the equation
where E_k is the average kinetic energy of the gas molecules
kB is the Boltzmann’s constant which is equal to
R is the universal gas constant which is equal to 8.314 J/K/mol
NA is the Avagadro’s constant which is equal to
Solved Example
Example:
- What is the average speed of hydrogen molecules on the earth?
Solution:
Let us consider the temperature on earth to be 300 K.
The mass of a hydrogen molecule is
The kinetic energy of the gas molecules is given by the equation
Substituting the values in the equation, we get
To find the velocity from the average kinetic energy we use the formula
By re-arranging the formula, we get
V2=(2)(6.21×10-21)/(2×1.67×10-27)
V = √(12.42 × 106)/3.34
V = √3.71× 106
V = 1.92× 103
Therefore, the average velocity is Vrms= 1928 m/s.
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