Here we consider a volume of an ideal gas comprising molecules in random motion.

If \( D (c_i)\hbox{d}c_{i} \) is the fraction of molecules within the fractional velocity range \( c_{i} {\text{ to }}c_{i} + \hbox{d}c_{i} \) then the product

$$ D (c_{1} )D(c_{2} )D(c_{3} ) $$
(11.1)

is the fraction (n) of molecules with velocity \( c = \sqrt {c_{1}^{2} + c_{2}^{2} + c_{3}^{2} } . \)

Thus

$$ n(c) = D(c_{1} )D(c_{2} )D(c_{3} ) $$
(11.2)

This equation can, by expressing it in logarithmic form and differentiating both sides, be recast in the form:

$$ \frac{ 1}{c}\frac{\hbox{d}(\ln n)}{\hbox{d}c} = \frac{1}{{c_{i} }}\frac{\hbox{d}(\ln D)}{{\hbox{d}c_{i} }} = {\text{constant}} $$

The ‘constant’ is necessary if the equation is to be true for all values of \( c_{i} . \)

The solution to this equation is:

$$ D (c_{i} ) = \frac{1}{{\sqrt {2\pi \tfrac{k}{\mu }T} }}\exp \left[ { - \frac{{\mu c_{i}^{2} }}{2kT}} \right] $$
(11.3)

where k is the Boltzmann constant and μ is the molecular weight for the gas. Hence given that the kinetic energy of a molecule is:

$$ E = \frac{{\mu c_{i}^{2} }}{ 2}{\text{ J}} $$
(11.4)

we can write

$$ D (c_{i} ) = \frac{1}{{\sqrt {2\pi \tfrac{k}{\mu }T} }}\exp \left[ { - \frac{E}{kT}} \right] $$
(11.5)