Most any thermal process is governed by the Boltzmann factor:
exp

−

E
kT

= e−
E/kT (5.41)
where k is the Boltzmann constant. Approximately speaking the Boltzmann factor expresses
the relative probability for an event requiring energy
E in a system at (absolute)
temperature T. Clearly if
E ≫ kT, the probability of the event happening is low. If an
electron requires an energy W (called the work function) to escape from the metal, The
Boltzmann factor suggests that this would happen with relative probability e−W/kT . Thus
you should expect that the current emitted by a heated metal would follow:
I ∼ e−W/kT

Read more: www.diplomaedu.com/2016/01/richardsons-law.html#ixzz3zejKoqvA