Paschen's Law reflects the Townsend breakdown mechanism in gases, that is, a cascading of secondary electrons emitted by collisions in the gap. The significant parameter is pd, the product of the gap distance and the pressure. Typically, the Townsend mechanism (and by extension Paschen's law) apply at pd products less than 1000 torr cm, or gaps around a centimeter at one atmosphere. Furthermore, some modifications are necessary for highly electronegative gases because they recombine the secondary electrons very quickly.

In general, an equation for breakdown is derived, and suitable parameters chosen by fitting to empirical data.

Here are three equations:

Breakdown voltage:
Vbreakdown = B * p * d / (C + ln( p * d))

Breakdown field strength:
Ebreakdown = p * ( B / ( C + ln ( p * d)))

where:
C = A / ln ( 1 + 1 / gamma)

where:
gamma is the (poorly known) secondary ionization coefficient.

For air:
A = 15 cm^-1Torr ^-1
B = 365 Vcm^-1 Torr^-1
and gamma = 10^-2
so
C = 1.18