Where V is the breakdown voltage in Volts, p is the pressure, d is the gap distance. The constants a and b depend upon the composition of the gas. For air at standard atmospheric pressure of 760 Torr, a = 43.6*106 and b = 12.8 , where p is the pressure in atmospheres and d is the gap distance in meters. The graph of this equation is the Paschen curve. By differentiating it with respect to (pd) and setting the derivative to zero, the minimum voltage can be found. This yields
and predicts the occurrence of a minimum breakdown voltage for (pd)=7.5*10-6 meter atmospheres. This is 327 Volts in air at standard atmospheric pressure at a distance of 7.5 micrometers. The composition of the gas determines both the minimum arc voltage and the distance at which it occurs. For argon, the minimum arc voltage is 137 Volts at a larger 12 micrometers. With sulfur dioxide, the minimum arc voltage is 457 volts at only 4.4 micrometers.
For air at STP, the intensity of the electric field needed to arc the minimum voltage gap is much greater than that necessary to arc a gap of one meter. For 7.5 micrometers, the field is 43 Megavolts per meter and for one meter it is only 3.4 Megavolts per meter. This is about 13 times greater. The phenomenon is well verified experimentally and is referred to as the Paschen minimum. The equation fails for gaps under a few micrometers in air at one atmosphere and incorrectly predicts an infinite arc voltage at a gap of about 2.7 micrometers.
Early vacuum experimenters found a rather surprising behavior. An arc would take place in a long irregular path rather than at the minimum distance between the electrodes. For example, at 1 Torr, the distance for minimum breakdown voltage is 5.7 millimeters or about 0.22 inch. The voltage required to arc that distance is 327 Volts and is greater for gaps above and below that point. For a 2.85 mm gap, the required voltage is 533 Volts, nearly twice as much. If 500 Volts were applied, it would not be sufficient to arc at the 2.85 mm distance, but would arc at a 5.7 mm distance.