The second Townsend coefficient is the number of electrons, which can be released on average from a surface by an incoming ion. Hence it is dependent on the work function of your surface (that determines how easily your electrons can leave the solid) and the kinetic energy with which the ion hits the surface.
The breakdown voltage depends on the pressure and the gap distance. The trend followed can be read from paschen's curve. The 2nd townsend coefficient depends on the work function of the cathode, kinetic energy of the impacting ion. The gas medium also decides this coefficient.