of parameters needed to describe the scattering process becomes very low. Basing on these ideas, in our previous work [@Jachymski2013] we were able to understand the reaction rates in Penning ionization of Ar by metastable He [@Narevicius] over several orders of magnitude in energy using just two parameters.

In this paper, we provide an extensive description of the results introduced in [@Idziaszek2010; @Jachymski2013]. We consider particles which can interact with arbitrary power-law potential $-C_n/r^n$ ($n>3$) at long range. Using a simple model in which the reaction channel has low threshold energy, we derive analytical formulas for the complex scattering length in the entrance channel, from which the elastic and reactive rates can be obtained. They can then be characterized using quantum defect functions, background scattering length and a single parameter which describes the short-range reactivity of the pair of particles. We extend the universal models in which the particles react at short range with unit probability $P^{re}=1$ [@Idziaszek2010; @Gao2010PRL; @Gao2011] to the case when $P^{re}<1$. We analyze in detail the behavior of the collision rates at high and low collision energies. Our results give the correct threshold behavior [@Idziaszek2010R; @Quemener2010] as well as classical high-temperature limits [@Langevin1905; @Gorin1938]. We discuss 