The Mie (λr, λa) intermolecular pair potential has been suggested as an alternative to the traditional Lennard–Jones (12–6) potential for modelling real systems both via simulation and theory as its implementation leads to an accuracy and flexibility in the determination of thermophysical properties that cannot be obtained when potentials of fixed range are considered. An additional advantage of using variable-range potentials is noted in the development of coarse-grained models where, as the superatoms become larger, the effective potentials are seen to become softer. However, the larger number of parameters that characterise the Mie potential (λr, λa, σ, ϵ) can hinder a rational study of the particular effects that each individual parameter have on the observed thermodynamic properties and phase equilibria, and higher degeneracy of models is observed. Here a three-parameter corresponding states model is presented in which a cohesive third parameter α is proposed following a perturbation expansion and assuming a mean-field limit. It is shown that in this approximation the free energy of any two Mie systems sharing the same value of α will be the same. The parameter α is an explicit function of the repulsive and attractive exponents and consequently dictates the form of the intermolecular pair potential. Molecular dynamics simulations of a variety of Mie systems over a range of values of α are carried out and the solid–liquid, liquid–vapour and vapour–solid phase boundaries for the systems considered are presented. Using the simulation data, we confirm that systems of the same α exhibit conformal phase behaviour for the fluid-phase properties as well as for the solid–fluid boundary, although larger differences are noted in the solid region; these can be related to the approximations in the definition of the parameter. Furthermore, it is found that the temperature range over which the vapour–liquid envelope of a given Mie system is stable follows a linear dependency with α when expressed as the ratio of the critical–point temperature to the triple–point temperature. The limit where potentials of the Mie family will not present a stable fluid envelope is predicted in terms of the parameter α and the result is found to be in excellent agreement with previous studies. This unique relation between the fluid range and the cohesive parameter α is shown to be useful to limit the pairs of Mie exponents that can be used in coarse-grained potentials to treat real systems in order to obtain temperature ranges of stability for the fluid envelope consistent with experiment.