Universal jump in slope of the chemical potential at second-order phase transitions

We point out that in systems exhibiting a second-order transition, the chemical potential has a jump in slope at the critical temperature, provided provided that ${\mathit{T}}_{\mathit{c}}$ depends on particle density. We derive a generally valid thermodynamic relation between the specific heat jump, the jump in dμ/dT and dln${\mathit{T}}_{\mathit{c}}$/dn, with μ the chemical potential and n the density of the particles causing the transition. This relation is similar, but not equivalent, to the more familiar Ehrenfest relation between expansivity, specific heat, and dln${\mathit{T}}_{\mathit{c}}$/dp, with p the externally applied pressure.