The mixed phase of a substance undergoing a first order phase transiti
on has entirely different behavior according to whether the substance
has more than one conserved charge or only one, as in the text book ex
amples. In the latter case the pressure and nature of the phases are c
onstants throughout the coexistence phase. For systems with more than
one conserved charge (or independent component) we prove two theorems:
(1) The pressure and the nature of the phases in equilibrium change c
ontinuously as the proportion of the phases varies from one pure phase
to the other. (2) If one of the conserved charges is the Coulomb forc
e, an intermediate-range order will be created by the competition betw
een Coulomb and surface interface energy. Their sum is minimized when
the coexistence phase assumes a Coulomb lattice of one phase immersed
in the other. The geometry will vary continuously as the proportion of
phases. We illustrate the theorems for a simple description of the ha
dron to quark phase transition in neutron stars and find a crystalline
phase many kilometers thick. However the theorems are general and per
tain to chemical mixtures, nuclear systems, either static as in stars
or dynamic as in collisions, and have possible applications to phase t
ransitions in the early universe. The theorems hold equally for the nu
clear gas-liquid transition.