  Models for spin systems known from statistical physics are used in
econometrics in the form of agent-based models. Econophysics research in
econometrics is increasingly developing general market models that describe
exchange phenomena and use the chemical potential $\mu$ known from physics in
the context of particle number changes. In statistical physics, equations of
state are known for the chemical potential, which take into account the
respective model framework and the corresponding state variables. A simple
transfer of these equations of state to problems in econophysics appears
difficult. To the best of our knowledge, the equation of state for the chemical
potential is currently missing even for the simplest conceivable model of an
ideal agent system. In this paper, this research gap is closed and the equation
of state for the chemical potential is derived from the econophysical model
assumptions of the ideal agent system. An interpretation of the equation of
state leads to fundamental relationships that could also have been guessed, but
are shown here by the theory.
