  Some existing approaches to modeling the thermodynamics of moist air make
approximations that break $\textit{thermodynamic consistency}$, such that the
resulting thermodynamics do not obey the 1st and 2nd laws or have other
inconsistencies. Recently, an approach to avoid such inconsistency has been
suggested: the use of $\textit{thermodynamic potentials}$ in terms of their
$\textit{natural variables}$, from which all thermodynamic quantities and
relationships are derived. In this paper, we develop this approach for
$\textit{unapproximated}$ moist air thermodynamics and two widely used
approximations: the constant $\kappa$ approximation and the dry heat capacities
approximation. The consistent constant $\kappa$ approximation is particularly
attractive because it leads to, when using virtual potential temperature
$\theta_v$ as the thermodynamic variable, adiabatic dynamics that depend only
on total mass, independent of the breakdown between water forms. Additionally,
a wide variety of material from different sources in the literature on
thermodynamics in atmospheric modelling is brought together. It is hoped that
this paper provides a comprehensive reference for the use of thermodynamic
potentials in atmospheric modelling, especially for the three systems
considered here.
