Calculated models of the chemistry of natural waters show how mass is distributed among aqueous species, minerals, and gases, whether individual minerals are undersaturated or supersaturated, and gas partial pressures within the waters. This chapter explores how to construct and interpret computed models of water chemistry, using seawater, river water, and deep-sea brines as examples.

The equilibrium state of a chemical system composed of an arbitrary number of thermodynamic components is described by a set of nonlinear equations that can be evaluated only using iterative methods. This chapter provides a detailed description of the process by which the Newton–Raphson method can be applied to solve for the equilibrium distribution of chemical mass in a multicomponent system composed of aqueous, solid, and gaseous phases.

Whereas determining the equilibrium point of a single chemical reaction is a straightforward application of thermodynamics, calculating the distribution of chemical mass among aqueous species, solids, and gases in a system composed of an arbitrary number of thermodynamic components requires that perhaps hundreds or thousands of reactions be evaluated at the same time. This chapter lays out a general set of equations by which a computer algorithm can quickly and reliably determine the equilibrium state of a multicomponent chemical system.

The thermodynamics potentials for describing matter at nonzero temperatures and densities or chemical potentials are summarized. Emphasis is put on the thermodynamically correct description within the canonical and grand canonical ensemble for dense matter. The notion of chemical equilibrium is introduced for several conserved quantities and used to describe matter in β-equilibrium where charge and baryon number are conserved. The limit for nonrelativistic and relativistic particles is worked out in detail. The concept of an equation of state is introduced and applied to free Fermi gases. The pressure integral is solved analytically and the nonrelativistic and relativistic limits for the equation of state are delineated. Finally, the properties of polytropes are discussed and connected to the limiting cases of the equation of state of a free Fermi gas.

The grand canonical ensemble applies to open systems that can exchange both energy and particles with their environment.The grand canonical partition function and its relation to the grand potential are derived, with an emphasis on the chemical potential.Examples in which the grand canonical ensemble applies are presented, including two-level systems, Langmuir adsorption isotherms, chemical equilibrium and the law of mass action.