# Invariant points for salt solutions

### Invariant points

Invariant points for salt solutions are points where several salts are in equilibrium with the same solution.

According to Gibbs phase rule, a system with C chemically independent components and P phases in equilibrium has F=C-P+2 degrees of freedom. For a system consisting of one pure species the phase rule is F=3-P. If two phases are in equilibrium with each other in a one-component system, the system has one degree of freedom. The system is univariant.

Pure water boils at 100°C if the pressure is one atmosphere. If the pressure is lower than one atmosphere, pure water will boil at a temperature below 100°C. If the pressure is higher than one atmosphere, pure water will boil at a temperature higher than 100°C.

A system consisting of water vapor in equilibrium with liquid water has only one degree of freedom. The pressure OR the temperature can be fixed. Fixing one will automatically determine the other. At the triple point of H_{2}O, ice, water, and steam are in equilibrium. According to the phase rule, a one component system has no degrees of freedom when three phases are in equilibrium (F=0).The system is invariant. The triple point of water is at 273.16 K and 612 Pa.

This phase diagram is from Wikipedia.

### Binary systems

An aqueous solution of a pure salt contains three species: water, cations and anions. Still there are only two chemically independent components as the charge of the cations has to be balanced with an equivalent charge of the anions. The solution is therefore considered a binary solution, a solution of two chemically independent components. The phase rule is F=4-P for this system. An invariant point in a binary system is a point where 4 phases are in equilibrium (two salts, liquid, and vapor). In a ternary system 5 phases are in equilibrium in an invariant point.

### Ternary systems

A system with three independent components has F=5-P degrees of freedom. An invariant point in a ternary system therefore contains 5 different phases in equilibrium with each other. An invariant point can for example consist of a vapor phase, a liquid phase, and three solid phases in equilibrium with each other. A ternary system with three phases (solid-liquid-vapor) in equilibrium with each other has two degrees of freedom. If the temperature is fixed, one degree of freedom remains. A phase diagram isotherm showing a ternary system with a vapor phase and a liquid phase requires therefore a line to mark the concentration range in which a solid phase is in equilibrium with the other two phases. A point is needed to mark concentrations where two solid phases are in equilibrium with liquid and vapor.

### Quaternary systems

A system with four independent components has F=6-P degrees of freedom. Invariant points for salt solutions with four ions therefore contain 6 different phases in equilibrium with each other. An invariant point can for example consist of a vapor phase, a liquid phase, and four solid phases in equilibrium with each other.

A quaternary system with three phases (solid-liquid-vapor, solid-liquid-liquid, or liquid-liquid-vapor) in equilibrium with each other has three degrees of freedom. If the temperature is fixed, two degrees of freedom remain.

A phase diagram isotherm for a quaternary system with a vapor phase and a liquid phase requires therefore an area to mark the concentration range in which a solid phase is in equilibrium with the other two phases. A line is needed to mark concentrations where two solid phases are in equilibrium with liquid and vapor. An isotherm for a quaternary system is therefore a three-dimensional diagram. It is customary to show this type of diagrams as Jänecke projections https://doi.org/10.1002/zaac.19060510109 .

Jänecke projections are contour plots, where the water content is shown as contours. Alternatively, as it is done on these pages, the water content can be shown as numbers (weight percent water) at grid intersections. For quaternary systems with two cations and two anions the charge fraction of each cation and each anion can be varied between zero and one. Thus the diagram becomes square. The quaternary sodium chloride – potassium sulfate – water system is an example of this. In systems with three cations and one anion or one cation and three anions, the Jänecke projection is triangular.