The Frost diagrams are used to represent evidently the disproportionation conditions of
ions, allowing to judge the possibility along with the extent of disproportionation
(coproportionation) of the valence states of the element [1]. In this diagram, the dependence of
the reduced standard energy of formation of ions (F is the Faraday number
and is equal to 98.487) on the degree of oxidation of an element (n) is represented. So, The
Gibbs energy change is expressed in eV / mol. The Frost diagrams, being simple to accomplish,
characterize clearly the disproportionation (dismutation) processes of ions in solution. The
disproportionation occurs if the value of the analyzed ion is situated above the straight
line joining the points of neighboring valence forms on the diagram. In the
case of oxygen-containing species (ions, molecules), on the ordinate axis the standard Gibbs free
energy of formation of ion ( ) minus the Gibbs energy of a number x of water molecules,
equal to the number of oxygen atoms in the examined species ( ), is
placed. When the valence form contains more than one element, which is subject to redox
transformations, the values is calculated per one atom. The standard redox potential
represents the negative slope of the straight line joining two points on the diagram,
corresponding to any two valence forms, since the equality is valid:
Here the n quantity coincides with the degree of oxidation of the ion of interest. The angle of
inclination of the curve for the given pair of the valence forms characterizes the ability of
interaction with the formation of the products with a lower Gibbs energy. By means of this
diagram is easy to establish whether a particular ion is stable against disproportionation. The
disproportionation occurs when of ion lies above the straight line joining the Gibbs
energies of two neighboring valence forms. This is explained by the fact that of the
products of disproportionation reaction corresponds to the point situated at the intersection of
this line with the vertical passing through the point corresponding to this ion. In addition, the
greater the gain in energy, the less stable the ion towards its disproportionation in solution. This
property of diagram can be fundamented using thermodynamic and linear algebra methods. We
present here a brief proof.
We will examine the redox system formed from the ions of an element, which are in three
valence states: a, b and c. Let the following reaction of disproportionation takes place in this
system:
(1)
and
(2)
The reaction (1) is characterized by the following sequence of the standard redox potentials:
In the case of reaction (1) occurring, the points a ( (a), a), b ( (b), b) and c ( (c), c)
on the diagram correspond to the subsequent valence states (Fig.1).

Figure 1. The Frost diagram for the case analyzed in this paper.
We will prove that if the point b ( (b), b), corresponding to the intermediate valence, lies
above the straight line ac joining the a and c points, which characterize respectively the forms
with lower and larger valences, then the disproportionation reaction (1) occurs. If the straight
line ac is drawn through the points a and c, then the point b is situated above the line. We will
draw the line perpendicular to the axis n through the point b. We will denote the intersection
point of the straight line perpendicular to straight line ac through x ( (x), x). The equation of
the straight line can be represented by the expression
(3)
from where we get
(4)
On the other hand, under the conditions needed for the reaction (1) to occur, from the inequality
(2) it follows that
(5)
Therefore, if the b point is located right above the ac straight line, the spontaneous
disproportionation of to and takes place. So, the proof is done!
References:
[1] Povar, I. In: Bul. ASM. St. Biol. Chim. 1993, 5(266), 64-67.