As mentioned in Section 8.2, the radioactive indicator applied in carrier-free or very low concentrations can coprecipitate with any macrocomponent of the system if they can form isomorphous crystals. As an example, barium chloride (macrocomponent) and radium chloride (microcomponent) have been mentioned.

In the initial step of coprecipitation, the macro — and microcomponents are mixed in solution. When mixing entropy becomes maximal, the solution will be homoge­neous. Then, there are two ways to achieve coprecipitation:

• By fast cooling of the solution. In this case, fine grains are produced. For example, (RaBa)Cl2 is precipitated by the addition of concentrated hydrochloric acid under cooling. The composition of the crystalline phase is the same in any small volume. This can be expressed by the Henderson—Kracek equation:

У = D-—У (9.118)

x a — x

where x and y are the quantities of the macro — and microcomponents in the crystalline phase, respectively; a and — are the quantities of the macro — and microcomponents in the whole system; and D is the fractionation coefficient. Fractionation by coprecipitation is possible when D ^ 1.

• The crystals are grown slowly; e. g., by the slow evaporation of the solvent. In this way, the composition of the solution continuously changes, and therefore the composition of the phases also change continuously. As a result, the composition of the crystal varies with the depth of the grains, as expressed by the Doerner—Hoskins equation:

a-

ln — = A ln — (9.119)

a— x -— y

where A is the fractionation coefficient and the other signs have the same meaning as in Eq. (9.118).

Equations (9.118) and (9.119) can be related, assuming a thermodynamical equi­librium for any infinitely short time of the crystal growing:

Separating the variables:

dx _ ^ dy

a — x b — y

Equation (9.121) can be solved for the whole period of crystal growth, assuming that the crystal growth is determined by the surface equilibria at any time:

dx _ ^ ‘ dy

a— x b— y

The solution is:

ln(a — x) _ A ln(b — y) + C (9.123)

In order to determine the integration constant (C), we assume that at the initial time of the crystal growth, (t _ 0), x _ 0 and y _ 0 (no crystal yet):

ln a _ A ln b 1 C (9.124)

From here:

C _ ln a — A ln b (9.125)

Substituting C into Eq. (9.123), the Doerner—Hoskins equation is obtained:

ln(a — x) _ A ln(b — y) 1 ln a — A ln b (9.126)

Experiments have shown that crystals that have a composition characterized by the Doerner—Hoskins equation can transform to crystals with uniform composition characterized by the Henderson—Kracek equation by isothermic transcrystallization.