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The separation factor, /3, is defined as the ratio between two elements in one fraction devided by their ratio in the other fraction and is expressed as
r Ml /Mi m
p = JtJ Ml (1)
where M1 and M2 are the concentrations of the two elements in one fraction, and M[ and M'2 are the concentration in any other fraction.
It becomes apparent that when precipitation is done slowly enough there will be an equilibrium between the molecules in the precipitate and those in solution. Effective separation per stage is often enhanced in this technique over the normal precipitation method. However, in a system of homogeneous precipitation, the slow equilibrium process necessary to
Methods of Separation of Individual Rare Earth Elements
produce a completely homogeneous solid phase is seldom realized in practice. The true equilibrium is generally between the solution and a very thin layer on the crystal surface. Hence it is necessary to modify the expression (1) as
F = log (Mi/M*) I log (MilM'*) (2)
Weaver , in a series of studies, found that mandelic acid is specially selective for rare earths. The ft value for a Sm—Nd mixture in oxalate precipitation is 1.4, and the value for mandelate is 3.8. A large /9 value of 14 was obtained for La—Nd mandelate. However, the rapidity and completeness of precipitation is dependent on the pH, temperature and concentration of both the rare earths and mandelic acid.
It has been found that from a homogeneous solution the oxalate of samarium is preferentially precipitated over oxalates of other rare earths and yttrium falls behind all of them. Thus, in a separation process Sm is concentrated to a greater extent at the head section and Y at the tail section .
The main difficulty with the classical fractionation process is that, as the fractionation progresses, the number of fractions increases, and their size becomes smaller. With fractional precipitation processes, the number of operations practicable is much smaller than the fractional crystallization, because of the trouble in redisolving the precipitates and following another reprecipitation. In fractional crystallization schemes sometimes the liquor and crystal fractions can be combined with the help of modem analytical techniques by determining their compositions, thus achieving multiplication of stages.
(iii) Fractional thermal decomposition.—Mixed nitrates when fused, and then leached rapidly with water yield Y in the more basic fractions.
The rare earth chlorides can be separated through sublimation but a very high temperature and good vacuum are required. Recently  Eu2+ has been obtained pure by the distillation of its halides using the fact that Eu2+-halides are less volatile than the halides of trivalent rare earths. Sm, Eu and Yb oxides can be reduced to the divalent state by carbon and volatilized selectively from a mixture with other rare earth oxides .
The separation of rare earth oxides at 2500° C in a Solar furnace has been attempted , and Ce4+-oxide was obtained in a pure state from its mixture with lanthanum oxide.
(i) Solvent extraction. — The separation of rare earths by the liquid-liquid extraction technique was first reported by Fischer et al. .
This technique usually exploits the distribution of rare earths between an aqueous phase, and an immiscible, nonaqueous, usually organic phase. The early works of Appleton and Selwood , and Templeton and Peterson [51, 52] on the partition of rare earth thiocyanates and nitrates between water and a normal alcohol merit a comment. Later Asseltn et al.  investigated the possibility of thiocyanate and nitrate systems from the separation point of view.
Phosphorous containing organic compounds are mainly used as extractants. These reagents may be classified as a) neutral orthophosphates and organophosphonates, b) monoadic orthophosphates and organophosphonates, c) diacidic orthophosphates and organophosphonic acids, and d) various neutral and acid organo-pyrophosphates.
Ware  was the first to report a preferential extraction of Ce4+ into tributyl phosphate (TBP) from a nitrate solution of other rare earths. In a system equilibrated between the aqueous phase containing nitrate ion and TBP, the extraction process may be represented by eq. (3).
M(aqu) + 3 TBP(org) + 3 NO 3 ^ M(N03)3(TBP)3(org) (3)
Although all factors involved in the TBP extraction process have not been fully evaluated, a successfull separation of rare earths on a large scale has been achieved by this method.
The third-power TBP dependence has been verified by Peppard et al. [55, 56] for lower Z rare earths and toluene diluent, and for Y and Ce by Scargill and his coworkers [57—59]. However, there is a lack of direct evidence for the third-power nitrate dependence. In a nitrate medium with TBP, using a countercurrent system Weaver, Kappel-mann and Topp  were able to prepare 95 per cent pure Gd in kilogram quantities. Additional equilibrium data on rare earth nitrate-TBP system has recently appeared .