9.1 The Thermodynamic Concept of Classification (Distribution of Radioactive and Stable Isotopes)

For radioactive indications, the most important factor that has to be considered is the distribution of the radioactive tracer (microcomponent) and the inactive carrier (macrocomponent). As mentioned in Section 8.2, the radioactive indicator has to be homogenously distributed in the studied system. In this chapter, the condition of the homogeneous mixing of the radioactive indicator and the inactive (stable) car­rier will be investigated via examining the change of the mixing entropy.

Let us assume two solutions with the same concentration C, each containing the macro — and microcomponent, respectively, as the same chemical species and have the same temperature. When the concentration C is the same, the dilution-free energy does not have to be taken into account; thus, the entropies of the two solu­tions are expressed as:

Sn = n(R ln T — R ln C 1 s0)	(9.1)
Sn = N(R ln T — R ln C 1 SN)	(9.2)

N and n are the atoms of the macro — and microcomponents, respectively; T is the temperature; R is the gas constant; and S0 is the absolute entropy. When the solu­tions are mixed, the entropy changes as follows:

Sn = n R ln T — R ln — C 1 sn

n n 1N n

Sn = N R ln T — R ln N C 1 SN n1 N

Nuclear and Radiochemistry. DOI: http://dx. doi. org/10.1016/B978-0-12-391430-9.00009-3

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When mixing the solution, it will be diluted for both components. The dilution can qualitatively be expressed by the molar fractions of the micro — and macrocomponents:

n — Xn n1 N n	(9.5)
N	(9.6)
Xn n 1N

where Xn and XN are the molar fractions. The partial molar concentration of the components is:

cn — c n n n1 N	(9.7)
Cn — C n N n 1N	(9.8)

The change of the entropy as a result of mixing can be expressed in Eqs. (4.1)—(4.4) as follows:

ASelegy — (Sn + SN) — (Sn + Sn) — — nR ln Xn — NR ln Xn

By dividing both sides of Eq. (9.9) by the value of (n + N), the molar mixing entropy is obtained for mixing:

when the system consists of i components (besides two), the molar mixing entropy for i components is:

ASMegy —-R£Xi ln Xi

The end of the mixing process is mathematically reached when the primary differ­ential quotient of the entropy is equal to zero:

^(ASMegy) — 0 (9.12)

This extremum of mixing entropy can be calculated as follows. Let us divide the whole mixture to elementary volumes containing (An 1 AN) atoms and choose an arbitrary kth element, in which the molar ratio of the ith component is:

From Eqs. (9.11)—(9.13), we obtain the following:

Equation (9.12) can be solved for any ith component, assuming that the total quan­tity of this component remains the same:

E S(ni)k = 0 (9.17)

k

From Eq. (9.16), we obtain the following:

Equation (9.18) is solved by using the method of the Lagrange multiplier, i. e., each member of Eq. (9.16) is multiplied by an undetermined constant (a) and added to Eq. (9.18):

In Eq. (9.19), the coefficients of S(Ani)k are equal to zero, i. e.:

From here:

— e—(1+a) — constant

An + AN

Equation (9.21) expresses the well-known fact that when mixing equilibrium, the radioactive indicator is homogeneously distributed in the whole system.

Previously, the system has been divided into elementary volumes containing (An + AN) atoms. When the total number of the elementary volumes is r, the moles of the ith components (any of these components can be the radioactive indicator) in the whole system are:

r(Ani)k = щ (9.22)

The total number of atoms of all components in the whole system is: r( An 1 AN) = п 1N (9.23)

Therefore, the molar fractions are as follows:

(Ani)k = ni = X

An 1 AN n 1N i

Equation (9.24) expresses that the ratio of the micro — and macrocomponents is the same in any elementary volume as it is in the whole system. It also means that in this case, the mixing entropy is maximal.

Equation (9.12) and its solution, Eq. (9.24), are valid even if the radioactive indica­tor is distributed among different chemical species of the macrocomponents. However, the value of a has not been restricted, so the different chemical species may be characterized by different constants. Therefore, Xi can be different for chemical species. As an example, the iron ions and hemoglobin are mentioned (see Section 8.2). Since there are no exchanges between iron ions dissolved in water and the iron(II) ions within the hemoglobin, when a radioactive iron isotope is added, it will be mixed with the iron ions dissolved in water, but it will not exchange with the iron ions of hemoglobin. Therefore, the specific activity of the two species of iron will be different. Depending on the chemical bonds, the distribution of a radioactive iso­tope can be different within the same molecules. For example, a 14C-labeled side- chain can be bonded to a 14C-labeled benzene; the specific activities of the two carbon atoms may be different, depending on the specific activities of the reactants. In these cases, mixing entropy has local maxima for the different species and bonds. When the radioactive indicator is homogeneously distributed among the different chemical spe­cies and bonds (e. g., isotope exchange can take place), the mixing entropy has reached its absolute maximum. In this case, the subsequent effects cannot change the homoge­nous distribution, as shown by Hevesy’s experiments for the separation of 210Pb (RaD) from lead chloride (PbCl2) (discussed in Section 8.1).

The molar fraction, Xi, can decrease because of the decay of the radioactive iso­tope; this effect, however, can be taken into consideration easily using decay kinetics.