In a homogeneous system containing isotope molecules of the different substances, an isotope exchange may occur. The rate of the isotope exchange depends on the binding energies. Through kinetic studies of the isotope exchange, the binding energy of the isotope in a given substance can be determined. The reaction is directed by the increase of the entropy and reaches equilibrium when the distribu­tion of the isotopes becomes homogeneous (maximum mixing entropy). This means that in equilibrium, the specific activities will be the same for each substance.

The McKay equation can be used for the kinetic description of the isotope exchange reaction. As an example, the isotope exchange reaction of ethyl iodide and sodium iodide labeled with 131I is mentioned. AX and BX mean the ethyl

iodide and the sodium iodide, respectively. The labeled forms of these compounds are indicated by 0. The exchange reaction is defined as:

AX 1BX0 з AX01BX (9.44)

C2H511 Na131IoC2H513111 NaI (9.45)

At a given time (t), the specific activity of the two compounds can be defined by the following:

aA 5 JA and aB 5 (9:46)

[A] [B]

where [A0] is the activity of the labeled AX0 and [A] is the quantity of AX. Similarly, [B0] is the activity of the labeled BX0 and [B] is the quantity of BX. For the isotope exchange reaction of ethyl iodide and sodium iodide, [A0] and [B0] are the activities of 131I in the ethyl iodide and sodium iodide, respectively. As postu­lated by the definition of the specific activity, the mass of the iodine has to be taken into account. (NB: Similar equations can be written when the molar activities of the two compounds are expressed where the moles of AX and BX are substituted into Eq. (9.46).)

The total activity of the system is constant at any time (a closed system):

[A0] 1 [B0] 5 [AN] 1 [BN] (9.47)

where [AN] and [BN] are the equilibrium activities of AX0 and BX0, respectively. In equilibrium, the mixing entropy has maximum value, and the distribution of 131I is directed by the ratio of the quantity of AX and BX:

IAN 5 [A]

[BN] [B]

that is, both the ratio of the activities and the ratio of quantities are the same.

To be able to solve the kinetic equation of the exchange reaction (see Eq. (9.51)), only one parameter should be variable. This means that the change of the activity of only one of the reactants is to be taken into account. To eliminate the activity of the other reactant, some equivalent mathematical transformations are necessary. At first, [BN ] is expressed from Eq. (9.48):

(9.49)

Let us substitute [BN] into Eq. (9.47) and then express [B0]:

0 1 [AN][B0]

nJ [A0

Since the system is closed, the change of the activity of AX0 ([A0]) as a function of time is as follows:

where R is the rate constant of the isotope exchange. Equation (9.51) expresses both the transmission of the labeling atom from BX0 to AX and the reversed pro­cess. After an equivalent mathematical transformation, we obtain:

By substituting the specific activities (Eq. (9.46)), after some transformation:

By substituting [B0] from Eq. (9.50), we obtain:

= ([A] [AN ] 1 [AN ] [B] — [A] [A0] — [B] [A0]) = ^ ([A’m ] — [A ])([A] 1 [B])

(9.54)

By the separation of the variates, the equation is given as follows:

d A 0

[AN] — [A0] [A] [B]

The solution of Eq. (9.55) is:

The integration constant (C) can be determined by assuming that at t = 0, [A0] = [A00]. Thus,

C = ln( [AN] — [A0 ]) The solution is:

when t = 0, [A00] = 0, there are no labeling atoms in AX (only sodium iodide is labeled with the 131I isotope), the solution is as follows:

02 ra)=(12 F)=2 eRt

The ratio of the activities at any time and in equilibrium is frequently denoted as:

This means that the ratio of exchanged activity to equilibrium activity is to be measured. This is a widely used technique in radiotracer experiments because the relative activity measurements are often much easier.

Equation (9.59) is similar to the kinetic equation of first-order chemical reac­tions; however, R’ depends on the quantities of the two compounds, AX and BX. The half-time of the reaction depends on the quantities of AX and BX. Thus, the reaction is only formally a first-order reaction; instead, it is a pseudo first-order reaction.

When [A] and [B] are constant, the activation-free energy (i. e., the binding energy of the labeling atom) can be calculated from the Rs determined at different temperatures using the Arrhenius equation.

An isotope exchange reaction can have three different kinetic profiles:

Figure 9.14 Isotope exchange with initial period.