In this section, the kinetics of a heterogeneous isotope exchange reaction will be shown, the rate-determining step of which is the isotope exchange. As an example, the isotope exchange of potassium ions between blood plasma and red blood cells will be mentioned.

Let us assume that the mass of the potassium ions is mj and m2 in plasma and red blood cells, respectively. Potassium ions are exchanged continuously between the two phases, plasma and red blood cells, reaching steady state. In order to deter­mine the rate of the exchange in steady state, radioactive potassium ions are added to the plasma in such a small quantity that the steady-state exchange is not dis­turbed. The reaction is directed by the increase of the entropy (see Section 9.3.3.1).

The activity of the radioactive potassium ions is I and can be expressed as:

where a01 is the initial (t = 0) specific activity of potassium ions in the plasma. After the addition of radioactive potassium ions to plasma, the radioactive potas­sium ions enter red blood cells as a result of the potassium exchange. After an arbi­trary t amount of time, the radioactivity of the plasma will be Ib and the specific activity of the potassium ions will be a1:

І1 = ma (9-62)

Similarly, for the radioactivity (I2) and the specific activity (a2) of the red blood cells:

І2 = m2a2 (9-63)

Since the system is closed for potassium ions (no potassium ions, including radio­active and inactive, are added later):

I = І1 112 (9-64)

In the time period dt, dm2’1 of potassium ions goes from the plasma to the red blood cells, and simultaneously, dm1’2 of potassium ions goes from the red blood cells to the plasma. As a result, the change of the potassium ions in the red blood cells is:

dm2 = dm2’1 — dm1′

and conversely, the change of the potassium ions in the plasma is:

dm1 = dm1’2 — dm2′ (9-66)

Mathematically, the change of the radioactivity both in the red blood cells and in the plasma can be expressed by the total differential quotient of the radioactivity:

dl2 = m2 da2 1 a2 dm2 (9-67)

dI1 = m1 da1 1 a1 dm1 (9-68)

Also, the change of the radioactivity can be expressed by the transport (Eqs. (9.65) and (9.66)) and the specific activities of potassium ions:

dl2 = a1 dm2’1 — a2 dm’2 (9-69)

dl1 = a2 dm1’2 — a1 dm2’1

By comparing Eqs. (9.67) and (9.69) with Eqs. (9.68) and (9.70), we arrive at the following two equations:

Я1 dm2’1 — «2 dm1’2 = m2 d«2 + a2 dm2 (9-71)

Я2 dm1’2 — «1 dm2’1 = m1 d«1 + «1 dm1 (9-72)

By equivalent mathematical transformations, we obtain: m2 d«2 = Я1 dm2’1 — «2(dm1’2 + dm2) (9-73) m1 dяl = Я2 dm1’2 — Яl(dm2’l + dm1) (9-74) Taking into consideration Eqs. (9.65) and (9.66), we get: m2 dя2 = Я1 dm2’1 — Я2 dm2’1 (9-75) m1 dяl = Я2 dm1’2 — Я1 dm1′ (9-76) From here: m2 dя2 = (Я1 — Я2)dm2’l (9-77) m1 dяl = (Я2 — Яl)dml’2 (9-78)

Now, let us study the change of the mass of the potassium ions in a period of time, dt, in the red blood cells:

dm2’1 m2 з d«2

dt Я1 — Я2 dt

and in the plasma:

dm1’2 m1 з d«1

dt Я2 — Я1 dt

As mentioned previously, the system is under steady-state conditions, i. e., the trans­port rate of the (inactive) potassium ions is the same in both directions, from the plasma to the red blood cells and vice versa. Let us denote this transport rate with C:

dm2’i dmi’2 „

— = — = C (9-81)

Using Eqs. (9.79) and (9.80), we obtain:

d^2 = — (a — Я2) (9.82)

dt m2

da^ = — (a2 — ai) (9.83)

dt m1

Since the system is closed for potassium ions (see Eqs. (9.62)—(9.64)), the total radioactivity at any time t is the sum of the radioactivities of the plasma and red blood cells:

I = miai 1 m2«2 (9.84)

Equation (9.84) can also be expressed to mean specific activity (a):

I = m{a 1 m{a (9.85)

From Eqs. (9.84) and (9.85), we get:

m1a1 1 m2a2 = m{a 1 m{a (9.86)

In Eqs. (9.82) and (9.83), there are two variables, a2 and ai, which depend on each other. For the solution of these equations, one of the variables must be eliminated. In order to do this, from Eq. (9.86), the specific activities, a2 and ai, and their differences, ai — a2 and a2 — ai, are expressed:

mi a 1 m2a — miai	(9.87)
m2
mi a 1 m{a — m2a2	(9.88)
mi
(mi 1 m2 )(a — a2) a2 =	(9.89)
mi
(mi 1 m2 )(a — ai) ai =	(9.90)

m2

Then, Eqs. (9.89) and (9.90) are substituted into Eqs. (9.82) and (9.83), thus:

da2 C mi 1 m2

= dt

a — a2 m2 mi

dai C mi 1 m2

= dt

a — ai mi m2

The solution of Eqs. (9.91) and (9.92) is:

— ln(a — Я2) = (—m11 mA 11K (9.93)

m2 m1

-ln(a — *1) = (11K (9.94)

m1 m2

where K and Kare integration constants.

For the calculation of the integration constants K and K, we assume that at t = 0, a2 = a20, and я1 = a10:

K = — ln(a — a20) (9.95)

K = — ln(a — a10) (9.96)

By substituting Eqs. (9.95) and (9.96) into Eqs. (9.93) and (9.94):

a —	— a2	C	m1 1 m2	(9.97)
	= ——	— -1
a —	a20	m2	m1
a —	a1	C	m1 1 m2	(9.98)
_		———— 1

ln ln

a — *10 m1 m2

The Kinetics of the Change of the Radioactivity in Red Blood Cells (a2) Since at t = 0 and a20 = 0 (radioactive potassium ions were added to the plasma), from Eq. (9.97), we obtain:

a — a2 C m1 1 m2 ln =—■ t

a m2 m1

After equivalent mathematical transformations:

C m1 1 m2

a — a2 = a exp —————— 1

m2 m1

C m1 1 m2

a2 = a — a exp — t

m2 m1

Since at t = 0, a20 = 0, and a1 = a0, from Eq. (9.86):

a0m1 = amj 1 am2

From here:

a (9.103)

m1 1 m2

Substituting Eq. (9.103) into Eq. (9.101), after mathematical transformation, we obtain:

Я0 m1 1 m2 m2 m1

The Kinetics of the Change of Radioactivity in Plasma (aa) As mentioned previ­ously, at t = 0, radioactive potassium ions were added only to the plasma; thus, a10 = a0. Substituting this into Eq. (9.98), we get:

in = _c_mL±m. t (9.105)

a — Я0 m1 m2

Similar to the transformations done previously for red blood cells, the change of radioactivity in the plasma can be expressed as follows:

(m1=m1 1 m2 )a0 — a1 (jm/m1 1 m2) —I

(m,1/m1 1 m2 )a0 — a0 (m1/m1 1 m2)

(m1/m1 1 m2) — (a1 /aQ) C m1 1 m2

(m1=m1 1 m2) — 1 m1 m2

m1 a1 m1 C m1

— = — 1 exp —

m1 1 m2 a0 m1 1 m2 m1

a1 m1 m1 C m1

= — — 1 exp —

a0 m1 1 m2 m1 1 m2 m1

when reaching equilibrium (t! 00), from Eqs. (9.104) and (9.109), we obtain: m1

a1 = a2 = ——- a0 (9.110)

m1 1 m2

This means that the specific activity is the same in the whole system; that is, the mixing entropy reached the absolute maximum.

Of course, this discussion can also be applied to other cases of heterogenous iso­tope exchange systems on the condition that the system has a component that can exchange freely between the phases.

Thus, we conclude that the heterogeneous isotope exchange allows us to study transport processes between phases under equilibrium conditions. Similar to the McKay equation (see Section 9.3.3.1) describing homogeneous isotope exchange, Eqs. (9.104) and (9.109) also show pseudo first-order kinetics, depending on the quantities in the two phases (mj and m2). The rate of the transport (C) can be mea­sured when the isotope exchange is the rate-determining process and the quantities in the two phases (mj and m2) are known. Moreover, it is enough to know only one of the two quantities (m1 and m2) because there are two kinetic equations, one for each phase. Thus, the quantity in the unknown phase also can be calculated. To provide an example of this latter application, we can mention the determination of the quantity of the exchangeable phosphate ions in soils. By the usual analytical methods, the total phosphate quantity present in the soil can be measured; however, only a portion of this phosphate can be dissolved in the soil solution. When the phosphate quantity of the soil solution is measured (it can be considered as m1), and the heterogeneous phosphate exchange between the soil solution and the soil is studied by adding radioactive phosphate to the soil solution, based on the kinetic studies, the quantity of the exchangeable phosphate of soil (m2) and the rate of iso­tope exchange (C) can be determined from Eqs. (9.104) and (9.109). The value of m2 can be calculated from the equilibrium-specific activities too (see Eq. (9.110)).

The heterogeneous isotope exchange clearly illustrates the most important aspects and advantage of the radiotracer methods: systems in thermodynamic equi­librium can be studied without disturbing equilibrium. The rate of processes in equilibrium can be given quantitatively. To do this, the isotope tracer method is the only option.

The rate of the isotope exchange reaction can also be determined at different temperatures. From the obtained values, the binding energy between the atoms par­ticipating in the isotope exchange can be calculated using the Arrhenius equation.

Nowadays, isotope exchange reactions are frequently used to study the rate of biological metabolisms and to determine the binding energy in heterogeneous cata­lytic reactions. In addition, the study of isotope exchange reactions provides useful information in geology (see Section 3.4).