Page 276 - Radiochemistry and nuclear chemistry

260 Radiochemistry and Nuclear Chemistry

very slow by a large energy of activation requirement in the formation of a necessary
transition state.
For the exchange reaction represented above, the rate of increase of AX* is equal to the
rate of formation minus the rate of destruction of AX*. The rate of formation is the product
of the rate of reaction k r, the fraction of reactions which occur with an active BX*, and the
fraction of reactions which occur with an inactive AX. Using the following notation

a = [AXI + [AX*] (9.17a)

b = [BX] + [BX*] (9.17b)

x = [AX*] (9.17e)

y = [BX'] (9.17d)
the rate of formation kf is equal to

kf = k r (y/b)(a-x)/a (9.18a)

In a similar fashion, the rate of destruction k d is equal to

k d = k r (x/a)(b-y)/b (9. lSb)

Therefore

dx/dt = kf - k d = k r (ay - bx)/(ab) (9.19)

The solution of this equation is

In(1 -F)= -k r t (a + b)/(ab) (9.20)

where F = x t Ix,, (x** is the value ofx t at t = co, i.e. equilibrium). The rate of exchange
k r is evaluated from the slope of a plot log (1 - F) versus t. If more than one rate of
exchange is present due to exchange with nonequivalent atoms in a reactant, it may be
difficult to resolve this curve sufficiently to obtain values for the reaction rates. Isotopic
exchange is a standard tool of the scientist studying the kinetics of chemical reactions whose
half-lives are longer than a minute.
One example of isotope exchange can be used to illustrate the value of these studies.
Consider the exchange between di- and trivalent chromium in HCIO 4 solutions. If the total
chromium ion concentration is 0.1 M, it takes 14 days for the exchange to reach 50%
completion at room temperature. Inasmuch as the di- and trivalent cations are both
positively charged, it is unlikely that they can approach each other closely enough to
exchange an electron directly to allow a reversal of oxidation state, and a more likely
mechanism is that an anion is involved as a bridge between the two cations such that the
intrusion of the anion reduces the repulsion between the two cations. If this model of the