Читать книгу Molecular Imaging - Markus Rudin - Страница 14

CHAPTER 2

Foundations of the Theory, Its First Formulation

In this chapter I have summarized the two Marcus’ landmark papers of 1956 – Refs. [1] (Part I of the theory) and [2], which laid the foundation of the whole theory. References have been given in keeping with the educational character of this work. The most important results have been emphasized and mathematical proofs have been given in separate starred sections following the results, so they can be omitted by the noninterested reader. In Chapters 5 and 6, the more general and more abstract formulations of the theory in terms of potential energy surfaces and statistical mechanics is dealt with, but the present earlier treatment in terms of the dielectric continuum theory represents an easier introduction to Marcus theory, with its detailed explanation of all the different steps involved in the ET process, and by itself allows an understanding of the applications given by M. in Part II and in Part III. Similarly, one deals with ideal gases before studying the real ones.

2.1.The Reaction Scheme for Bimolecular ET Reactions

The usual way of calculating a reaction rate is that of first determining the free energy of activation and of then introducing it in the absolute reaction rate theory formula [3–7] for the rate constant. But the ET overall reaction scheme corresponds in general to a sequence of steps, some of which may be slow and so must be considered in calculating the overall rate constant to be compared with the observed rate constant of the reaction sequence. The reaction scheme considered by M. for the bimolecular ET reactions is the following:

The reverse step of Eq. (2.3) is not considered because we are interested in calculating the rate constant of the overall forward reaction and, moreover, the concentration of the products, starting from a very dilute solution of reactants, would be very low.

The process relative to the constant k₋₁ corresponds to the probability of “a disorganizing motion of the solvent, destroying the polarization appropriate to the intermediate state” so leading to deactivation. Likewise, X can go back to X^∗ and so a rate constant k₋₂ must be considered. Because of the small electronic interaction that was supposed to exist between reactants, the X^∗ → X process can be slow and is the reason why two electronic structures of the TS were considered.

Note that A and B are not necessarily the actual compounds introduced in the reaction system, they may rather be “active entities formed from them” (M.).

The overall rate constant of the reaction sequence is k_bic_ac_b where the c’s denote concentrations and k_bi is the observed bimolecular rate constant. According to Eq. (2.3), the rate is also given by k₃c_x, so that:

The steady-state equations [3–7] for the concentrations of X^∗ and X are given by Eqs. (2.5) and (2.6):

The meaning of Eq. (2.5) (standard chemical kinetics. . .) is the following: if during the reaction the concentration of X^∗ remains constant, this means that the effect of the two reactions leading to formation of X^∗, with velocities k₁c_ac_b and k₋₂c_x is equal to the effect of the reactions leading to depletion of it with velocities and The same argument is valid for Eq. (2.6). The two equations allow for the determination of those values of and c_x for which the conditions and dc_x/dt = 0 are valid.

Introducing the value obtained for c_x in Eq. (2.4), Eq. (2.7) is obtained [1]:

M. has shown that when the probability of forming X from X^∗ is ≥ than the probability of X^∗ reforming A and B, then

If the above condition is not valid, the more complex Eq. (2.7) is to be used. A more detailed analysis, given in Marcus’ later papers [8–10] uses a nonadiabatic/adiabatic transition probability, replacing the factor 1/[1 + (1 + k₋₂/k₃)k₋₁/k₂].⁽¹⁾

In an impressive tour de force, M. estimated in his landmark paper [1] all of the rate constants appearing in Eq. (2.7). The description of their calculation is the main purpose of this chapter.

2.2.A Model for Reactants, the Medium, and Their Interactions

We begin with a model for reactants that is the simplest and the first one used in the theory.

M. assumes that each reactant may be treated as a sphere which, if the reactant is an ion, may be surrounded by a concentric spherical shell of saturated dielectric, that is, of dielectric completely oriented in the ion’s electric field, with which is in equilibrium. Outside this spherical shell the solvent medium is supposed to be dielectrically unsaturated, in the 1956 paper [1]. The ion plus the rigid saturated dielectric region is treated as a conducting sphere of radius a,⁽²⁾ and in a more general manner in later papers. Let us now have two reacting ions of radii a₁ and a₂. In the first treatment of M., the radii are supposed to be of a fixed value before, during, and after ET. Each reactant is made up also by all of the surrounding solvent molecules (in theory, considering the long-range nature of the electrostatic interactions). The solution is supposed to be dilute enough that other couples of reactants do not interact with the pair we are considering. Moreover, in this first formulation of the theory no ionic atmosphere is considered.

Let us now consider the free energy of the system made up of two ions with charges q₁ and q₂ at distance R from each other. There are four distinct contributions to it:

(i)The free energy of interaction of all atoms within the first sphere with each other and with the central ionic charge

(ii)The same for the second sphere

(iii)The free energy of interaction of all molecules outside of the two spheres with each other and with the charges of the spheres

(iv)The interaction of the two ionic spheres with each other

If we assume that the atoms within the spheres do not change their average position during the mutual approach of the ions, the first two contributions to the formation of the TS are independent of the interionic distance R and therefore do not contribute to the free energy of its formation from the reactants.

In order to calculate contributions (iii) and (iv), it is necessary to consider the properties of the dielectric outside the spheres.

2.3.Electrostatic Characteristics of the Transition State

State X^∗ is considered as a macroscopic system, made up by x^∗ microstates, in which the dielectric medium surrounding the spheres is a continuum characterized by a definite value of the macroscopic polarization at each point of the system. In order to calculate contribution (iii), we need to determine the polarization function in the volume outside that occupied by the spheres. The electronic polarization for X^∗ will be denoted as and the atomic plus orientation as where r is the position vector with respect to an arbitrarily chosen origin. The total polarization function is P^∗(r), that is:

For state X, the corresponding polarization functions are P_e(r), P_u(r), and P(r). These functions have the following properties:

(i)As previously seen, the X^∗ and X states have the same atomic and orientation polarization functions, that is,

(ii) and P_e(r)⁽³⁾ are always in equilibrium with the respective electric field strengths, that is, they are determined by the fields:

where α_e is the polarizability associated with the E-type polarization.

(iii)P_u(r) is unrelated to the electric field strengths in the nonequilibrium states.⁽⁴⁾

The meaning of the macroscopic polarization functions in relation to the microscopic structure of the dielectric media will be now defined following Ref. [11] where it is very clearly explained.

The classical electrostatics definition of P is

The macroscopic P(r) is the result of very many microscopic dipoles pointing along the direction of the polarizing electric field. What does it mean to represent discrete molecular dipoles by a continuous macroscopic density function P(r)? The microscopic electric field inside matter varies abruptly from point to point and also so does in time: the field can be, for instance, very great near an electron at a certain instant and it can be completely different an instant later as the atoms move about because of thermal agitation. One is unable to calculate this wildly varying microscopic field which is also in general of no interest. The problem would be similar to that of keeping track of the instantaneous microscopic density at a point in a liquid or gas. What one does is to calculate the macroscopic field inside the matter defined as the average field in a region large enough to contain thousands of atoms, say, so as to allow a smoothing out of the uninteresting microscopic fluctuations and be describable by a P(r). The region must also be small enough to follow the large-scale variations of the field. This is what is meant by macroscopic average field inside matter. So this is the meaning of the E(r) and P(r) fields we are considering.

We now want to determine the contribution to the electric field caused by the polarization itself.

We know from electrostatics that the individual microscopic dipole p exerts an electric potential:

What is now the potential caused by the polarization P(r)? In each tiny volume element dV we have

so that the potential exerted in r by the polarization P will be

Observing that [11, p. 15]:

we finally have:

The complete potential when dielectrics are present [12, p. 281] is in general the sum of a contribution from volume charges with volume density ρ(r), of surface charges with surface density σ(r) plus the contribution earlier considered of the polarized volume elements dV. On the overall we have [1]:

Note that the integrals run over the entire volume of the dielectric and overall surfaces present.

The problem is now that of determining the free energy of a system whose “solvent configuration” is not in equilibrium with the ionic charges of the TS and so cannot be determined by standard electrostatics, just knowing the ionic charge distribution, the dielectric constant and the overall radius of the TS considered as a sphere whose charge is the sum of the charges of the reactants, as is usually done when the polarization is in equilibrium with the charges [5]. Ref. [2] was devoted to finding an expression for this free energy in terms of equilibrium and nonequilibrium electrostatic macroscopic functions for a two spheres system.

For the description of systems with nonequilibrium electrical polarization, M. uses three vectors [2], the electric field strength E, the polarization P, and the electric field strength E_c which the charge distribution would exert “if it were in a vacuum rather than in a polarized medium” [2]. The electric field strength in the absence of polarized medium is the negative gradient of the potential due to the polarized spheres and so

Eq. (2.11a) can be expressed in the form:

which is valid for any system, equilibrium or not. Only in an equilibrium system can P be expressed in terms of E.

where α is the total polarizability of the medium and considering that E = −∇ψ. The values of E_c, P_e, P_u, P, and E which obtain in the intermediate state X^∗ will be designated by an asterisk, while those characteristic of state X will bear no asterisk. Since the U-type polarization is the same in both states, equals P_u.

The electrostatic free energy of any state is generally defined as the reversible work to charge up that state [1, 13, 14]. The electrostatic free energy of the nonequilibrium systems X^∗ and X will be derived in the next starred paragraph.

We give here the final results of the derivation:

Where the dot denotes the dot product of two vectors and where α_u is the polarizability of the U-type polarization. α_u can be expressed [2] in terms of the static dielectric constant D_s and the optical constant D_op. D_op is the square of the refractive index in the visible region of the spectrum [1, 12]:

If the solvent is water, D_s = 78.5 and D_op ~ 1.8 at 25^∘C.

“The electrostatic contribution ΔF^∗ to the free energy of formation of the intermediate state X^∗ from the reactants in the dielectric medium” [1] is found by subtracting from F^∗ the reversible work required to charge them up when they are isolated, that is, far apart in the dielectric medium:

Similarly for X:

The reversible work required to charge up a conducting sphere of radius a in vacuum with charge e is e²/2a and e²/2aD_s in the dielectric medium [15, p. 204 ff.].

In Fig. 1^∗, the uncharged system, the charged one with reactants at infinite distance from each other and the charged system X^∗ with reactants at distance R, are represented using schematic orientational polarization diagrams.

Fig. 1^∗.

2.4.^∗The Electrostatic Free Energy for the Nonequilibrium Systems X^∗ and X

Derivation of the Formulas

In [2], there are two derivations of Eq. (2.14). For Marcus’ famous expression of the nonequilibrium polarization expression by a reversible two steps charging process see Chapter 3; a second one, more intuitive, is reported here.

We have just seen how much free energy is stored in a charged sphere. Likewise, the free energy stored in an induced dipole p is given by:

where α₀ is the electronic polarizability. It is the energy necessary to create the induced dipole in a field E. The free energy of interaction of the dipole with the field is [12, p. 285]:

If we now imagine turning off the field E but of holding fixed the dipole moment, the energy of interaction (2.19) would become zero, but the energy (2.18) stored in the dipole would remain there.

We are interested in a system made up of a distribution of charges in a polarized medium with polarizabilities P_u(r) and P_e(r). The free energy of the whole system is the sum of the free energies of each volume element, considered isolated, and the free energy of interaction among different volume elements. It follows from Eq. (2.18) that the free energy stored in each volume element is

summing up the two contributions to the total polarization. This follows from Eq. (2.18) successively substituting in it p = P_udV and p = P_edV, α₀ = α_udV and α₀ = α_edV. We have α₀ = p/E = P_edV/E = (P_e/E)dV = α_edV. The same for α_u. We determine now the contributions arising from the various interactions among different volume elements. To this aim, we consider the contribution to the whole field strength E given by the fields: E_c, arising from the charges in vacuum, E_u, arising from the U-type polarization and E_e from the E-type, that is,

We see from Eq. (2.19) that the free energy of interaction of the entire field with the dipole of E-type polarization in a volume dV is:

Similarly, for the U-type polarization in the same volume dV we have the interaction energy:

Finally, one should consider the free energy of interaction among charges. This term is the same as that among charges in vacuum because the terms describing the interactions among charges and polarizations have already been considered. We have, for unit volume [12, p. 288]:

The expression for the electrostatic free energy F of the system will be the sum of the earlier terms integrated over the whole volume of the system. In performing the integration, some terms must be divided by two in order not to count the interactions twice. Let us consider, for example, the integral

The differential term P_e(r) · E_e(r)dV is the interaction between the dipole P_e(r)dV and the electric field E_e(r) which is the resultant in r of the electric field strengths due to all the electronically polarized volume elements dV′ ≠ dV. At the same time, the dipole P_e(r)dV will participate in the electric field E_eon all dV′ ≠ dV. This means that the interaction between two dipole elements, P_e(r₁)dV₁ and P_e(r₂)dV₂ say, is counted twice in the integral, which is then to be divided by two in order to correctly take into account the interactions among different electronically polarized volume elements.

The terms which are to be divided by a factor of two are those in the integral:

Marcus thus obtained:

Using Eq. (2.20) and the relations P_e = α_eE, P = P_e + P_u into Eq. (2.21), he finally obtained:

which is identical with Eq. (2.14).

2.5.Restraints Imposed on the Forms X^∗ and X of the Transition State

In the framework of the 1956 theory, we have seen that the two forms X^∗ and X of the TS have the following properties:

(i)Same atomic configuration

(ii)Same total electronic energy

(iii)Same U-type polarization, that is,

Moreover, the transition X^∗ → X is of a Franck–Condon type [16] and so

(iv)The momentum distribution of the atoms in X^∗ and X is the same

Properties (i) and (iv) say that the configurational and the thermal entropy [17] of X^∗ and X are the same so that the only possible entropy difference between them can arise from a possible difference in electronic degeneracy between products and reactants, denoted by M. as ΔS_e. Designating with ^∗ the product of the electronic degeneracies of the reactants and with the same for the products, ΔS_e is given by [1, 18]:

ΔS_e is usually equal or close to zero [1].

Since F^∗ = U^∗ − TS^∗, F = U − TS and U^∗ = U for X^∗ and X, we have then F − F^∗ = −T ΔS_e.

In his first treatment of the kinetics of ET reactions, M. considers a further restriction on the reactants: all atoms within the sphere of radius a “maintain their relative positions throughout the reaction” so that consequently the radii a₁ and a₂ remain fixed throughout the reaction, as seen earlier.

Because of all of the earlier restrictions, the overall standard free energy of formation of the products from the reactants ΔF⁰ can be written as sum of three terms:

(i) free energy of formation of state X^∗ from the reactants

(ii)−T ΔS_e, free energy of formation of X state from X^∗

(iii) free energy of formation of the products from X state:

This equation “summarizes the restraints imposed upon the two intermediate states.”

2.6.Minimization of the Free Energy Subject to the Free Energy Restriction (2.24)

The Formula for ΔF ^∗

There is an infinite number of pairs of intermediate states X^∗ and X which could satisfy the free energy restriction (2.24). All of the pairs have the same charge distribution but the r-dependent P_u(r) is different for different pairs. The problem is now of determining the P_u(r) of that pair of “groups of atomic configurations” that “contribute to the macroscopic or ‘thermodynamic’ properties of the transition state TS” [19]. Such a pair, subject to the free energy restriction (2.24), is characterized by the minimum free energy F^∗, because we are looking for that thermodynamic state which has the maximal probability of formation from the reactants.⁽⁷⁾ In terms of atomic configurations, one can say that many of them can satisfy the energy restriction, but only a group of atomic configurations contribute to the macroscopic or “thermodynamic” properties of the TS, namely those which minimize its free energy of formation from the reactants [19]. I give here the final results for P_u and F^∗. They allow the calculation of ΔF^∗, which is one of the major results of the Marcus theory.

The work required to charge up two conducting spheres of charges and and of radii a₁ and a₂ in the dielectric medium is given by⁽⁸⁾:

Similarly for

Marcus obtained for the P_u(r) and the F^∗:

Using the preceding results Marcus obtained the famous formula for ΔF^∗:

where we have used the conservation of charge relation and R is the mean activation distance in the activated complex.

In order to properly appreciate the earlier astonishing final result, the reader should consider that it was obtained by three very clever steps. In the first one, M. discovered the beautiful general formulas (2.13) and (2.14) (appearing in the demonstration earlier in the form of Eq. (2.22)) for the electrostatic free energies F^∗ and F. He then devised the minimization procedure to get the formulas for the free energies of the activated complexes. Finally, starting from an expression for F^∗ depending on E^∗, P_u, and E_c he ended up—as if by magic—with a formula of great simplicity and elegance and of immediate applicability depending on the charges the geometry of the system, (a₁, a₂, R), the dielectric properties of the medium (D_op and D_s), and the thermodynamic quantities ΔF⁰ and T ΔS_e which appear through the Lagrangian multiplier m obtained from the formula:

Let us use it to calculate the value of m for the electron exchange reaction:

We see that for this case each term on the RHS of Eq. (2.28) is zero. On the LHS, the terms in parenthesis are different from zero, Δe = 1 and so Eq. (2.28) can only be satisfied if 2m + 1 = 0, that is, if

To understand what this means, let us consider again formula (2.26) for the P_u which minimizes F^∗.

We see that if

that is, the value of P_u which minimizes F^∗ is the one that would be in equilibrium with a hypothetical electric field equal to the arithmetic average of the fields E^∗ and E. And is consistent with the intuitive idea that the P_u of the TS would be in equilibrium with fictitious charges on the ions which would be averages between initial and final charges!

^∗ Proof of Eqs. (2.26), (2.27), and (2.28)

At the minimum of F^∗, we shall have δF^∗ = 0 (M. uses here the Calculus of Variations, elements of which can be found in books of Advanced Calculus or of Mathematical Methods of Physics and Chemistry, for example in Refs. [20–24] and, more extensively, in Ref. [25]).

M. expresses the variation δF^∗ as function of the variations δP_u of the function P_u.

The charges are fixed, so is fixed and Computing δF^∗ from Eq. (2.13) one obtains:

We have seen that so that:

δF^∗ depends on the variations of P_u and E^∗. But we know that P_u and E^∗ are not independent because E^∗ is the negative gradient of a potential dependent on the charges but also on the polarization P_u, as we have seen in Eq. (2.11b). In [2] it is demonstrated that Eq. (2.29) can be expressed as:

Putting δF^∗ = 0 we have:

Note that if there were no constraints on δP_u(r), i.e. if its arbitrary variations would not affect E^∗, Eq. (2.24) would be satisfied for arbitrary variations δP_u(r) only if

that is, if

which is the usual electrostatic relation between P_u and E^∗ for systems with equilibrium U-type polarization. In our nonequilibrium system, the variations δP_u must satisfy the equation of constraint (2.24) where from we see that:

The equation for the variation δF is (see Eq. (2.30a)):

Subtracting Eqs. (2.30b) from (2.30a) one has:

Equations (2.30a) and (2.31b) are to be satisfied simultaneously. Using the method of Lagrangian multipliers (see, e.g., Refs. [20–25]), the second condition multiplied by a Lagrangian multiplier m is added to the first one to have:

This is an identity valid for every arbitrary variation of P_u in each volume element. It can be equal to zero only if the term in braces is equal to zero and this condition gives:

So this is the P_u we were looking for. The first term is the usual electrostatic equilibrium relation between P_u and E^∗. A physical interpretation of Eq. (2.26) was given in the preceding section.

M. expresses now E^∗ and E in terms of and E_c which can be easily calculated from the known charge distribution. The formulas for E^∗ and E are [1]:

With the aid of these equations, Eq. (2.26) for P_u becomes:

Introducing the equations for E^∗, E, and P_u into Eq. (2.13) for the electrostatic free energy of state X^∗, we obtain:

We also have:

where from m can be calculated.

We designate now the charges of the reactants 1 and 2 in state X^∗ by and The corresponding charges in X will be e₁ and e₂. The radii a₁ and a₂ are assumed not to change essentially during the course of the reaction.

The vector is the negative gradient of the potential which the reacting ions in state X^∗ would exert if they were in a vacuum rather than in a polarized medium. The potential exerted at point r by two ions of charges e₁ and e₂ is:

where r₁ and r₂ are the distances of the field point r from the centers of the ions. The vector E_c is −∇ψ, ψ being given by the earlier equations. We thus have:

The expressions for E_c and are introduced into Eq. (2.32) and into Eq. (2.33) and the integrations are performed. The following integrals are used for this purpose:

where i can be 1 or 2, R is the distance between the centers of the ions and the integration volume excludes the volume physically occupied by the two ionic spheres, that is, r₁ ≥ a₁ and r₂ ≥ a₂ simultaneously.

With the aid of Eqs. (2.24), (2.32), (2.34), (2.35), (2.25), M. obtained Eq. (2.27) and with the aid of Eqs. (2.33), (2.34), (2.35), (2.25) he obtained Eq. (2.28).

2.7.Rate Constants of the Elementary Steps

(a) Estimation of k₋₁ and k₃

The rate constants k₋₁ and k₃ are associated with the unimolecular reactions of dissociation of X^∗ to reform the reactants and of X to form the products, respectively. M. considers a model for X^∗ in which the collision complex in solution is made up of the two reactants contained in a solvent cage [26, pp. 156, 403–404]. The reactants are supposed to vibrate in the solvent cage striking the walls about 10¹³ times a second. The chance of one of them escaping from the cage is α per collision with the walls of the cage, with α < 1. In a first mode of dissociation of X^∗, this reaction happens every time one of the reactants escapes from the cage, and so the unimolecular rate constant for this mode of X^∗ decomposition is 10¹³α sec⁻¹.

A second mode of decomposition is “a disorganizing motion of the solvent destroying the polarization appropriate to the intermediate state.” We have seen that the relaxation times for orientation and atomic polarizations are from 10⁻¹¹ to 10⁻¹³ seconds. The atomic polarization is an appreciable fraction of the total U-polarization P_u(r) so that if it reverts to an unsuitable value, this is enough to destroy the X^∗ state. The unimolecular rate constant for this mode of decomposition would then be 10⁻¹³ sec⁻¹. This is no less than the value for the solvent escape mechanism and therefore it is to be considered a prevalent mode of decomposition. What has been said for X^∗ is also valid for X, so that:

(b) Estimation of k₁

The equilibrium constant of reaction (2.1) is k₁/k₋₁. M. calculates the equilibrium constant using the recipe given by Statistical Mechanics, using in this case elementary collision theory and partition functions (see, e.g., Ref. [27], Ref. [17, p. 382], and Ref. [7, p. 93]) and, knowing k₋₁ from the previous section, obtains for k₁ the formula:

where Z is the collision number in the gas phase, see Ref. [7, p. 93, 130], approximately the number of collisions occurring between two neutral species in unit volume in unit time at the mean separation distance in the TS [9]. Z is about 10¹¹ liter mole⁻¹ sec⁻¹, which is adequate also for the majority of reactions in solution, see Ref. [7, p. 130]. Marcus will later correct this number, using the more precise value 10¹² he obtained in Ref. [28].

^∗ Proof of Eq. (2.36)

Each of the two reactants A and B has three translational degrees of freedom. In the intermediate state X^∗, they become three translational degrees of freedom of the center of gravity of the two reactants, two rotational degrees of freedom about this center and one degree of freedom of the reactants vibrating with respect to each other in the solvent cage. The partition function of the three translational degrees of freedom of reactant 1, a rigid sphere A say, in the gas phase would be

The corresponding factors for reactant 2 (B), and those for the state X^∗, are obtained from the preceding formula replacing m₁ by m₂ and (m₁+m₂) respectively. The rotational partition function is [27, p. 32]:

where μ is the reduced mass:

and R is the distance between the centers of gravity of the reactants. The vibrational partition function for motion within the cage is equal to one, within a factor of three.

Introducing the earlier results into a statistical mechanical expression for the equilibrium constant of reaction (2.1) [27, 7, p. 93], one then gets:

It now so happens that k₋₁ has the value of 10¹³ sec⁻¹, which is also the value of kT/h. Substituting then k₋₁ with kT/h one gets, after cancellations,

This expression is Eq. (2.36) with Z the collision number in solution taken as approximately equal to that in the gas phase [7, p. 130]. ΔF^∗, the free energy of formation of the intermediate state X^∗ from the reactants

is the free energy for the reaction A + B → products.

(c) Estimation of k₂ and k₋₂

The free energy difference between states X^∗ and X is −T ΔS_e where ΔS_e is given by Eq. (2.23).

The equilibrium constant for the interconversion reaction of X^∗ and X, Eq. (2.2), is then:

This ratio will in general be approximately or exactly equal to 1.

In order to estimate k₂ and k₋₂, some model is to be assumed for the electronic jump process. Such process was treated as an electronic tunneling process in Ref. [29].⁽¹¹⁾ The probability κ_e of an electron tunneling through a barrier from one reactant to the other was estimated to depend exponentially on the tunneling distance r_ab between A and B in X^∗, that is:

For the ferrous-ferric isotopic exchange reaction in water a value of β = 1.23Å⁻¹ was estimated. The tunneling distance between the ions is taken by M. as about twice the diameter of a water molecule, that is, 5.5Å, because each ion is supposed to be strongly bound to an innermost layer of water molecules. M. calculated κ_e = 10⁻³ for this distance.

For k₂ we have the following:

The above number of times per second is given by the frequency of motion of the valence electron in the ground state of the ferrous ion, which is of the order of the frequency of excitation of this electron to the next higher principal quantum number (for a brief discussion of the correlation between classical frequency of revolution in electron orbits and quantum frequency of energy emission see, e.g., Ref. [30]). From data on the energy levels of the ferrous ion [31] M. estimated the frequency to be 2 × 10¹⁵ sec⁻¹, so that:

which is of the same order of magnitude as k₋₁ considering the approximations which have been made.

2.8.Validity of the Assumed Small-Overlap Transition State

In the Marcus’ theory of ET reactions, states X^∗ and X are supposed to be isoenergetic with weak electronic interaction of the reacting particles, that is, with a small overlap of their electronic orbitals.

A fundamental limitation to the statement of equal energy for X^∗ and X comes from the energy–time uncertainty relation (a very nice discussion of it can be found in Ref. [32]. See also, e.g., Refs. [33, 34]). The energy–time uncertainty relation relates the lifetime τ of a state to the uncertainty δε of its energy:

which means that the energy of state X^∗ is broadened by an amount δε and the same is true for state X. The energies of the two states can then be equal only within an energy interval of 2δε prescribed by the above uncertainty principle. The energy broadening 2δε is then like a minimal splitting of PESs in the neighborhood of the nuclear configurations space region where the ET process may happen. Such a splitting is related, as we have seen, to the orbitals’ overlap in the activated complex and “The greater the overlap the shorter will be the lifetimes of X^∗ and X.” But 2δε is related to τ in Eq. (2.38) and τ depends, on its turn, on the rate constants k₂, k₋₂, k₃, and k₋₁ making up the factor 1/[1 + (1 + k₋₂/k₃)k₋₁/k₂] in Eq. (2.7). Their knowledge allows then—through knowledge of the lifetime of the activated complex—an estimate of the splitting 2δε and so of the extent of the overlap in the complex.

From Eq. (2.38), we see then that overlaps and lifetimes are inversely proportional. Let us consider the lifetime of X^∗ which can disappear through the reactions:

Its lifetime is then equal to 1/(k₋₁ + k₂) and is essentially the same as that of state X. For sec mole⁻¹ and mole⁻¹. In Eq. (2.24):

which was derived supposing exactly equal energies for X^∗ and X, we must replace ΔF⁰ with ΔF⁰ ± 0.15kcal mole⁻¹, a negligible correction with negligible effects on the calculated ΔF^∗.

Even if, as a result of large values of k₂ and k₋₂, the lifetimes were as small as 10⁻¹⁴ sec, 2δε would be only 1.5kcal mole⁻¹, with a relatively small effect on ΔF^∗.

A large-overlap activated complex would be characterized by a switching of electronic structures from X^∗ to X of the order of the electronic frequency in molecules. These frequencies are of about 10¹⁵ sec⁻¹. The lifetime τ would be about a femtosecond, 10⁻¹⁵ sec and 2δε would have the value of 15kcal mole⁻¹, that is, 0.65eV, a large value. In Eq. (2.24), ΔF⁰ should be replaced by ΔF⁰ ± 15kcal mole⁻¹, and the restraint involved in the equation would not be very strong anymore.

As a result of the earlier discussion, we have now a quantitative measure to distinguish a small-overlap activated complex from a large overlap one. If the lifetimes of the intermediate states X^∗ and X are greater than 10⁻¹⁴ sec, we have a small-overlap activated complex. “On the basis of the calculations for k₃ and k₂ given previously we infer that a small-overlap activated complex complex may well prevail for many ET reactions.” (M.)

NOTE: This first formulation of the theory was later replaced by a later derivation. In this first paper, M16, there is a quasiequilibrium between X^∗ and X with rate constants in the opposite directions of the reaction. In the later derivation in M53 of 1965, Marcus has a more modern formalism which applies the Transition State Theory in the spirit of Wigner in his paper in the Transactions of the Faraday Society of 1938 and of Eyring, Walter, and Kimball in their book “Quantum Chemistry” of 1944. There is there a ballistic treatment and a nonadiabatic formalism. The k₋₁ or the equivalent for the electron going through is the rate constant for the fastest velocity at which an ET could occur, with the ET happening in 100 femtoseconds according to the more recent experiments.

2.9.The Interionic Distance R

The interionic distance R of the reactants in the intermediate states X^∗ and X affects the overall reaction rate because it appears in (i) the tunneling probability κ_e, Eq. (2.37), and (ii) in ΔF^∗, Eq. (2.27).

(i)κ_e determines the rate constant k₂ in (2.37b). We saw that when k₂ ≥ 10¹³ sec⁻¹, the overall reaction rate is independent of k₂. But κ_e depends exponentially on R (r_ab in Eq. (2.37)) and, for large R, k₂ will become small and will then affect the overall reaction rate. Because of the exponential decrease of the electron jump probability with R, the reaction rate k₂ will be maximal for minimum R.

(ii)R appears in ΔF^∗, Eq. (2.27), in the first term, the one of the Coulombic interaction between ions, and in the second, related to the barrier due to difference in solvation of the ions. For an electron exchange between positive ions, we see that a decrease of R increases the contribute to ΔF^∗ in the first term and decreases it in the second. If the second term in Eq. (2.27) is greater than the first, so that the major barrier to reaction lies in the difference of solvation about reactants, the reaction will occur most readily for the smallest possible R. As a matter of fact, until now, when the ions come close to each other, we have only considered the overlap of their electronic orbitals. But another kind of overlap should be considered: that of their solvation atmospheres, because, according to M., when the solvation atmospheres overlap they become more similar to each other so as to lower the solvation barrier to reaction and to favor the ET process.

In summary, while in theory the most appropriate value of R is to be found maximizing the overall rate constant with respect to R, limitations in the quantitative knowledge of the electronic jump process suggested to Marcus in 1956 to consider as the most suitable value for R its minimum value, that is, the sum of the radii of the separated reactants:

2.10.The Radius a

The radii a₁ and a₂ of the reacting ions enter the expression for rate constant k₁ through ΔF^∗, Eq. (2.27). M. considers in its first paper the radii for simple cations and anions and for complex cations and anions.

In the case of simple cations, the radius is taken as the sum of the crystallographic radius plus the diameter of a solvent molecule, water in particular, since the cations have a fairly tightly bound hydration layer. The recipe follows the previously assumed model of a reactant considered as a sphere formed by the ion inside a spherical shell made up by an innermost layer of dielectrically saturated solvent molecules, outside of which the solvent is assumed to be dielectrically unsaturated.

In the case of simple anions, this recipe needs to be modified because the solvent in the first layer appears in this case not to be completely saturated. After this warning, M. did not give a recipe for this case in Ref. [13]. The topic of effective polarizing radius will be introduced in the “Effective Radii” and “Ionic Radii” sections of Chapter 3.

In the case of complex ions such as or the degree of dielectric saturation in the first solvation layer is much less than for monoatomic ions because the orienting effect of the ions’ electric field varies roughly as the inverse square of the distance from the center of the ion. In this case then the radius of the ion is to be considered as only that of the naked ion.

NOTES

1.M. will give in his 1965 paper an expression for k_bi given by

M: “The 1965 expression is derived from statistical mechanics, the other, is an approximate macroscopic expression. In the 1965 expression there are all sorts of microscopic details that are not present in the 1956 paper.”

2.M’s correction of the statement on p. 970 of [1], “We shall treat an ion plus its rigid, saturated dielectric region as a conducting sphere of radius a.”:

M: “I think it is unfortunate that I called it a conducting sphere. . . in the later papers of course I didn’t have the system as a conducting sphere.”

NOTE: M. believes that he used that model in the first two papers because he was under the influence of Born’s work on the charging of ions, where they were considered as conducting spheres.

3.M: “Both α_e and α_u depend on temperature. α_e is really related to an optical dielectric constant, so it depends very little on temperature, but it depends a little. α_u depends upon a difference of dielectric constants, one of which is the static dielectric constant and so it definitely depends on temperature.”

4.M: “The polarization in an equilibrium system is always proportional to the electric field, so that means that as soon as you specify the polarization in an equilibrium system, you specify the electric field everyplace. In a non-equilibrium system that is not true.”

5.Q: The macroscopic system is formed by the couple of the reactants plus all the other solvent molecules and ions surrounding them, so in theory the potential is exercised by all of the 10²³ molecules and ions which make up the macroscopic system. But in practice how many are the solvent molecules and ions interacting with the couple of electron exchanging reactants?

M: “I’d say, maybe a few thousands, because Coulomb forces are long range, and of course it depends on how dilute the solution is. But it is true that usually you have solvent there which dampens the Coulomb forces, so the distance of the solvent molecules and ions affecting the reacting pair doesn’t go beyond a few Debye lengths, and you can call mesoscopic the system that would cover most of the relevant part of the system. The main advantage of considering it would be for numerical calculations, you wouldn’t have to use 10²³ coordinates.”

6.Q: In the M16 paper, you say nothing about the nature of the reaction coordinate but looking at the nonequilibrium free energy formula one sees that at constant E_c the functional F depends on the function P_u, that is, apparently you tacitly consider the P_u polarization as the ET reaction coordinate.

M: “In the 1956 paper the reaction coordinate is really P_u(m). m is really the reaction coordinate. You see, P_u varies all over the place, every point has a different value of P_u, but those P_u’s are connected in a way in a form of reaction coordinate with that m, the Lagrangian multiplier, so really if one would ask what really was the reaction coordinate in the 1956 paper, a single coordinate, you could say that m was the coordinate. m = 0 when the function P_u is everywhere appropriate to the reactants, if it’s m = −1 then the function everyplace is appropriate to the products, but in the TS has a value given by that (2m + 1)λ formula. . . so m is really the reaction coordinate in the first paper, although I didn’t say it explicitly, and it describes how P_u changes along the reaction coordinate everyplace. . . m can serve as a reaction coordinate, in the 1956 paper, but taking on for the actual system the value given by −(2m + 1)λ = ÄG⁰.”

Q: But what exactly is a reaction coordinate?

M: “Every coordinate that leads from reactants to products, and of course that means that it could be some terrible reaction coordinate and the whole idea is to try to find the best within some framework of reaction coordinates. Nothing unique. I mean, there are different reaction coordinates, then better ones, then better ones. . . and getting the best one is a big question and one may not be able to do it. So one does the best one can. One should speak of the best possible reaction coordinate that the writer can think of.”

Q: My ideas are much more clear now.

M: So are mine.

Q: Through these discussions things become more clear. M: Yes, and that’s the beauty of discussions.

7.M. “If you are thinking of each pair X^∗ and X as occupying some section of the 10²³–1 dimensional hypersurface that constitutes the [whole] TS and if you are assigning to each pair, to each member of the pair a free energy, then the one that has the minimum free energy, that would be the TS [considered in the paper]. You don’t have to consider the full thing, you can consider just an important part of it. In other words, it is wise to consider the subset of the hypersurface which is associated with the minimum free energy. The different parts of the hypersurface correspond to different thermodynamic systems and you pick the one that has the minimum free energy. . . you think of that 10²³–1 dimensional hypersurface effectively broken into different systems. . . each polarization function being specific of each system.”

8.M: “The 1956 theory neglects interactions of other pairs of reactants on this particular pair, it neglects the effects of other reactants, it focuses only on this pair. . . it treats the different pairs as independent of each other. A later paper treats the effects of other ions, of the ionic atmosphere, on the particular pair.”

9.M. used the variational condition δP_udV = 0 to demonstrate that P_u = α_uE. He did so because he wanted to be sure that there were no other constraints that would give the earlier standard equilibrium result P_u = α_uE, that is, he wanted to be sure that if he didn’t impose other conditions he would get the usual expected result.

10.M. points out that when one considers the condition F^∗ = F, that means that one is at the intersection of the potential energy surfaces and one should consider that F^∗ is a function of P_u and that F^∗ equals F as function of P_u and that when one considers δF one means δF obtained by varying δP_u. It is from the condition F^∗ = F that δF^∗ − δF = 0 follows. If one plots the free energies of reactants and products as functions of P_u, one has δF^∗ = 0 at the bottom of the reactants free energy well, δF = 0 at the bottom of the free energy well of the products with different values of P_u corresponding to those two minima. One has δF^∗ − δF = 0 at the TS at the same common value of P_u.

11.M. comments on the mistake that was made by the authors in Ref. [27] in the calculation of tunneling probability:

M: “They calculated the probability of tunneling, but that’s tunneling of the electron, what they should have done is multiply that tunneling by the number of times the electron is hitting that tunneling barrier, in this classical picture you have sort of classical frequency, but that’s 10¹³ per second, so [overlooking that] they made a big error.”

References

1.R. A. Marcus, J. Chem. Phys. 24, 966–978, (1956).

2.R. A. Marcus, J. Chem. Phys. 24, 979–989, (1956).

3.K. J. Laidler, Chemical Kinetics, 3rd Edition, Harper Collins Publishers (1987).

4.W. C. Gardiner, Jr., Rates and Mechanism of Chemical Reactions, W.A. Benjamin, Inc., New York, Amsterdam (1969).

5.S. Glasstone, K. J. Laidler, H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York and London (1941).

6.R. E. Weston, Jr., H. A. Schwarz, Chemical Kinetics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1972).

7.A. A. Frost, R. G. Pearson, Kinetics and Mechanism, 2nd Edition, John Wiley & Sons, Inc., New York, London, Sydney (1961).

8.R. A. Marcus, Discuss. Faraday Soc. 29, 21–31, (1960).

9.R. A. Marcus, Ann. Rev. Phys. Chem. 15, 155–196, (1964).

10.R. A. Marcus, J. Chem. Phys. 43, 679–701, (1965).

11.D. J. Griffiths, Introduction to Electrodynamics, 3rd Edition, Prentice Hall International Inc. (1999).

12.G. Joos, Theoretical Physics, 3rd Edition, Dover Publications, Inc., New York (1986).

13.W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism, 2nd Edition, Addison-Wesley, Reading, MA (1962).

14.C. Kittel, Thermal Physics, John Wiley & Sons, Inc., New York, London, Sidney, Toronto (1969).

15.J. O’ M. Bockris, A. K. N. Reddy, Modern Electrochemistry 1, 2nd Edition, Plenum Press, New York and London (1998).

16.G. Herzberg, Molecular Spectra and Molecular Structure, I. Spectra of Diatomic Molecules, 2nd Edition, Van Nostrand Reinhold Company, New York, Cincinnati, Toronto (1950).

17.K. Denbigh, The Principles of Chemical Equilibrium, 4th Edition, Cambridge University Press (1981).

18.E. A. Guggenheim, Thermodynamics, 6th Edition, North-Holland Publishing Company, Amsterdam, New York, Oxford (1977).

19.R. A. Marcus, Can. J. Chem. 37, 155, (1959).

20.W. Kaplan, Advanced Calculus, 2nd Edition, Addison-Wesley Publishing Company, Reading, Massachusetts, Menlo Park, California, London, Don Mills, Ontario (1973).

21.H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry, 2nd Edition, D. Van Nostrand Company, Inc., Princeton, NJ (1956).

22.J. Mathews, R. L. Walker, Mathematical Methods of Physics, 2nd Edition, W.A. Benjamin Inc., New York (1970).

23.M. Boas, Mathematical Methods in the Physical Sciences, 3rd Edition, Wiley, New York (2005).

24.G. Arfken, H.-J. Weber, Mathematical Methods for Physicists, 6th Edition, Elsevier Academic Press (2005).

25.F. B. Hildebrand, Methods of Applied Mathematics, Dover Publications, Inc., New York (1992).

26.J. I. Steinfeld, J. S. Francisco, W. L. Hase, Chemical Kinetics and Dynamics, Prentice Hall (1989).

27.B. Widom, Statistical Mechanics, A Concise Introduction for Chemists, Cambridge University Press (2002).

28.R. A. Marcus, Int. J. Chem. Kinet 13, 365, (1981).

29.R. J. Marcus, B. J. Zwolinski, H. Eyring, J. Phys. Chem. 58, 432, (1954).

30.H. E. White, Introduction to Atomic Spectra, McGraw-Hill Book Company Inc., New York, St. Louis, San Francisco (1934).

31.C. E. Moore, Atomic Energy Levels (National Bureau of Standards, 1952), circular 467, Vol. II.

32.F. Mandl, Quantum Mechanics, John Wiley & Sons, Chichester, New York, Brisbane (1992).

33.B. H. Bransden, C. J. Joachain, Quantum Mechanics, p. 75, 2nd Edition, Prentice Hall (2000).

34.J. J. Sakurai, Modern Quantum Mechanics, The Benjamin/Cummings Publishing Company Inc., Menlo Park California, Reading, Massachusetts (1985).