MODULE 23 (701) REACTIONS IN SOLUTION: DIFFUSION AND CONDUCTION In the liquid phase, free space is very limited. Each molecule interacts strongly with neighboring molecules. “Mean free path" is almost meaningless. (In gas phase the average distance between collisions is large compared to molecular radius.) In solution phase, molecules move down concentration gradients. When ionic species are involved, potential gradients become important.

MODULE 23 (701) DIFFUSION Consider a dilute, homogeneous, isotropic solution of molecules A and B in an inert solvent. Imagine a single molecule of A as at the center of a small volume of solution in which species B (and solvent) predominate.

MODULE 23 (701) At A, [B] is zero and a gradient exists which favors B molecules to approach A through normal diffusive processes (Brownian motion). Diffusion tends to equal out concentration gradients. Molecules at a point of high concentration will tend to move towards regions of lower concentration (Fick’s first law).

MODULE 23 (701) Where n i is the excess of species moving through a plane of area A cm 2 in a direction x, perpendicular to the plane. dc i /dx is the concentration gradient along the linear dimension x. The flux (  i ) in units of (quantity) cm -2 s -1 is given by For a spherically symmetrical situation, where dc i /dr is the spherically symmetrical concentration gradient about molecule A and D i is the diffusion coefficient (in cm 2 s -1 ) of the species i.

MODULE 23 (701) This is Fick's law for a radial distribution. The negative sign assures that D i is a positive quantity. This law applies to all species, neutrals and ions, but for the latter another factor arises. IONIC CONDUCTANCE Another definition of flux is given by the product of c i (in molecules cm -3 ) and velocity (in cm s -1 ) For a spherically symmetric potential gradient dE/dr about a central ion of (field dependent) mobility u i (cm 2 V -1 s -1 )

MODULE 23 (701) where z i is the ionic charge and the factor -z i /lz i l accounts for the direction of the ionic motion in the field. In liquids, the mobility of ions is not dependent on the potential gradient they experience since the energy gained by an ion from the field between collisions with the surrounding molecules is rapidly dissipated in the next few collisions. TOTAL MOTION An ion in solution has motion derived from both concentration gradients and potential gradients.

MODULE 23 (701) In a chemical reaction both motions are clearly important since without the diffusion-driven component, interaction between ions of like charge would never occur. Using the two identities: where e is the electronic charge, k B is the Boltzmann constant, T is absolute temperature, and V(r) is the potential energy of an ion of charge z i e Coulombs in a potential field of strength E volts,

MODULE 23 (701) RAPID BIMOLECULAR REACTIONS The maximum rate at which a bimolecular reaction can occur in solution is limited by the rate at which the reactants can get together, i.e., by diffusion. Once the molecules approach to their reaction distance, the reaction itself can become instantaneous. Consider a photochemical reaction, initiated by a brief flash of light. Initially only A is present in solution, and then suddenly the flash produces a low concentration of B species in a homogeneous distribution.

MODULE 23 (701) When the concentrations of both A and B are low, we can imagine that immediately after the flash the concentration of B in a small volume that contains a single A is zero (our earlier condition). Thus a flux of B molecules towards A will be set up. If A is also an ion, we can regard each A species as being at the center of radial concentration and potential gradients. Let the total flux of B species moving towards A be  B [B] r is the local concentration of B at a distance r from A

MODULE 23 (701) A steady state condition will be achieved when the rate of transport of B molecules into the environment of an average A molecule becomes equal to the rate at which they are removed by the chemical reaction. The total inward flux through a spherical shell of radius r centered on A, of area 4  r 2 is -4  r 2  B. For all A molecules this is -4  r 2  B [A]. Thus the reaction rate is given by

MODULE 23 (701) integrating between the limits of r = R and r = ∞ where R is the contact distance  has the dimensions of length and the properties of a reaction radius. Removing the substitution

MODULE 23 (701) k (the overall rate constant) exhibits a maximum value of k D even if k R  ∞ (i.e., the reaction is instantaneous at r = R). Thus k D is the rate constant of a bimolecular reaction in solution that is governed solely by the diffusion together of the reactants. i.e. the rate is infinite once the reactants are within the collision radius.

MODULE 23 (701) In M -1 s -1 units where N 0 is Avogadro's number. DIFFUSION-LIMITED REACTIONS The above equation can be used in a predictive way when we know R and the form of the potential V(r). We examine the situation for charged and uncharged reacting species.

MODULE 23 (701) (i)Uncharged reactants for V(r)  0 this reduces to Thus k Diff can be computed for non-ionic species when D i and R are known (see later).

MODULE 23 (701) (ii) Ionic reactants: The potential energy term, V(r), is given by a Coulomb term  is the static dielectric constant of the medium. The radius r 0 is the critical radius around an ion where the Coulomb and the thermal energies are equal. In H 2 O (  at 25 o C, r 0 = 0.71 nm; in hexane (  = 2) at 25 o C, r 0 = 28.0 nm.

MODULE 23 (701) The second term on the right hand side is the Debye factor. It takes into account the Coulomb effect that modifies the neutral- neutral diffusion controlled rate constant. For z A = 1, z B = -1 (z A z B = -1). For small ions (R << r 0 ) in low  media   r 0 and

MODULE 23 (701) CALCULATION OF D i An Einstein equation tells us that where  is the viscous resistance of the medium. Stokes showed that for spherical particles of radius r in a continuous medium of viscosity  where  v is the coefficient of sliding friction. For large solute molecules in small solvent molecules,  v  inf and

MODULE 23 (701) For neutral species we showed and putting R = r A + r B and and the diffusion limited rate constant for neutral reactants depends only on the solvent viscosity and the temperature. See Tables in module 23

MODULE 23 (701) The following table shows k Diff values calculated for some liquids Solvent SOLVENTη 20 /cP k Diff /M -1 s -1 DIETHYL ETHER N-HEXANE ACETONITRILE METHANOL BENZENE WATER GLYCEROL

MODULE 23 (701) The next two tables show a comparison between calculated and experimental rate constants for some chemical reactions. Solvent ReactionSolvent10 5 D A a 10 5 D B a RbRb k calc c k obs c I+I  I 2 CCl 4 4.2__ OH+OH  H 2 O 2 H2OH2O2.6__ OH+C 6 H 6  C 6 H 6 OHH2OH2O C 2 H 5  C 4 H 10 C2H6C2H6 0.7__ a in cm 2 s -1 b in pm c in M -1 s -1

MODULE 23 (701) Solvent Reaction k (calc) k (obs) REACTION k calc /M -1 s k obs /M -1 s -1 e aq - +Co(NH 3 ) e aq - +Cr(H 2 O) H 3 O + +SO e aq - +Ag OH - +H 3 O