## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 14. Some Special Topics: Reactions in Solutions

### 14.7. Ionic Solutions

In __Chapter 11__ we derived the activity coefficients for ionic solutions using the Debye–Hückel theory, which can be used immediately if the structure of the activated complex is known. For this structure, two extremes have been proposed: (i) a *double-sphere model*; and (ii) a *single-sphere model*. Each of these models will be discussed in the following sections (__Figure 14.5__).

** Figure 14.5** The double-sphere and single-sphere models for the activated complex in an ionic reaction.

**14.7.1 The Double-Sphere Model**

For the double-sphere model consider the reaction A + B → M^{‡}, where the reactants have charges *z*_{A}*e*, *z*_{B}*e* and (*z*_{A} + *z*_{B})*e*, respectively. The activated complex is considered to be a double-sphere with *r** the distance between A and B. The concentration of the activated complex is taken as proportional to the bulk concentration of A multiplied by the average concentration of B at a distance *r**. The latter is given by

(14.57)

where *c*_{B} is the bulk concentration of B, *ψ* is the mean electrostatic potential at a distance *r** from A, and *e* is the unit charge. Hence, the concentration of activated complex is

(14.58)

with *k*′ the proportionality constant. Using the Debye–Hückel expression for *ψ* (Eq. 12.48), the expression becomes

__(14.59)__

where *a* is the mean distance of closest approach of A and B. The parameter *κ* is defined by the Debye length *κ*^{2} = 2*N*_{A}*e*^{2}*I*_{c}/*ε*_{0}*ε*_{r}*kT*, where *N*_{A} is Avogadro's constant and the ionic strength *I*_{c} is defined by . At high dilution, __Eq. (14.59)__ becomes

__(14.60)__

and since the activity coefficient *γ* is equal to *c*^{0}/*c*, we obtain

(14.61)

or, using __Eq. (14.59)__ and __(14.60)__

__(14.62)__

The activity coefficient is with reference to the infinitely diluted solution, but we need this with respect to the gas phase. Here, we use *α* for the activity coefficient of the solution with respect to the gas phase, *β* for the activity coefficient of an infinitely diluted solution with respect to the gas phase, and *γ* for the activity coefficient of the solution with respect to the infinitely diluted solution. So

__(14.63)__

where the quantity involving the *β*-terms is given by

(14.64)

and the subscript “gas” indicates the ideal gas state with permittivity *ε*_{r} = 1. Applying the Debye–Hückel expression, this reads

__(14.65)__

Eliminating the activity coefficients *α* using __Eq. (14.62)__, __(14.63)__ and __(14.65)__, the final result for the double-sphere model employing *k*_{AB} = *k*_{AB,0}(*α*_{A}*α*_{B}/*α*_{‡}) is

__(14.66)__

**14.7.2 The Single-Sphere Model**

For the single-sphere model, one uses essentially the Born hydration model (see Section 12.2). The activated complex is considered to be a single-sphere of radius *r*, which is transferred from the gas phase to the liquid phase. For this process we showed in __Chapter 12__ that

(14.67)

and because for this process Δ*G* = *kT* ln*β* (with *β* as the activity coefficient defined in the previous section), we obtain

(14.68)

Hence, it follows that

__(14.69)__

where *r*_{‡} is the radius of the (spherical) activated complex with charge (*z*_{A} + *z*_{B})*e*. The activity coefficient *γ* is as given by the Debye–Hückel model (see Section 12.5)

(14.70)

and, if a mean distance *a* is assumed for the distance of closest approach of the ions, it follows that

__(14.71)__

Now, employing again *k*_{AB} = *k*_{AB,0}(*α*_{A}*α*_{B}/*α*_{‡}) and introducing __Eq. (14.69)__ and __(14.71)__ in the *α*-terms of __Eq. (14.63)__, the final result becomes

__(14.72)__

The expressions for the single-sphere and double-sphere models become identical when *r*_{A} = *r*_{B} = *r*_{‡}. The expressions derived in this and the previous section can be tested in two ways: first, via the dependence on ionic strength while keeping the permittivity constant; and, second, via the dependence on permittivity of the solvent using data for infinite dilution, that is, extrapolated to zero ionic strength. This will be discussed in the next two sections.

Problem 14.4

Verify __Eq. (14.66)__ and __(14.72)__.

Problem 14.5

Discuss the difference in dependence on ionic radii between the single-sphere and double-sphere models for the rate constant. To that purpose, express *r*_{A}, *r*_{B} and *r*_{‡} for the single-sphere model in *r**, and consider when the resulting expression differs significantly from the corresponding expression for the double-sphere model. Do you expect that experimental data on the rate constant can help to distinguish between the two models?

**14.7.3 Influence of Ionic Strength**

For both __Eq. (14.66)__ and __(14.72)__ the influence of the ionic strength is given by

(14.73)

where is the rate constant for infinite dilution. Inserting = 2*e*^{2}*N*_{A}*I*_{c}/*εkT* (Eq. 12.48) and neglecting *κa* in the term 1 + *κa* in comparison with unity, this expression reduces to

(14.74)

Introducing all numerical factors it appears that *A* = 1.02 (using mol l^{−1} for molarity) for aqueous solutions at 25 °C. In a plot of *k*_{AB} versus , one should thus find a straight line with slope 1.02*z*_{A}*z*_{B}. The influence of the ionic strength for various reactions is shown in __Figure 14.6__a, where the experimental slopes nicely match the expected 1.02*z*_{A}*z*_{B}, including the zero slope if either *z*_{A} or *z*_{B} is zero.

** Figure 14.6** Dependence of the specific rate constants on (a) ionic strength

*I*and (b) permittivity

_{c}*ε*

_{r}. Panel (a): I. Co(NH

_{3})

_{5}Br

^{2+}+ Hg

^{2+},

*z*

_{A}

*z*

_{B}= +4; II. ,

*z*

_{A}

*z*

_{B}= +2; III. ,

*z*

_{A}

*z*

_{B}= +1; IV. CH

_{3}CO

_{2}C

_{2}H

_{5}+ OH

^{−},

*z*

_{A}

*z*

_{B}= 0; V. H

_{2}O

_{2}+ H

^{+}+ Br

^{−},

*z*

_{A}

*z*

_{B}= −1; VI. Co(NH

_{3})

_{5}Br

^{2+}+ OH

^{−},

*z*

_{A}

*z*

_{B}= −2. Panel (b): The reaction between bromoacetate and thiosulfate in various media. Data from Ref. [11].

Of course, deviations occur when the solution is too concentrated, that is, if *I _{c}* is too high. This occurs first, due to the further (mathematical) approximation made in the Debye–Hückel expression and, second, due to the approximate nature of the Debye–Hückel model itself. Moreover, for solvents with a low permittivity, the association of the ions may cause discrepancies, as has been shown for the reaction of sodium bromoacetate (Na

^{+}BrAc

^{−}) with sodium thiosulfate . At high dilutions BrAc

^{−}reacts with so that

*z*

_{A}

*z*

_{B}= +2. However, at higher concentrations the complexes Na

^{+}···BrAc

^{−}, and are formed, and the effective value of

*z*

_{A}

*z*

_{B}changes to zero. The repulsion between the negatively charged ions BrAc

^{−}and is decreased by the formation of complex ions, such that the specific reaction rate increases.

**14.7.4 Influence of Permittivity**

For a solution with ionic strength equal to zero, the terms in in __Eq. (14.66)__ and __(14.72)__ disappear and we are left with only the effect of the permittivity. Plotting ln*k*_{AB}, extrapolated to zero ionic strength, versus 1/*ε*_{r} results for __Eq. (14.66)__ in

__(14.75)__

while for __Eq. (14.72)__ one obtains

__(14.76)__

Both expressions give the same order of magnitude answer because the values of *r**, *r*_{A}, *r*_{B} and *r*_{‡} are not too different. From __Eq. (14.75)__ it is immediately clear that the slope has a negative sign if the ions are similarly charged, and positive if they are oppositely charged. This result can also be obtained from __Eq. (14.76)__.

The influence of the permittivity on the rate constants extrapolated to zero ionic strength for the reaction between bromoacetate and thiosulfate in various media is shown in __Figure 14.6__b. Here, the line corresponds to a value of *r** = 0.51 nm, but a good fit can also be obtained by using *r*_{A} = 0.33 nm (bromoacetate), *r*_{B} = 0.17 nm (thiosulfate) and *r*_{‡} = 0.50 nm (activated complex), respectively. The use of mole fractions instead of volume fractions (which are actually used in the single-sphere model) leads also to a good fit with *r** = 0.56 nm. Yet, deviations can also occur, in particular for low-permittivity solvents.

Problem 14.6: The dissociation of AgNO_{3}

The dissociation constants *K* for AgNO_{3} in H_{2}O and C_{2}H_{5}OH are 1170 × 10^{3} mol l^{−1} and 4.42 × 10^{3} mol l^{−1}, respectively. Estimate the value for *K* in CH_{3}OH using the double-sphere model, and compare the result with the experimental value *K* = 20.5 × 10^{3} mol l^{−1}. Estimate the value for 2*r** and compare the result with the sum of the Stokes radii for Ag^{+} and . Use, where necessary, data from __Appendix E__.