Chapter-3- Ionic Interaction...Chapter-3- Ionic Interaction 3/5/2017 College Of Science and...

Chapter-3- Ionic

Interaction

3/5/2017 1 College Of Science and humanities, PSAU

Dr/ El Hassane ANOUAR

Chemistry Department, College of Sciences and Humanities, Prince

Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharij 11942, Saudi

Arabia.

(The slides are summarized from:

Principles and applications of Electrochemistry (D. R. Crow)

3/5/2017 College Of Science and humanities, PSAU 2

3.1 The nature of electrolytes

When electrolytes dissolved in a solvent:

The ions become free to move

The highly ordered lattice structure characteristic of crystals is

almost entirely destroyed.

Crystal structures have high lattice energies

Dissolution

In solvent

(e.g., water)



Lattice energy (Elattice): The energy required to completely separate one

mole of a solid ionic compound into gaseous ions. It is always

endothermic.

Hess’ Law (Born-Haber cycles) is used to

calculate lattice energy (Elattice)

ΔH = -Elattice = ΔHf − ΔHsub − ½ ΔHBE − IE − EA

Lattice energy increases as

Q increases


In a crystal:

Energies of a large number of component ion-pairs contribute to the

total lattice energy which is effectively the energy evolved when the

lattice is built up from free ions.

A large amount of energy is required to break down the ordered

structure and liberate free ions.




Explanation of easy dissolution of lattice structure => Occurrence of

another process (Exothermic reactions of individual ions with the

solvent), which produces sufficient energy (Heat of solvation) to

compensate for that lost in the rupture of the lattice bonds.


From the First Law of Thermodynamics:

The algebraic sum of the lattice

and solvation energies is the heat of solution.


This explains both:

Why the heats of solution are usually fairly small

Why they may be endothermic or exothermic -

depending upon whether the lattice energy or solvation

energy is the greater quantity.

Heat of solution = Elattice + Solvation energy


Interionic and ion-solvent interactions

are so numerous and important in

solution that, no ion may be regarded

as behaving independently of others.


Certain dynamic properties such as ion conductances, mobilities and

transport numbers may be determined, although values for such

properties are not absolute but vary with ion environment.

In the most dilute cases, ion may be regarded as behaving independently

of others/


3.2 Ion activity

In electrolytic solution, the properties of one ion species are affected by

the presence of other ions with which they interact electrostatically.

Thus, the concentration of a species is an unsatisfactory parameter to

use in attempting to predict its contribution to the bulk properties of a

solution.

We use the activity (a):

𝑥𝑖 mole fraction

𝑐𝑖 molar concentration

𝑚𝑖 molal concentration

𝛾𝑥 rational activity coefficient

𝛾𝑐, 𝛾𝑚 are practical activity coefficients

𝐚𝐢 = 𝛄𝐜 𝐜𝐢 = 𝛄𝒙 𝒙𝐢 = 𝛄𝒎 𝒎𝐢


3.2 Ion activity

Example

The chemical potential μi of a species i may be expressed in the forms

𝛍𝐢 = 𝛍𝐢𝟎𝐱 + 𝐑𝐓 𝐥𝐧 𝐱𝐢𝛄𝐱 = 𝛍𝐢

𝟎𝐜 + 𝐑𝐓 𝐥𝐧 𝐜𝐢𝛄𝐜 = 𝛍𝐢

𝟎𝐦 + 𝐑𝐓 𝐥𝐧𝐦𝐢𝛄𝐦

To determine properties of an electrolyte:

Mean ion activities (a±) and mean ion activity coefficients ( 𝑎±)

Note: Both forms takes account of both types of ions characteristic of an electrolyte

𝛍±𝛎 = 𝐚+

𝛎+ × 𝐚−𝛎− and 𝛄±

𝛎 = 𝛄+𝛎+ × 𝛄−

𝛎−

𝜈+ Number of cations deriving from each 'molecule' of the electrolyte

𝜈− Number of anions deriving from each 'molecule' of the electrolyte

𝜈 = 𝜈+ + 𝜈−


3.3 Ion-ion and ion-solvent interactions

In strong electrolytes, ions are not entirely free to move independently of

one another through in solution. Ions will move randomly with respect to

one another due to fairly violent thermal motion. Coulombic forces will

exert their influence to some extent with the result that each cation and

anion is surrounded on a time average by an 'ion atmosphere' containing

a relatively higher proportion of ions carrying charge of an opposite sign

to that on the central ion.



Application of an electric field:

Movement of ions will be very slow and subject to disruption by

the thermal motion.

Movement of the atmosphere occurs in a direction opposite to that

of the central ion => Breakdown symmetric ion atmosphere

As the ion moves in one direction through the solution => re-

formation of the ion atmosphere



The time-lag between the restructuring of the atmosphere and the movement of

the central ion causes the atmosphere to be asymmetrically distributed around

the central ion causing some attraction of the latter in a direction opposite to that

of its motion. This is known as the asymmetry, or relaxation effect.

central ions experience increased viscous hindrance to their motion on

account of solvated atmosphere ions which, on account of the latter's

movement in the opposite direction to the central ion, produce movement

of solvent in this opposing direction as well. This is known as the

electrophoretic effect.



The central ions experience increased viscous hindrance to their motion

on account of solvated atmosphere ions which, on account of the latter's

movement in the opposite direction to the central ion, produce movement

of solvent in this opposing direction as well. This is known as the

electrophoretic effect.

These interactions increase in significance with increasing

concentration of the electrolyte.


2.4 The electrical potential in the vicinity of an ion

The electrical potential, 𝜓 , at some

point is the work done in bringing a

unit positive charge from infinity

(where 𝜓 = 0) to that point.


The concentration of positive and negative ions (N+,N-) at the point P,

where the potential is ψ may be found from the Boltzmann distribution

law, thus.


𝐍+ = 𝐍+𝟎𝐞− 𝐳+𝛜𝛙 𝐤𝐓

𝐍− = 𝐍−𝟎𝐞+ 𝐳−𝛜𝛙 𝐤𝐓

where k = Boltzmann constan

Ni = Number of ions of either kind per unit volume in the bulk



The electrical density (𝜌) at the point P where the potential is 𝜓 is the

excess positive or negative electricity per unit volume at that point.

𝛒 = 𝐍+𝐳+𝛜 − 𝐍−𝐳−𝛜

𝛒 = 𝐍+𝟎𝐳+𝛜𝐞

− 𝐳+𝛜𝛙 𝐤𝐓 − 𝐍−𝟎𝐳−𝛜𝐞

+ 𝐳−𝛜𝛙 𝐤𝐓

N+ = N+0e− z+ϵψ kT and N− = N−

0e+ z−ϵψ kT

Where

Thus,



Example 1 : 1 electrolyte => z+ = z− = 1 𝑎𝑛𝑑 and N+0 = N−

0 = Ni Thus,

𝜌 = Niϵ e−ϵψ kT − eϵψ kT (𝟐. 𝟕

Assume that 𝜖𝜓/𝑘𝑇 ≪ 1

=> e−ϵψ kT = 1 − ϵψ kT and eϵψ kT = 1+ ϵψ kT

𝜌 ~ Niϵ 1 − ϵψ kT − 1+ ϵψ kT = Niϵ − 2ϵψ kT

𝝆 ~ − 𝟐𝑵𝒊𝛜𝟐𝝍 𝒌𝑻

For electrolytes, 𝑧+, 𝑧− ≠ 1,

𝛒 ~ − 𝐍𝐢𝐳𝐢𝟐 𝛜𝟐𝛙

𝐤𝐓


Poisson equation: Electrostatic potential and charge density relation

𝝏𝟐𝝍

𝝏𝒙𝟐+𝝏𝟐𝝍

𝝏𝒚𝟐+𝝏𝟐𝝍

𝝏𝒛𝟐=𝟒𝝅𝝆

𝑫

where D is the dielectric constant of the solvent medium

x, y, z are the coordinates of the point at which the potential is ψ.

In terms of polar coordinates, Poisson equation becomes

𝟏

𝒓𝟐𝝏

𝝏𝒓𝒓𝟐𝝏𝝍

𝝏𝒓= −𝟒𝝅𝝆

𝑫 where 𝛒 ~ − 𝐍𝐢𝐳𝐢

𝟐 𝛜𝟐𝛙

𝐤𝐓

Thus,

𝟏

𝒓𝟐𝝏


𝝏𝒓=𝟒𝝅

𝑫 𝑵𝒊𝒛𝒊

𝟐 𝝐𝟐𝝍

𝒌𝑻




𝟏

𝒓𝟐𝝏


𝝏𝒓=𝟒𝝅


𝟐 𝝐𝟐𝝍

𝒌𝑻 = 𝒌𝟐𝝍

where

𝒌 = 𝟒𝝅


𝟐 𝝐𝟐

𝒌𝑻

𝟏/𝟐

Thus, 𝟏

𝒓𝟐𝝏


𝝏𝒓= 𝒌𝟐𝝍

A general solution of Equation is

𝛙 =𝐀𝐞−𝐤𝐫

𝐫+𝐀′𝐞𝐤𝐫

𝐫

in which A, A' are integration constants.

Since as 𝑟 → ∞, 𝜓 → 0 (A' = 0)

Therefore, by substitution of 𝜓 into 𝜌 expression, we obtain:

𝛒 = −𝐀 𝐍𝐢𝐳𝐢

𝟐 𝛜𝟐𝐞−𝛋𝐫

𝐤𝐓𝐫= 𝐀𝛋𝟐

𝟒𝛑𝐃𝐞−𝛋𝐫

𝐫


For electro-neutrality:

The total negative charge of the atmosphere about a given positively

charged central ion = −𝑧𝑖𝜖

The total charge of the atmosphere is determined by considering the charge

carried by a spherical shell of thickness dr and distance r from the central

ion and integrating from the closest distance that atmosphere and central

ions may approach out to infinity.


Spherical

shell of

thickness

dr

Central

ion

𝟒𝛑𝐫𝟐𝛒𝐝𝐫∞

𝐚

= 𝑨𝜿𝟐𝑫 𝒓𝒆−𝜿𝒓𝒅𝒓∞

𝒂

= −𝐳𝐢𝛜

𝐀 =𝐳𝐢𝛜

𝐃

𝐞𝛋𝐚

(𝟏 + 𝛋𝐚)

Integration by parts

Hence,

𝛙 =𝐳𝐢𝛜

𝐃

𝐞𝛋𝐚

(𝟏 + 𝛋𝐚) 𝐞−𝛋𝐫

𝐫



Spherical

shell of

thickness

dr

Central

ion In general

𝛙 = ±𝐳𝐢𝛜

𝐃𝐚±𝐳𝐢𝛜

𝐃

𝛋

𝟏 + 𝛋𝐚

When r approaches a, the distance of

closest approach:

𝛙 =𝐳𝐢𝛜

𝐃𝐚

𝟏

(𝟏 + 𝛋𝐚)=𝐳𝐢𝛜

𝐃𝐚−𝐳𝐢𝛜

𝐃

𝛋

(𝟏 + 𝛋𝐚)

𝛙 is composed of two contributions: 𝒛𝒊𝝐

𝑫𝒂 Due to the ion itself

𝒛𝒊𝝐

𝑫

𝜿

𝟏 + 𝜿𝒂 Represents the potential on the ion due to its atmosphere

𝛙 =𝐳𝐢𝛜

𝐃

𝐞𝛋𝐚

(𝟏 + 𝛋𝐚) 𝐞−𝛋𝐫

𝐫

𝜿

𝟏 + 𝜿𝒂=𝟏

𝒂 𝒂 =

𝟏 + 𝜿𝒂

𝒌

The effective radius- that

of the ion atmosphere


3.5 Electrical potential and thermodynamic functions: The

Debye-Hückel equation

For an ideal solution, the chemical potential 𝜇𝑖 of an ion i in is given by

𝝁𝒊 = 𝝁𝒊𝟎 + 𝐑𝐓 𝒍𝒏𝒙𝒊

xi : Mole fraction of ion i

For non-ideal solutions Equation

𝝁𝒊 = 𝝁𝒊𝟎 + 𝐑𝐓 𝒍𝒏𝒂𝒊 = 𝝁𝒊

𝟎 + 𝐑𝐓 𝒍𝒏𝒙𝒊 + 𝐑𝐓 𝒍𝒏𝜸𝒊

By definition μi is the change in free energy of the system which would occur if 1

g-ion of species i were added to a large quantity of it

𝒂𝒊 = 𝒙𝒊 𝜸𝒊

RT ln γi may be regarded as the contribution that the ion atmosphere makes to the

total energy of the ion.


The contribution per ion is 𝐤𝐓 𝒍𝒏𝜸𝒊 and may be equated to the work which must

be performed to give an ion of potential ψi (due to its atmosphere) its charge 𝑧𝑖𝜖).



The work done, dw, in charging the ion by an increment of charge, dϵ , is

𝐝𝐰 = 𝛙𝐢𝐝𝛜

so that the work, w, required to give the ion its charge 𝑧𝑖𝜖 is

𝑤 = ψi

𝑧𝑖𝜖

0

𝑑𝜖 = − 𝑧𝑖𝜖

𝐷

𝜅

1 + 𝜅𝑎

𝑧𝑖𝜖

0

𝑑𝜖 = −𝑧𝑖2𝜖2𝜅

2𝐷 1 + 𝜅𝑎

Therefore,

𝐤𝐓 𝐥𝐧 𝛄𝐢 = −𝐳𝐢𝟐𝛜𝟐

𝟐𝐃

𝛋

𝟏 + 𝛋𝐚




𝐤𝐓 𝐥𝐧 𝛄𝐢 = −𝐳𝐢𝟐𝛜𝟐

𝟐𝐃

𝛋

𝟏 + 𝛋𝐚

In terms of the mean ion activity coefficient for the electrolyte the becomes

𝐥𝐧 𝛄± = −𝛜𝟐

𝟐𝐃𝐤𝐓𝐳+𝐳−

𝛋

𝟏 + 𝛋𝐚

𝛋 =𝟒𝛑𝛜𝟐

𝐃𝐤𝐓 𝐍𝐢𝐳𝐢

𝟐

𝟏 𝟐

𝐍𝐢 =𝐍𝐜𝐢𝟏𝟎𝟎𝟎

𝐤 = 𝐑/𝐍

𝛋 =𝟒𝛑

𝐃

𝛜𝟐

𝐤𝐓

𝐍

𝟏𝟎𝟎𝟎 𝐜𝐢𝐳𝐢

𝟐

𝟏 𝟐

=𝟖𝛑

𝟏𝟎𝟎𝟎

𝛜𝟐𝐍

𝐤𝐓

𝟏

𝟐 𝐜𝐢𝐳𝐢

𝟐

𝟏 𝟐

𝐤 =𝟖𝛑

𝟏𝟎𝟎𝟎

𝛜𝟐𝐍

𝐃𝐤𝐓𝛍

𝟏 𝟐


𝐤 =𝟖𝛑

𝟏𝟎𝟎𝟎

𝛜𝟐𝐍

𝐃𝐤𝐓𝛍

𝟏 𝟐



𝛍 =𝟏

𝟐 𝐜𝐢𝐳𝐢

𝟐 , 𝜇 is the ionic strength of the solution

𝐥𝐧 𝛄± = −𝛜𝟐

𝟐𝐃𝐤𝐓𝐳+𝐳−

𝛋

𝟏 + 𝛋𝐚= −

𝝐𝟐

𝟐𝑫𝒌𝑻𝒛+𝒛−

𝟖𝝅𝟏𝟎𝟎𝟎

𝝐𝟐𝑵𝑫𝒌𝑻𝝁

𝟏 𝟐

𝝁

𝟏 +𝟖𝝅𝟏𝟎𝟎𝟎

𝝐𝟐𝑵𝑫𝒌𝑻𝝁𝟏 𝟐

𝒂 𝝁

Hence

− 𝐥𝐧𝛄± =𝐳+𝐳− 𝐀 𝛍

𝟏 + 𝐁𝐚 𝛍

𝐵 =8𝜋𝑁𝜖2

1000𝑘

1 2 1

𝐷𝑇 1/2=50.29 × 108

𝐷𝑇 1/2

𝐴 =𝜖2

2.303

2𝜋𝑁𝜖2

1000𝑘3

1 2 1

𝐷𝑇 3/2=1.825 × 106

𝐷𝑇 3/2


For water at 298 K,

A = 511 and

B = 3.29 × 107


2.6 Limiting and extended forms of the Debye-Hückel

equation

For very dilute solutions, the so-called 'Limiting Law' holds, viz.

The equation

In practice, activity coefficients show a

turning point at some value of 𝜇, after

which they progressively increase.

It is thus seen to be necessary to modify

Equation by the addition of a further term

which is an increasing function of 𝜇, i.e

𝐥𝐧 𝛄± = −𝐀 𝐳+𝐳− 𝛍

𝟏 + 𝐁𝐚 𝛍 + 𝐛𝛍

− 𝐥𝐧𝛄± = 𝐳+𝐳− 𝐀 𝛍

− 𝐥𝐧𝛄± =𝐳+𝐳− 𝐀 𝛍

𝟏 + 𝐁𝐚 𝛍⇒ −𝐀 𝐳+𝐳− 𝛍

𝐥𝐧 𝛄±= 𝟏 + 𝐁𝐚 𝛍

Hückel equation


2.7 Applications of the Debye-Hückel equation

The various forms of the equations resulting from the Debye-Hückel theory find

practical application in:

The determination of activity coefficients

The determination of thermodynamic data

2. 7.1 Determination of thermodynamte equilibrium constants

Consider dissociation of a 1: 1 weak electrolyte AB

AB ⇌ A+ + B−

The thermodynamic dissociation constant KT is given by

𝐾𝑇 =A+ B−

𝐴𝐵

𝛾A+𝛾B−

𝛾𝐴𝐵= 𝐾𝛾±2

𝛾𝐴𝐵

𝐾𝑇~𝐾𝛾±2

K: The concentration, or

conditional dissociation

constant

In dilute solution

γAB = 1



2. 7.1 Determination of thermodynamte equilibrium constants

AB ⇌ A+ + B− 𝐾𝑇~𝐾𝛾±2

log𝐾𝑇 = log𝐾 + 2 log 𝛾±

Taking

logarithms

− ln 𝛾± = 𝑧+𝑧− 𝐴 𝜇

From the

limiting law

expression

log𝐾𝑇 = log𝐾 − 2𝐴 𝜇

log𝐾 = log𝐾𝑇 + 2𝐴 𝜇

KT may therefore be determined from

measured values of over a range of ionic

strength values and extrapolating the K

versus 𝜇 plot to 𝜇 = 0.

This is a general technique for the determination of all types of thermodynamic

equilibrium constants, e.g., solubility, stability and acid dissociation constants.



2. 7.2 Effect of ionic strength on ion reaction rates in solution

Consider the reaction equilibrium

AZA + BZB ⇌ X ZA+ZB#→ Products

X ZA+ZB#: Critical complex

AZAand BZB : Reactant ions

For the pre-equilibrium

𝐾 =X#

A B

𝛾#

𝛾𝐴𝛾B

Note: omitting charges for clarity

The rate, 𝑣, of pre-equilibrium reaction:

𝑣 = 𝑘 A B = 𝑘0 A B𝛾𝐴𝛾B𝛾# where 𝑘 = 𝑘0

𝛾𝐴𝛾B𝛾#

𝑘0 being the specific rate constant in infinitely di1ute solution, where

𝛾𝐴𝛾B𝛾#= 1




𝑘 = 𝑘0𝛾𝐴𝛾B𝛾#

log k = log k0 + log γA + log γB − log γ#

− ln 𝛾± =𝑧+𝑧− 𝐴 𝜇

1 + 𝐵𝑎 𝜇

Express activity

coefficients as

Taking

logarithms

log 𝑘 = log 𝑘0 +𝐴 𝜇

1 + 𝐵𝑎 𝜇 𝑧𝐴2 + 𝑧𝐵

2 − 𝑧𝐴 + 𝑧𝐵2

= log𝑘0 +2𝐴𝑧𝐴𝑧𝐵 𝜇

1 + 𝐵𝑎 𝜇

Or in very dilute solution,

log 𝑘 ~ log 𝑘0 + 2𝐴𝑧𝐴𝑧𝐵 𝜇

For water as solvent at 298 K log 𝑘 ~ log𝑘0 + 1.02𝑧𝐴𝑧𝐵 𝜇




The above equations take -account of the salt effect observed for reactions

between ions, the slopes of graphs of log 𝑘 𝑘0 versus 𝜇 being very close to

those predicted at low concentrations.


At higher concentrations, deviations from linearity occur and these are

particularly noticeable for reactions having 𝑧𝐴𝑧𝐵 = 0, e.g. for a reaction

between an ion and a neutral molecule.

According to Equation:

log 𝑘 ~ log 𝑘0 + 1.02𝑧𝐴𝑧𝐵 𝜇

such reactions should show no variation of rate with ionic strength and this is

indeed the case up until about 𝜇 = 0.1. Above this point, increasing ionic strength

does cause the rate to vary. The reason for this is the bμ term of the Hückel

equation. In this case it may be readily shown that

𝑙𝑜𝑔 𝑘 = 𝑙𝑜𝑔 𝑘0 + 𝑏𝐴 + 𝑏𝐵 − 𝑏# 𝜇 (𝟐. 𝟓𝟎)

so that log 𝑘 𝑘0 in this case becomes a linear function of 𝜇 rather than of 𝜇. This

has been experiment ally verified.




2.8 Ion association

It is found that in many cases:

The experimental values of conductances do not agree with theoretical

values predicted by the Onsager equation:

𝜆𝑖 = 𝜆0 −

8𝜋𝑁𝜖2

1000𝐷𝑘𝑇

1

2 𝑧+𝑧−

3𝐷𝑘𝑇 𝜆0𝑞

𝑞+𝐹2𝑧𝑖

6𝜋𝜂𝑁

𝑧+ + 𝑧−

2 𝐶1

2

The mean ion activity coefficients cannot always be properly predicted by

the Debye-Hückel theory:

− ln𝛾± =𝑧+𝑧− 𝐴 𝜇

1+𝐵𝑎 𝜇

Under certain conditions, Bjerrum suggested that, oppositely charged ions of an

electrolyte can associate to form ion pairs. In some circumstances, even

association to the extent of forming triple or quadruple ions may occur.


The most favourable situation for association:

Smaller ions with high charges

Solvents of low dielectric constant

2.8 Ion association

Association leads:

To a smaller number of particles in a system

Associated species have a lower charge than non-associated ones.

Diminish the magnitudes of properties of a solution which are dependent on

the number of solute particles and the charges carried by them.


2.8 Ion association

Bjerrum's basic assumption :

The Debye-Hückel theory holds so long as the oppositely charged

ions of an electrolyte are separated by a distance q greater than a

certain minimum value given by

Note: When the ion separation is less than q => Ion pairing is

regarded as taking place.

q =zizjϵ2

2DkT

The above equation may be derived

from a consideration of the

Boltzmann distribution of i-type ions

in a thin shell of thickness dr at a

distance r from a central j-type ion.

Model for determination

of distribution of i-ions

within shells of specified

dimensions about j-ions

Relation of a

to q for ion

pairing


2.8 Ion association

The number of i-type ions in such a

shell is given by

𝐝𝐍𝐢 = 𝐍𝐢𝐞𝐱𝐩 ±𝐳𝐢𝛜𝛙𝐣

𝐤𝐓





Relation of a

to q for ion

pairing

The potential at a small distance from

the central j-ion may be assumed to

arise almost entirely from that ion and

is given by

𝛙𝐣 = ±𝐳𝐣𝛜

𝐃𝐫

Thus,

𝐝𝐍𝐢 = 𝟒𝛑𝐍𝐢𝐞𝐱𝐩 ±𝐳𝐢𝐳𝐢𝛜

𝟐

𝐃𝐤𝐓𝐫𝐫𝟐𝐝𝐫


2.8 Ion association

One expects to find decreasing

probability of finding i-type ions per

unit volume at increasing values of r.

However, the volumes of the

concentric shells increase outwards

from a j-ion so that, in fact, the

probability passes through a minimum

at some critical distance.





Relation of a

to q for ion

pairing


𝟐


Figure shows shapes of probability

curves for distribution of

(a) i-ions about j-ions

(b) j-ions about j-ions.


2.8 Ion association

From the minimum condition

𝐝𝐍𝐢

𝐝𝐫= 𝟎 ⇒ 𝐪 =

𝐳𝐢𝐳𝐣𝛜𝟐

𝟐𝐃𝐤𝐓


𝟐


For a 1:1 electrolyte in aqueous solution at 298 K, q has the value 0.357 nm.

Should the sum of the respective ionic radii be less than this figure then ion pair

formation will be favoured.


It is evident that for a given electrolyte at constant temperature, lowering of

dielectric constant will encourage association.

Example:

For tetraisoamylammonium nitrate the sum of ion radii is of the order of 0.7 nm.

This gives a value of about 42 for D and implies that, for solvents of greater

dielectric constant than 42, there should be no association but rather complete

dissociation, i.e., the Debye-Hückel theory should hold good.

Conductance measurements have verified that, in fact, virtually all ion pairing has

ceased for D > 42.

2.8 Ion association


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