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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)
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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)
3/5/2017 College Of Science and humanities, PSAU 3
3.1 The nature of electrolytes
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
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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.
3.1 The nature of electrolytes
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3.1 The nature of electrolytes
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.
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From the First Law of Thermodynamics:
The algebraic sum of the lattice
and solvation energies is the heat of solution.
3.1 The nature of electrolytes
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
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Interionic and ion-solvent interactions
are so numerous and important in
solution that, no ion may be regarded
as behaving independently of others.
3.1 The nature of electrolytes
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/
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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
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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−
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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.
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3.3 Ion-ion and ion-solvent interactions
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
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3.3 Ion-ion and ion-solvent interactions
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.
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3.3 Ion-ion and ion-solvent interactions
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.
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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.
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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.
2.4 The electrical potential in the vicinity of an ion
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
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2.4 The electrical potential in the vicinity of an ion
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,
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2.4 The electrical potential in the vicinity of an ion
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
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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𝝏
𝝏𝒓𝒓𝟐𝝏𝝍
𝝏𝒓=𝟒𝝅
𝑫 𝑵𝒊𝒛𝒊
𝟐 𝝐𝟐𝝍
𝒌𝑻
2.4 The electrical potential in the vicinity of an ion
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2.4 The electrical potential in the vicinity of an ion
𝟏
𝒓𝟐𝝏
𝝏𝒓𝒓𝟐𝝏𝝍
𝝏𝒓=𝟒𝝅
𝑫 𝑵𝒊𝒛𝒊
𝟐 𝝐𝟐𝝍
𝒌𝑻 = 𝒌𝟐𝝍
where
𝒌 = 𝟒𝝅
𝑫 𝑵𝒊𝒛𝒊
𝟐 𝝐𝟐
𝒌𝑻
𝟏/𝟐
Thus, 𝟏
𝒓𝟐𝝏
𝝏𝒓𝒓𝟐𝝏𝝍
𝝏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
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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.
2.4 The electrical potential in the vicinity of an ion
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
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3.4 The electrical potential in the vicinity of an ion
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
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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.
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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).
3.5 Electrical potential and thermodynamic functions: The
Debye-Hückel equation
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
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3.5 Electrical potential and thermodynamic functions: The
Debye-Hückel equation
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
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𝐤 =𝟖𝛑
𝟏𝟎𝟎𝟎
𝛜𝟐𝐍
𝐃𝐤𝐓𝛍
𝟏 𝟐
3.5 Electrical potential and thermodynamic functions: The
Debye-Hückel equation
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
Debye-Hückel equation
For water at 298 K,
A = 511 and
B = 3.29 × 107
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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
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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
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2.7 Applications of the Debye-Hückel equation
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.
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2.7 Applications of the Debye-Hückel equation
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
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2.7 Applications of the Debye-Hückel equation
2. 7.2 Effect of ionic strength on ion reaction rates in solution
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
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2.7 Applications of the Debye-Hückel equation
2. 7.2 Effect of ionic strength on ion reaction rates in solution
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.
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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.
2.7 Applications of the Debye-Hückel equation
2. 7.2 Effect of ionic strength on ion reaction rates in solution
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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.
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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.
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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
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2.8 Ion association
The number of i-type ions in such a
shell is given by
𝐝𝐍𝐢 = 𝐍𝐢𝐞𝐱𝐩 ±𝐳𝐢𝛜𝛙𝐣
𝐤𝐓
Model for determination
of distribution of i-ions
within shells of specified
dimensions about j-ions
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
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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.
Model for determination
of distribution of i-ions
within shells of specified
dimensions about j-ions
Relation of a
to q for ion
pairing
𝐝𝐍𝐢 = 𝟒𝛑𝐍𝐢𝐞𝐱𝐩 ±𝐳𝐢𝐳𝐢𝛜
𝟐
𝐃𝐤𝐓𝐫𝐫𝟐𝐝𝐫
Figure shows shapes of probability
curves for distribution of
(a) i-ions about j-ions
(b) j-ions about j-ions.
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2.8 Ion association
From the minimum condition
𝐝𝐍𝐢
𝐝𝐫= 𝟎 ⇒ 𝐪 =
𝐳𝐢𝐳𝐣𝛜𝟐
𝟐𝐃𝐤𝐓
𝐝𝐍𝐢 = 𝟒𝛑𝐍𝐢𝐞𝐱𝐩 ±𝐳𝐢𝐳𝐢𝛜
𝟐
𝐃𝐤𝐓𝐫𝐫𝟐𝐝𝐫
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.
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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