Electrolyte Solutions Review of Electrical Quantities

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1
Electrolyte Solutions
Review of Electrical Quantities
Charge
Charge is a fundamental is a fundamental characteristic of matter and (similar to mass)
it is understood better for what it does (it attracts and repels) rather than what it is.
Charge is measured with many different units. Modern chemists usually use three
units to describe charge
SI units
In the SI system, the fundamental unit of charge is the Coulomb, itself defined in
terms to the Ampere, the unit of electrical current.
1 C = 1 A⋅s
When referring to macroscopic amount of charge, the Coulomb is usually most
convenient.
Atomic units
When referring to microscopic amounts of charge, such as the charge of an electron
or the charge of a nucleus, atomic units are often convenient (and often taken for
granted).
e = 1 au = 1.6022 × 10-19 C
Faraday
The connection between the microscopic and the macroscopic amount of charge is often
made using the unit of Faraday which is the charge in Coulombs for one mole of
electrons.
F = eNA = (1.6022 × 10-19 C)(6.0221 × 1023 mol-1) = 96,485 C/mol = 96485 J/V⋅mol
2
Force (via Coulomb’s Law)
The electrical force between two charges is proportional to the product of the charges
and inversely proportional to the square of the distance between them.
F=
1 Q1Q 2
4πε 0ε r 2
where ε0 is the permittivity of free space = 8.854 × 10-12 C2/J⋅m and is necessary to
equate the mechanical definition of force (1 N = 1kg⋅m2/s2) to the electrical
definition.
ε is the dielectric constant of a material and is different for each material. The
dielectric constant is a measure the effective attraction between charges is reduced
when the charges are immersed in a material.
The dielectric constant of water is very high (ε = 78) compared to that of other
solvents. The strongly polar nature of the water “dilutes” the strength of a charge
immersed in water. Thus attractions between charges in water are significant over
smaller ranges than in other solvents. In a nutshell, water greatly affects the
interaction between ions in solution.
Electric Field
An alternative way of considering the electrical force with the interaction between a
charge and an electric field.
F = Q1E
Comparing this equation with the force equation from Coulomb’s law yields the
expression for the electric field of a charge.
E=
F
1 Q1
=
Q 2 4πε 0 ε r 2
The electric field has several sets of equivalent units in the SI system. Usually volts
per meter are the most convenient
1N
1J
1V
=
=
C C⋅m m
3
Energy
The force between objects is found from the derivative of their potential energy with
respect to distance.
F=−
∂V
∂x
⇒ V = − ∫ F ⋅ dx
Using electrical quantities, the energy of interaction becomes
F=−
∂W
∂r
⇒ W = −∫
1 Q1Q 2
1 Q1Q 2
⋅ dr =
2
4πε 0ε r
4πε 0 ε r
The energy of interaction is inversely proportional to the distance between the
charges. (We are usually interested in energy more than force in chemical systems.)
Again note how the dielectric constant reduces the energy of interaction between two
charges compared to the interaction energy in a vacuum.
Potential (Voltage)
Just as using the electric field yields an alternative way of considering the electrical
force, the potential (electrostatic potential) can be used in an alternative way of
viewing the electrical interaction energy.
W = Q1φ
The potential can be written as
φ=
W
1 Q1
=
Q 2 4πε 0ε r
Just as there is a differential relationship between force and energy,
F=−
∂W
∂r
⇒ W = − ∫ F ⋅ dr
there is a differential relationship between the electric field and the potential.
E=−
∂φ
⇒ φ = − ∫ E ⋅ dr
∂r
The electrostatic potential can be very convenient for chemists to use when thinking
about how charges are arranged with molecules (molecular geometry) and among
molecules (intermolecular forces), since positive charges will move to a region with
the lowest potential and negative charges will move to a region with the highest
potential.
4
Molar Conductivity
Definitions
Ohm’s law
Recall Ohm’s law, where the proportionality constant that describes a material’s
relationship between voltage and current is the resistance.
φ = V = IR ⇒ I =
V
R
Conductance
An alternative way of describing the current/voltage is using the conductance, G, in
instead of the resistance.
1
G=
⇒ I = GV
R
Conductivity
The conductance is an extensive property. Thus in order to make comparisons
between materials, the intensive property, conductivity, κ, should be examined. The
conductance of a material is directly proportional of the material’s surface area
through which the current flows. Also the conductance is inversely proportional to
the length through which the current flows. Thus the definition of conductivity
reflects both of these dependencies.
G=κ
A
l
⇒ κ=G
l
A
Molar conductivity
The conductivity of water is a constant value (at a given temperature). When solutes
are added to the water, the conductivity of the solution may change. When the
conductivity changes, the conductivity of the solution may have a concentration
dependence. To better examine the nature of the solutes dissolved in a solution, the
concentration dependence is controlled by dividing the conductivity by the molar
concentration to yield the molar conductivity, Λ.
Λ=
κ
c
When a solute does not affect the conductivity of a solution, the solute is called a
nonelectrolyte. If the solute affects the conductivity of a solution, it is an
electrolyte.
5
Weak Electrolytes
Svante Arrhenius in his doctoral dissertation of 1884 examined the nature of solutions
by testing their conductivities. He proposed that electrolytes dissociate into ions
when they dissolve in water. (He almost failed in his dissertation. Dissociated ions
in solution were too controversial!)
His results were odd and needed explanation.
1. Some electrolyte solutions had a relatively constant molar conductivity
especially at low concentration.
2. The conductivity of other electrolyte solutions changed significantly with
concentration especially at low concentration.
The first attempt at explanation failed. The argument depended on considering that
the movement of the electrical charge through the solution depended on the
concentration. Kohlrausch’s law of independent migration of ions (1879)
demonstrated that the movement (mobility) of the ions was independent of
concentration (especially at low concentration). Thus the simplest explanation for the
different molar conductivities was inadequate.
Arrhenius Theory
To explain how the molar conductivity changes with concentration, Arrhenius
assumed that the electrolyte dissociated into ions.
1. All molecules fully dissociate at infinite dilution
2. The degree of dissociation, α, for an electrolyte can be predicted from its
conductivity in solution. Λ0 is the molar conductivity at infinite dilution.
α=
Λ
Λ0
0 < α <1
Ostwald’s Dilution Law
Ostwald was able to use Arrhenius’ results to find a relationship between the molar
conductivity of some electrolytes and their concentrations.
⎛ Λ ⎞
⎜Λ ⎟
c ⎝ 0⎠ =K
⎛ Λ ⎞
1− ⎜
⎟
⎝ Λ0 ⎠
2
The relationship seems to be a mess; however, a key feature is that the value K was
constant over large ranges of concentration.
It is worth noting that Ostwald’s dilution law is useful only for weak electrolytes. It
completely fails for strong electrolytes.
6
Law of Mass Action
Two decades earlier (1864), Norwegian chemists, Cato Guldberg and Peter Waage,
examined the concentration dependence of reversible reactions and found an
expression very similar to our modern formulation of an equilibrium constant. For
the reaction, A + B U C + D, such as an esterification, Guldberg and Waage put forth
the following relationship (in simplified form) known as the law of mass action.
(a 0 − x)(b 0 − x) = K ( c0 + x ) (d 0 + x)
Note that the expression relates the initial concentrations of the species with a change
away from initial conditions, “x”.
Arrhenius’ ideas of ionic dissociation fit very nicely with the law of mass action. For
his theory of ionic dissociation, Arrhenius won the third Nobel Prize in Chemistry in
1903.
Combined with Ostwald’s dilution law, we get a view as to how examining the
conductivity of solutions yields the evidence to need to confirm our modern view of
the behavior of weak electrolytes in solution as a reversible reaction between
dissociation and association.
⎛ Λ ⎞
⎛ Λ ⎞
c2 ⎜
2
⎜Λ ⎟
( cα )2
Λ 0 ⎟⎠
c2 ( α )
0 ⎠
⎝
⎝
K=c
=
=
=
⎛ Λ ⎞
⎡ ⎛ Λ ⎞ ⎤ c [1 − ( α )] c (1 − α )
1− ⎜
⎟ c ⎢1 − ⎜ Λ ⎟ ⎥
⎝ Λ0 ⎠
⎣ ⎝ 0 ⎠⎦
2
2
In the modern view, the constant that Ostwald found was the equilibrium constant
(acid-dissociation constant).
Initial
Change
Equilibrium
HA (aq) U H+ (aq) + A- (aq)
c
0
0
-cα
cα
cα
c(1 – α)
cα
cα
Arrhenius’s theory, together with van’t Hoff’s work on osmosis (remember the van’t
Hoff factor) was solid evidence that electrolytes split into ions when dissolved in
water.
7
Strong Electrolytes
The Arrhenius theory of ionic dissociation did an excellent job of explaining the
conductivity of solutions like acetic acid solutions. However for solutions like
hydrochloric acid solutions, the theory didn’t fit at all since no constant (for
Ostwald’s dilution law) could be found.
For dilute strong electrolyte solutions, the molar conductivity increases linearly with
concentration (conductivity is directly proportional to the number of ions in solution).
However, for more concentrated solutions (above 0.001 M) the conductivity of the
solution increases less than expected. The interaction of the solvated ions with other
solvated ions affects the conductivity especially at high ion concentration. Thus,
understanding the conductivity of a solution begins with understanding the
environment surrounding the ion. To address these discrepancies, Peter Debye and
Erich Hückel in 1923 constructed a model of electrolyte solutions
Debye – Hückel Theory
Finding the potential of an ion in solution
Debye and Hückel wanted to find the potential that an ion experiences in solution due
to the other ions in solution. Finding the potential would be simple if one knew the
charge density. However to find the charge density, one needs the potential. Thus,
our situation is like a dog chasing its tail!
We start by considering the fundamental relationship between charge density, ρ, and
potential, φ, that comes from Coulomb’s law.
∇2φ =
ρ
ε0
The ∇2 operator, called the Laplacian, is a three-dimensional second derivative. In
Cartesian coordinates the operator is
∂2
∂2
∂2
∇2 = 2 + 2 + 2
∂x
∂y ∂z
Because of the nature of the electrical force, the spherical polar coordinates are easier.
(Truly!)
1 ∂⎛ ∂ ⎞
1
∂ ⎛
∂ ⎞
1
∂2
∇2 = 2 ⎜ r2
+
sin
θ
+
⎟
⎜
⎟
∂θ ⎠ r 2 sin 2 θ ∂Φ 2
r ∂r ⎝ ∂r ⎠ r 2 sin θ ∂θ ⎝
The potential is spherically symmetric (no θ, Φ dependence); therefore the second
and third terms become zero for ∇2φ.
1 ∂⎛ 2 ∂
⎜r
r 2 ∂r ⎝ ∂r
4πρ
⎞
⎟φ =
ε0ε
⎠
8
Again this equation is not useful to find the potential until we have the charge density.
Let us consider the electrical environment surrounding an ion labeled “j”. If the charge
of this ion is Zje, then the total charge for the rest of the ions in solution is –Zje. Also
the electric potential that emanates from the charge can be given the subscript “j” as φj.
To calculate the potential, we will attempt to find the charge density surrounding ion
“j” by the finding the distribution of charges surrounding the ion. The distribution of
charges will depend on the electrical interaction between the charges (which, on
average, have the opposite charge of ion “j”) and the potential of ion “j” which
imparts to the system the tendency to arrange ions in specific places as well as the
thermal energy which imparts the tendency of the ion to spread. This distribution of
ions can be approximated using the Boltzmann distribution (remember the MaxwellBoltzmann distribution of gas velocities).
N i′ = N i e
−
Zi eφ j
kT
Note the distinction being made between Zi and φj.
The above expression gives the probability of one ion. To find the charge density
surrounding ion “j”, we need to sum of all of the other charges in the solution.
ρ j = ∑ Zi eN i′ = ∑ Zi eN i e
i
−
Zi eφ j
kT
i
Using a Taylor series to expand the exponential function yields,
1 2 1 3
x − x +"
2!
3!
2
3
Zi eφ j
Z
e
φ
Z
e
φ
Z
e
φ
−
⎛
⎞
⎛
⎞
1
1
i
j
i
j
i
j
+ ⎜
e kT = 1 −
⎟ − ⎜
⎟ +"
kT
2! ⎝ kT ⎠ 3! ⎝ kT ⎠
2
⎡ Zi eφ j 1 ⎛ Zi eφ j ⎞ 2
⎤
Zi eφ j
Zi eN i ⎛ Zi eφ j ⎞
ρ j = ∑ Zi eN i ⎢1 −
+ ⎜
+∑
⎟ + "⎥ = ∑ Zi eN i − ∑ Zi eN i
⎜
⎟
kT
2! ⎝ kT ⎠
kT
2! ⎝ kT ⎠
⎣
⎦ i
i
i
i
e− x = 1 − x +
Summing over all charges “i”, the first summation is zero because of the conservation
of charge. If Coulombic interaction is much less than the thermal energy, i. e.,
Zi eφ j kT , then the third (and other) summations can be neglected. Thus the charge
density becomes
ρ j = −∑
N i Zi2 e 2 φ j
i
kT
=−
e2φ j
kT
∑N Z
i
2
i
i
Using this charge density, we substitute into the earlier relationship between potential
and charge density.
1 ∂⎛ 2 ∂
⎜r
r 2 ∂r ⎝ ∂r
4πe 2 φ j
4πρ
⎞
φ
=
=
−
N i Zi2
∑
⎟ j
ε0ε
ε 0εkT i
⎠
9
To simplify the equation (and to help our later discussion), let us define the constant κ as
κ2 =
4πe 2
N i Zi2
∑
ε 0εkT i
We will also define the ionic strength, I, as
I=
1
N i Zi2
∑
2 i
In calculating the ionic strength of a solution, all of the ions in the solution must be
included in the sum.
Thus the differential equation that needs to be solved to find the potential surrounding
the ion “j” is
1 ∂⎛ 2 ∂ ⎞
2
⎜r
⎟ φj = κ φj
r 2 ∂r ⎝ ∂r ⎠
The solution to this equation (the electric potential surrounding ion “j” is
φj =
Zi e e κa e−κr
ε 0 ε 1 + κa r
where a is the “distance of closest approach” between ion “j” and any other ion.
“a” is of the order of 25 nm to 100 nm. (Higher ionic charge yields higher “a”.)
The constant κ has some physical content as the inverse of the “thickness of the ionic
atmosphere”. That is,
ε εkT
1
= 0 2
8πe I
κ
The thickness of the ionic atmosphere can be considered as the maximum distance
that ion “j” can attract other ion or the effective range for the potential of ion “j”.
The total potential surrounding ion “j” comes from the potential of the ionic
environment, φ″, as well as the potential from the ion itself, φ′.
φ = φ′ + φ′′
The potential of the ion itself has the simple form of
φj =
1 Zi e
ε0ε r
10
Thus to find the potential of the ionic environment, we need to subtract the potential
of the ion itself from the total potential.
φ′′ = φ − φ′ =
Zi e e κa e−κr
1 Zi e Zi e ⎛ e κa −κr ⎞
−
=
e − 1⎟
⎜
ε 0 ε 1 + κa r
ε0ε r
ε 0εr ⎝ 1 + κa
⎠
The above potential is valid for all distances r > a. The potential when r = a (that is ,
when solvent cages of the ions interacts) becomes,
φ′′ =
=
⎞ Ze ⎛ 1
Zi e ⎛ e κa −κr ⎞
Z e ⎛ e κa e−κa
⎞
− 1⎟ = i ⎜
− 1⎟
e − 1⎟ = i ⎜
⎜
ε0 εr ⎝ 1 + κa
⎠ r =a ε 0εa ⎝ 1 + κa
⎠ ε 0εa ⎝ 1 + κa ⎠
Zi e ⎛ 1 − (1 + κa ) ⎞ − Zi e ⎛ κa ⎞ − Zi e ⎛ κ ⎞
⎜
⎟=
⎜
⎟=
⎜
⎟
ε 0 εa ⎝ 1 + κa ⎠ ε 0εa ⎝ 1 + κa ⎠ ε 0ε ⎝ 1 + κa ⎠
φ′′ =
− Zi e ⎛ κ ⎞
⎜
⎟
ε0 ε ⎝ 1 + κa ⎠
1
, that is,
κ
the distance of closest approach of the ion plus the thickness of its ionic atmosphere.
This potential is the same as the Coulomb potential, φ′, at a distance r = a +
1/κ – thickness of the ionic atmosphere
(The thickness of the ionic atmosphere is on the order of a few nanometers.)
Nonideality in electrolytic solutions (calculation of the activity coefficient)
Consider the Gibbs free energy of an ion in solution
ΔG i = ΔG *i + RT ln a i
By redefining the standard state, ΔG*i , we can relate activity to the molar
concentration rather the mole fraction. Such a redefinition of the standard state
implies a redefinition of the activity coefficient.
a i = γ i ci
Thus the Gibbs free energy of an ion is solution becomes
ΔG i = ΔG*i + RT ln ci + RT ln γ i
The nonideal behavior of the ion in solution is expressed through the activity
coefficient.
11
ΔG inonideal − ΔG iideal = ΔG*i + RT ln ci + RT ln γ i − ( ΔG*i + RT ln ci ) = RT ln γ i
The major contribution to the nonideal behavior is the interaction between the ion and
the ionic environment. The interaction is described simplest by considering the work
needed to bring the ion from infinity to within the “distance of closest approach”.
a
RT ln γ i = w = ∫ φ dQ
∞
Recall the potential we found surrounding ion “j” that was due to the ionic
environment.
− Zi e ⎛ κ ⎞
⎜
⎟
ε 0 ε ⎝ 1 + κa ⎠
φ′′ =
The charge, Q, is Zi⋅e. Thus an expression for the activity coefficient can be written
as
−1 ⎛ κ ⎞
−Q 2 ⎛ κ ⎞
Q
dQ
=
⎜
⎟
⎜
⎟
2ε 0εRT ⎝ 1 + κa ⎠
ε0 εRT ⎝ 1 + κa ⎠ ∫0
Q
Q
RT ln γ i = w = ∫ φ dQ ⇒ ln γ j =
0
ln γ j =
− Z2j e2 ⎛ κ ⎞
⎜
⎟
2ε0 εRT ⎝ 1 + κa ⎠
Often, the activity coefficient is written in terms of common (base 10) logarithm
rather than natural logarithm.
log b x =
ln x
ln b
⇒ log10 x =
ln x
ln10
⎛
⎞⎛ κ ⎞
e2
log γ j = − Z ⎜
⎟
⎟⎜
⎝ 2ε 0 εRT ln10 ⎠ ⎝ 1 + κa ⎠
2
j
Recall that κ =
8πe2 I
= B′ I
ε0 εkT
⎛ B′ I ⎞
⎛
⎞
I
2
log γ j = − Z2j B′′ ⎜
⎟ = −Z j B ⎜
⎟
⎝ 1 + B′a I ⎠
⎝ 1 + B′a I ⎠
where B = 0.5091 M-½ at 25 °C and B′ = 0.0328 M-½⋅nm-1.
This expression for the activity coefficient is known as the extended Debye-Hückel
law. For 1:1 electrolytes, the extended Debye-Hückel law can be used solutions up to
0.1 M.
12
-3
If a solution is dilute (<10 M for a 1:1 electrolyte); then on average, r >> a. Thus
the potential of the ion can be simplified to
φ′′ =
− Zi e ⎛ κ ⎞ − Zi eκ
⎜
⎟≈
ε0 ε ⎝ 1 + κa ⎠
ε0ε
Using the simplified potential yields a simplified expression of the activity coefficient
known as the Debye- Hückel limiting law
⎛ B′ I ⎞
2
log γ j = − Z2j B′′ ⎜
⎟ ≈ −Z j B I
⎝ 1 + B′a I ⎠
The extending Debye-Huckel law can be improved by adding a correction factor.
⎛
⎞
I
log γ j = − Z2j B ⎜
⎟ + cI
⎝ 1 + B′a I ⎠
The coefficient for the correction factor is found empirically (that is, from
experimental data) rather than theoretically. If a is assumed to be the inverse of B′
such that a × B′ = 1, then equation becomes known as the Davies equation.
⎛ I ⎞
log γ j = − Z2j B ⎜
⎟ + cI
⎝ 1+ I ⎠
The Davies equation can be used for some electrolytes up to 1 M concentration.
13
Thermodynamics of Ions
It is impossible to measure absolute thermodynamic quantities for a single ion since
we never isolated cations from anions.
On the other hand because ions are always paired in solution, finding relative values
for the thermodynamic quantities of ions is perfectly acceptable.
To find relative values, we need to choose a standard. The standard chosen is
1 M (H+) at 25 °C such that,
ΔH 0f ( H + ) = 0
ΔG 0f ( H + ) = 0
ΔS0f ( H + ) = 0
With our standard in place, we can consider the thermodynamics of other ions by
examining the thermodynamics of solution with the hydrogen ion. For example,
consider an experimentally determined value for the enthalpy of formation for
hydrochloric acid (at 1 M and 25 °C).
ΔH 0f ( HCl ( aq ) ) = ΔH 0f ( H + ( aq ) ) + ΔH 0f ( Cl− ( aq ) )
⇒ ΔH 0f ( Cl− ( aq ) ) = ΔH 0f ( HCl ( aq ) ) − ΔH 0f ( H + ( aq ) ) = ΔH 0f ( HCl ( aq ) )
Next we could determine the enthalpy of formation of the sodium ion from the
experimentally determined quantity, ΔH 0f ( NaCl ( aq ) )
ΔH 0f ( NaCl ( aq ) ) = ΔH 0f ( Na + ( aq ) ) + ΔH 0f ( Cl− ( aq ) )
⇒ ΔH 0f ( Na + ( aq ) ) = ΔH 0f ( NaCl ( aq ) ) − ΔH 0f ( Cl− ( aq ) )
Next we could determine the enthalpy of formation of the bromide ion from the
experimentally determined quantity, ΔH 0f ( NaBr ( aq ) )
ΔH 0f ( NaBr ( aq ) ) = ΔH 0f ( Na + ( aq ) ) + ΔH 0f ( Br − ( aq ) )
⇒ ΔH 0f ( Br − ( aq ) ) = ΔH 0f ( NaBr ( aq ) ) − ΔH 0f ( Na + ( aq ) )
Next we could determine the enthalpy of formation of the potassium ion from the
experimentally determined quantity, ΔH 0f ( KBr ( aq ) )
ΔH 0f ( KBr ( aq ) ) = ΔH 0f ( K + ( aq ) ) + ΔH f0 ( Br − ( aq ) )
⇒ ΔH 0f ( K + ( aq ) ) = ΔH 0f ( KBr ( aq ) ) − ΔH 0f ( Br − ( aq ) )
Et cetera, et cetera, et cetera …
14
Concentration dependence of chemical potential
As with other solutions, the chemical potential of a solution component depends on
its activity. In examining the chemical potential of the solute, we choose a new
standard state of c = 1 M. (rather than a mole fraction of 1). With a new standard
state , concentration dependence of the chemical potential of an ion is expressed as
μi ( aq ) = μi0 ( aq ) + RT ln a i = μi0 ( aq ) + RT ln γ i ci
Because ions cannot be separated in solution, the activity coefficient for a single ion,
γi, is not sensible. Thus an average activity coefficient (an average for both ions) is
used. The average used is a geometric average rather than arithmetic average. Thus
for a 1:1 electrolyte the mean ionic activity coefficient is
γ± = γ+ γ−
In general, for an electrolyte with “a” cations and “b” anions, the mean ionic activity
coefficient is
γ ± = a + b γ a+ γ −b
For example, for a solution of aluminum sulfate
3
2
3
5 γ 3+ γ 2−
γ ± = 2+3 γ 2Al3+ γ SO
2− =
Al
SO
4
4
or
3
γ 5± = γ 2Al3+ γ SO
2−
4
0
0
μ Al2 (SO4 ) = μ 0Al2 (SO4 ) = 2μ 0Al3+ + 3μSO
2− = RT ln a i = μ i ( aq ) + RT ln γ i ci
3
3
4
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