1 - Lecture notes 1 PDF

Preview

Summary

advavanced electrochemistry...

Description

1.

The Electrical Double Layer

(Ref.:Delahay: Double Layer and Electrode Kinetics; Mohilner in Bard, Ed.: Electroanalytical Chemistry, A Series of Advances, Vol. 1; Sparney: The Electrical Double Layer; Bockris and Reddy: Modern Electrochemistry, Vol. 2) 1.1 The electrified interface Interface between metal electrode and electrolyte solution:

−

+

− − − −

+ + +

+

Electrical double layer : the arrangement of charges and oriented dipoles constituting the interphase region at the boundary of an electrolyte Electrochemical Potential Outer (Volta) potential) of a given phase, Ψ : the work done to transport a unit test charge from infinity to a point just outside a charged but dipole-free phase (just outside the reach of the image forces from the phase) Surface potential of a given phase, χ : the work done to carry the unit test charge across the dipole layer at the surface of an uncharged phase Inner (Galvani) potential, φ : work done to transport the unit test charge from infinity across the charged surface covered with a dipole layer to a point inside the phase

φ=Ψ + χ

(1)

Chemical potential, μi : work done to bring a mole of i particles from infinity into the bulk of an uncharged, dipole-layer- free material phase.

1

−

Electrochemical potential, μ : total work done to take a mole of charges from infinity in vacuum into the bulk of the material phase (it is the sum of potentials, one for material phase without either charges or dipole layer on the surface, and another involving only

ψ

→

+

+

→ →

χ

φ

→

+

→ →

the charges and the dipole layer). Thus, −

μ = μ + zFφ = μ + zF (ψ + χ )

(2)

Note: electrical work to bring a unit charge = φ electrical work to transport one particle bearing a charge zieo = zieo φ electrical work to bring an Avogadro number (NA) of particles inside a material phase = NAzieoφ = ziF φ −

Total driving force for the flow of a particular species is:

d μi / dx

−

For an interface to be at equilibrium,

( d μ i / dx ) = 0 ,

2

i.e. the electrochemical potential of a species i must be the same on both sides of the −

interface, or,

−

μ i ( metal ) = μ i ( solution ) −

The electrochemical potential μi of a species i in a particular phase is the change in the (electrochemical) Gibbs energy of the system resulting from an introduction of a mole of i particles into the phase (while keeping other conditions constant), i.e. ⎛ − ⎞ ⎜ ∂G ⎟ μi = ⎜ ⎜ ∂ n i ⎟⎟ ⎝ ⎠T , P , n j −

(3)

Thus, equality of electrochemical potentials on either side of phase boundary implies that the change in Gibbs energy of the system resulting from the transfer of particles from one phase to the other should be the same as that due to transfer in the other direction; i.e. −

d G = 0 . This implies a free flow of species across the interface. An interface which maintains an “open border” is a non-polarizable interface. Thus, thermodynamic equilibrium exists at a non-polarizable interface.

For a non-polarizable interface: S

or,

S

−

ΔM μ i = −

ΔM μ i =

S

ΔM (μ i + z i Fφ ) = 0

(4)

S

ΔM μ i + zi F S ΔM φ = 0

(4’)

where i is the species exchanged across the non-polarizable interface. Rearranging, S

or,

ΔM φ = −

1 ziF

d ( S ΔMφ ) = −

S

ΔM μ i

RT 1 d ln ai dμi = − zi F zi F

(5)

(5’)

1.2 Thermodynamics of the electrified interface The combined statement of the first and second law of thermodynamics (where only PdV type of wok is considered) for an open system is: dG = VdP – SdT + Σμidni

(6)

If other types of work are included, e.g. work of expansion of the interface, of interfacial tension γ , by an area dA, which is given by γ dA, or work involved in altering the charge 3

on the metal by an amount dqM upon the application of a potential φ, which is given by φ dqM , then the complete expression for the change in Gibbs energy becomes: dG = VdP – SdT + Σμidni + φ dqM + γdA

(7)

dG = Σμidni + φ dqM + γdA

(8)

G = Σμi ni + φ qM + γ A

(9)

At constant T and P,

Integrating:

Taking the total differential, dG = Σμidni + φ dqM + γ dA + Σ ni dμi + qM d φ + A d γ

(10)

Comparing with Eq. (8),

Σni d μi + qM dφ + A dγ = 0

(11)

Rearranging, (and taking unit area ,A = 1) dγ = - Σ ni dμi - qM dφ

(12)

Surface excess, Γ i , is defined as the amount of material over and above that which would have existed had there been no double layer. If ni is the actual number of moles of species i in the interfacial region and nio is the number of moles that would have been there if there had been no double layer, then (per unit area)

Γ i = ni - nio and therefore

(13)

o ni = Γ i + ni

ni dμi = Γi d μi + nio dμI

(14)

Σ ni d μi = Σ Γi dμ i + Σ nio dμi

(15)

Summing over all i :

From the Gibbs-Duhem equation (for bulk material phase, without surface effects):

Σ nio d μi = 0

(16) 4

Thus,

Σ ni d μi = Σ Γi dμ i

(17)

Substituting Eq.(17) in Eq. (12): dγ = - Σ Γi dμi - qM dφ Note that dφ represents the change in the inner or Galvani potential difference across the interface under study. It may be written as d ( M1 ΔS φ ) , where M1 is the metal electrode under study and S is the solution. Thus, dγ = - Σ Γi dμi - qM d ( M1 ΔS φ )

(17’)

Although the inner potential is not measurable, the change in inner potential can be measured provided that the M1/S interface is polarizable and that it is connected to a nonpolarizable interface M2/S to form an electrochemical cell. If such a cell is connected to an external source of electricity, V =

M1

Δ Sφ + S ΔM 2 φ +

M2

'

ΔM 1 φ

(18)

where M1’ is the connecting metal wire that completes the circuit. Since the last term in Eq. (18) does not depend upon the potential V supplied from the external source, or the solution composition, dV = d( or

d(

M1

M1

ΔS φ ) + d( S ΔM 2 φ )

S Δ Sφ ) = dV - d( ΔM 2 φ )

(19) (20)

Substituting in Eq. (17’): dγ = - Σ Γi d μi - qM [dV - d( S ΔM2 φ )]

(21)

Since there is thermodynamic equilibrium at the non-polarizable interface, (see Eq. (5)), d ( S ΔM 2 φ ) = −

1 dμj zjF

where the non-polarizable interface (reference electrode) is reversible with respect to ion j. Eq. (21) then becomes:

dγ = −q M dV −

qM z jF

dμ j − ∑ Γi dμ i

(22)

i

5

[Note: The reference electrode is reversible with respect to ion j. However, M2/S refers to an electrode M2 dipped into the same solution S as M1; i.e. the cell is represented as: M1/S/M2. Thus, when potential V is read, this value refers to a potential with respect to a reference electrode consisting of metal M2 immersed in solution of the same composition as that of the surrounding test electrode. Potentials referred to those reversible to ions of a given varying concentration are denoted by V+ or V-] For a solution of fixed composition (and where i = j ), d μi = 0 so that

dγ = − q M dV ⎛ ∂γ ⎞ ⎜ ⎟ = − qM ⎝ ∂V ⎠ μ

or,

(22)

Eq.(22) is referred to as the Lippmann equation. The potential at which the derivative in Eq. (22) becomes zero (i.e. qM = 0 ) is known as the potential of zero charge (p.z.c.). The plot of interfacial tension γ versus potential is known as the electrocapillary curve.

γ

p. z. c V

The electrocapillary curve

The differential capacitance of the interface is defined as: ⎛ ∂2 γ ⎞ ⎛ ∂q ⎞ ⎟ C = ⎜ M ⎟ = − ⎜⎜ 2⎟ ⎝ ∂V ⎠ μ ⎝∂V ⎠ μ

(23)

Consider the cell Hg⎜HCl⎜AgCl⎜Ag with varying composition using a reversible silversilver chloride electrode. This reference electrode is reversible with respect to the anion Cl-. At constant potential, Eq. (22) becomes:

6

dγ = −

qM zjF

d μ j − ∑ Γi dμ i

(24)

i

For i = j = -1, expanding the summation over all i :

dγ =

qM dμ − − Γ+ dμ + − Γ− dμ − F

(25)

Since μ = μ+ + μ- , then dμ = dμ+ + d μ -, or dμ + = d μ - dμ- , so that

dγ =

qM dμ − − Γ+ dμ + Γ+dμ − − Γ−dμ − F

(26)

or

⎛ q + FΓ + − FΓ − ⎞ dγ = −Γ+ dμ + ⎜ M ⎟d μ− F ⎝ ⎠

(27)

Due to electrical neutrality across the interface, Fn+ - Fn- = - qM

(28)

whereas in the absence of the double layer Fn+o - Fn-o = - qM

(28’)

Combining these two equations: o o q M + F (n + − n + ) − F (n − − n − ) = 0

(29)

Introducing the definition of surface excess, (Eq.(13), qM + FΓ + - FΓ- = 0

(30)

The numerator in the quantity in brackets in Eq. (27) is thus zero, and therefore

or,

dγ = - Γ + dμ

(31)

⎛ ∂γ ⎞ − Γ+ = ⎜⎜ ⎟⎟ ⎝ ∂μ ⎠ V −

(32)

Since 7

μ = μ + + μ − = (μ + o + RT ln a + ) + (μ − o + RT ln a − ) = (μ + o + μ −o ) + RT ln a + a − and 2

a± = a+ a−

μ = ( μ+ + μ− ) + 2RT ln a ±

(33)

⎛ ∂γ − Γ+ = ⎜⎜ ⎝ 2RT∂ ln a±

(34)

o

then

o

so that ⎞ ⎟ ⎟ ⎠V−

Excess charge densities in solution side of interface (due to surface excesses of positive and negative ions) are given by q+ = z+FΓ+

and

q- = z-FΓ-

qS = q+ + q- = -qM

(36)

-q M = z+FΓ+ + z-FΓ-

(37)

positively ch arg ed electrode

q

q + = F Γ+

+

q S = − qM

0

q − = − FΓ−

−

(35)

p.z.c.

potential difference

1.3 The Structure of Electrified Interfaces 1.3.1 The Helmholtz Model: electrified interface consists of two sheets of charge, one on the solution side, and another on the electrode side – similar to a parallel plate condenser.

8

d

q+ + +

− − −

ε

V =

4π d

ε

q

(38) V = potential difference across the condenser, q = charge on the parallel plates, d = distance between plates, ε = dielectric constant of material between the plates. Assuming ε is a constant, and that the charge on the metal is qM dV =

4π d

ε

From the Lippmann equation,

d γ = -qM dV

Substituting for dV,

d γ = - qM

γ

Integrating,

∫ dγ = −

γ max

4π d

ε

(39)

dq M

4π d

ε

dq M

(40)

q

∫q

M

dq M

0

γ − γ max = −

4π d qM ε 2

γ − γ max = −

ε 1 2 V 4π d 2

2

(41)

or, in terms of the potential,

(42)

This is an eqation of a parabola symmetrical about γ max. Experimental electrocapillary curve, however, is not symmetrical. Also, according to the parallel plate model, the capacitance of the condenser is given by: C=

ε dq = = cons tan t dV 4π d

(43)

(for constant ε and d). The differential capacitance of the interface, however, is not a constant but depends on potential in a complex manner.

9

1.3.2 Gouy-Chapmann Theory

Consider a lamina in the electrolyte parallel to electrode and at a distance x from the electrode. The charge density in the lamina can be expressed as follows (Poisson’s equation):

ρx + + + + + +

ψ

ρx = −

ε d 2ψ x 4π dx 2

(44)

where ψ x is the outer potential difference between the lamina and the bulk of the solution for which ψ x→∞ = 0 . Note that since Δχ for two points in the same phase is zero,

ψ x = φ x. From the Boltzmann distribution law:

ρ x = ∑ ni ( z i eo ) = ∑ nio ( z ie o ) exp[− ( z ie o )ψ x / kT ] i

(45)

i

where ni and nio are the concentrations of the ith species in he lamina (at a distance x from the electrode) and in the bulk of the solution, respectively; zi is the valence of species i ; eo is the electronic charge. Combining the Poisson and Boltzmann equations:

d 2ψ 4π o =− ni (z i e o ) exp[− (z ie o )ψ x / kT ] ∑ 2 ε i dx

(46)

Note that:

10

2

d ⎛ dψ ⎞ d ⎛ dψ ⎞ ⎛ dψ ⎞ ⎛ d ⎞ ⎛ dx ⎞ ⎛ d ψ ⎞⎛ dψ ⎞ ⎟⎟ ⎜ ⎜ ⎟ =2 ⎜ ⎟⎜ ⎟ = 2 ⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ dψ ⎝ dx ⎠ dψ ⎝ dx ⎠ ⎝ dx ⎠ ⎝ dx ⎠ ⎝ dψ ⎠ ⎝ dx ⎠⎝ dx ⎠ ⎛ dx = 2 ⎜⎜ ⎝ dψ

⎞⎛ d 2ψ ⎟⎟⎜⎜ dx 2 ⎠⎝

⎞⎛ dψ ⎞ ⎛ d 2ψ ⎟⎟⎜ ⎟ = 2 ⎜⎜ 2 ⎠⎝ dx ⎠ ⎝ dx

⎞ ⎟⎟ ⎠

2

d 2ψ 1 d ⎛ dψ ⎞ or, ⎜ ⎟ = dx2 2 dψ ⎝ dx ⎠ Using this in the Poisson-Boltzmann equation: 2

d ⎛ dψ ⎞ 8π ⎜ ⎟ =− ε dψ ⎝ dx ⎠

8π ⎛ dψ ⎞ d⎜ ⎟ =− ε ⎝ dx ⎠ 2

Rearranging,

(47)

∑n

o

i

zi e oexp[ − z i eoψ x / kT]

(48)

i

∑n

o

i

z e ψ x / kT]

z ie o e[ − i o

dψ

(49)

i

Integrating, ⎛ d ψ ⎞ = − 8π ⎜ ⎟ ε ⎝ dx ⎠ 2

∑

∑n

o i

z ie o exp[ − z i e oψ x / kT ] + const

i

(− zi eo / kT )

i

(50)

Since, for x → ∞ , ψ = 0 and (d ψ / dx ) = 0 , then const = − 2

Therefore,

8π kT

ε

∑n

8π kT ⎛ dψ ⎞ ⎜ ⎟ = ε ⎝ dx ⎠

o

(51)

i

i

∑n

o i

{exp[ − z i e o ψ x / kT ] − 1}

(52)

For a z-z valent electrolyte, where ⎜z+⎜ = ⎜z-⎜ = z and n+o = n-o = no 2

8π kTn o ze oψ x / kT ⎛ dψ ⎞ e − 1 + e − ze oψ x / kT − 1 ⎜ ⎟ =− ε ⎝ dx ⎠ 2

or,

(

8π kTn o zeo ψx ⎛ dψ ⎞ = e ⎜ ⎟ ε ⎝ dx ⎠

(

)

/kT

+ e − zeo ψx

/kT

−2

)

(53)

This may be rewritten as:

11

2

(

2

(

8π kTn o ze oψ x / kT ⎛ dψ ⎞ e + e − ze oψ x / kT − 2 ( e ze oψ x / kT )( e − ze oψ x / kT ) ⎜ ⎟ = ε ⎝ dx ⎠ 8π kTn o zeoψ x / 2 kT ⎛ dψ ⎞ − e − zeoψ x / 2 kT e ⎟ = ⎜ ε ⎝ dx ⎠

)

2

(54)

Since ex – e-x = 2 sinh x 32π kTn o ⎛ dψ ⎞ sinh ⎜ ⎟ = ε ⎝ dx ⎠ 2

2

ze oψ x 2 kT

1/ 2

⎛ 32π kTn o ⎛ dψ ⎞ ⎜ ⎟ = ± ⎜⎜ ε ⎝ dx ⎠ ⎝

⎞ ⎟⎟ ⎠

⎛ 32π kTn o ⎛ dψ ⎞ ⎜ ⎟ = ± ⎜⎜ ε ⎝ dx ⎠ ⎝

⎞ ⎟⎟ ⎠

sinh

ze oψ x 2 kT

(55)

Assume that sinh x ≈ x; then

or,

where

1/ 2

⎛ dψ ⎞ ⎜ ⎟ = ± bψ x ⎝ dx ⎠

⎛ 8 π n o ( ze o ) 2 ⎞ ⎛ ze oψ x ⎞ ⎟ ⎜ ⎟ = ± ⎜⎜ ⎟ ε kT ⎝ 2 kT ⎠ ⎠ ⎝

1/ 2

ψx

(56)

⎛ 8 π no ( zeo) 2 ⎞ b = ⎜⎜ ⎟⎟ ε kT ⎠ ⎝

1/ 2

(57)

At a positively charged electrode, ψ > 0, but ( dψ / dx) < 0, and at a negatively charged electrode, ψ < 0, while (dψ / dx ) > 0. [Note also Gauss’s theorem where 4π q M = −ε (dψ / dx ).] Thus, only the negative root of Eq. (56) corresponds to the physical situation: ⎛ dψ ⎞ ⎜ ⎟ = − bψ x ⎝ dx ⎠

The integrated form is : as shown below.

ψ x = ψ o exp( −bx)

(58) and the potential - distance profile is

12

)

ψ

x

From Gauss’s law, 1/ 2

⎛ dψ ⎞ ⎛ 32π kTno ⎞ −ε ⎜ ⎟ ⎟=ε⎜ ε ⎝ dx ⎠ ⎝ ⎠ and

qM =

ε 4π

sinh

⎛ 32π kTno ⎞ ⎜ ⎟ ε ⎝ ⎠

1/ 2

⎛ 2kTno ε ⎞ q M = −q s = ⎜ ⎟ ⎝ π ⎠

where

⎛ kTn oε ⎞ A =⎜ ⎟ ⎝ 2π ⎠

zeoψ x = 4π q M 2kT

sinh

(59)

zeoψ x = −q s 2 kT

1/ 2

sinh

zeo ψ x ze ψ = 2 A sinh o x 2 kT 2kT

(60)

1/ 2

(61)

For x = 0, ψ x = ψ o = potential at x = 0 relative to the bulk of the solution where the potential is zero, or ψ ∞ = 0 . The diffuse charge density (qd = qs) is therefore

⎛ 2kTno ε ⎞ qd = − ⎜ ⎟ ⎝ π ⎠

1/ 2

sinh

zeoψ x 2kT

(62)

The differential capacity may be evaluated from ⎛ ∂q C = ⎜⎜ M ⎝ ∂ψ M

2 ⎛ ∂q ⎞ ⎛ z 2 e o n o ε ⎞⎟ ⎞ ⎟⎟ = − ⎜⎜ d ⎟⎟ = ⎜ ⎝ ∂ψ o ⎠ ⎜⎝ 2π kT ⎟⎠ ⎠

1/ 2

cosh

ze oψ o 2 kT

(63)

The cosh function gives an inverted parabola, and the Gouy-Chapmann theory therefore predicts an inverted parabola dependence of C on potential across the interface. This is approximated experimentally only at very dilute solutions.

13

C

G − C Theory

C

Exp ' t ( dil. solution )

Exp 't (conc. solution ) Potential

Potential

1.3.3 Stern’s modification of the Gouy-Chapmann theory

Stern: ions cannot reach electrode beyond “plane of closest approach” – which is same for cations and anions. Double layer divided into two regions: compact double layer (between electrode and plane of closest approach, and a diffuse double layer, from plane of closest approach to bulk of solution. Due to charge separation, a potential drop results:

ψ M = ψ M − ψ S = (ψ M − ψ 2 ) + (ψ 2 − ψ s )

(64)

Differentiating the above:

or,

∂ψ M ∂ (ψ M −ψ 2 ) ∂ψ 2 = + ∂q M ∂ qM ∂qd

(65)

1 1 1 = + C C M −2 C2 −s

(66)

Model of double layer: two capacitors in series

⎯⎜⎜⎯⎯⎜⎜⎯ CM-2 C2-S

14

plane of closest approach (2)

− − − − −

Electrode

scattered ions

−

−

−

−

−

− compact double layer

diffuse double layer

Note: for large concentrations, no is large and from the Gouy-Chapman expression, for capacitance, C2-S becomes large and hence 1/C2-S becomes small compared to 1/CM-2 . Therefore C ≈ CM-2. Hence the disagreement of the G-C theory with experiment for large electrolyte concentrations. But for low concentrations, C ≈ C2-S in agreement with experiment. CM-2 is not experimentally accessible. However, C can be measured, and q can be calculated from the C versus Potential curve...