Lecture #11 of 17 - chem.uci.eduardo/echem/UCI-CHEM248-2019S_lectur… · Debye–Hückel–Onsager...

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766

Lecture #11 of 17

767

Q: What’s in this set of lectures?

A: B&F Chapter 2 main concepts:

● “Section 2.1”: Salt; Activity; Underpotential deposition

● Section 2.3: Transference numbers; Liquid-junction

potentials

● Sections 2.2 & 2.4: Donnan potential; Membrane potentials;

pH meter; Ion-selective electrodes

768

an ISE (for nitrate ions)an SCE

… two general liquid junctions that we care about (the most)…RECALL FROM LAST TIME…

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when two ionic solutions are separated across an interface that

prevents bulk mixing of the ions, but has ionic permeability, a

potential (drop) develops called the liquid junction potential.

Bard & Faulkner, 2nd Ed., Wiley, 2001, Figure 2.3.2

same salt;

different conc.one ion in common;

same conc.everything else

… liquid junctions:

770

same salt;

different conc.


… example “1”: ● starting at the side with larger ion concentration

● the ion the with larger mobility will impart its

charge to the opposite side of the junction

771

same salt;

different conc.


… example “1”: ● starting at the side with larger ion concentration



… conceptually, think about a condition in

the limit where tH+ 1 (say tH+ ≈ 0.9)…

… as H+ diffuses down its concentration

gradient, an electrostatic force is exerted

on Cl– to pull it along (at a large flux) while

at the same time slowing transport of H+

… this happens until ti–effective = 0.5 for both

H+ and Cl–, and at which time the system

has attained steady-state mass transport

and has generated a maximum liquid-

junction potential.

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772

same salt;

different conc.


FYI, in semiconductor physics this same process

results in a Dember potential … and the transport

process is called ambipolar diffusion

● starting at the side with larger ion concentration



… example “1”:

… conceptually, think about a condition in

the limit where tH+ 1 (say tH+ ≈ 0.9)…

… as H+ diffuses down its concentration

gradient, an electrostatic force is exerted

on Cl– to pull it along (at a large flux) while

at the same time slowing transport of H+

… this happens until ti–effective = 0.5 for both

H+ and Cl–, and at which time the system

has attained steady-state mass transport

and has generated a maximum liquid-

junction potential.

773

equivalent ionic

conductivity

774

/////////////////

= F

equivalent ionic

conductivity

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one ion in common;

same conc.

● compare dissimilar ions (cations or anions)

● the ion with the larger mobility will impart its



776

… the sign of the liquid-junction potential

is obvious for Types 1 and 2 (but not

Type 3) based on the mobilities of the

individual ions…

… and so when in doubt, think logically

about the sign of the potential to verify

answers.


one ion in common;

same conc.

● compare dissimilar ions (cations or anions)

● the ion with the larger mobility will impart its



777

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778Recall that transference number, ti, (or transport number) is based on …

… and that the ionic conductivity, κ or σ, is defined as…

… so ti is the fraction of the solution conductivity attributable to ion "i"

units: S/cm or 1/(Ω cm)

units: cm2/(s V)

(Stokes' law)

Siemens





units: cm2/(s V)

(Stokes' law)

Siemens

Λ = a(C)1/2

The Kohlrausch law (empirical) and

Debye–Hückel–Onsager equation

(theoretical) predict that the equivalent

molar conductivity is proportional to the

square root of the salt concentration

from Wiki

Physicist

Friedrich Wilhelm Georg Kohlrausch

(1840–1910)





units: cm2/(s V)

(Stokes' law)

Siemens

Λ = a(C)1/2

The Kohlrausch law (empirical) and

Debye–Hückel–Onsager equation

(theoretical) predict that the equivalent

molar conductivity is proportional to the

square root of the salt concentration

from Wiki

P-Chemist & Physicist

Lars Onsager

(1903–1976)

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781we use ti values, which are based on kinetic transport to determine

the liquid-junction potential (for derivations, see B&F, pp. 70 – 72)…

Type 1 Type 2 Type 3

same salt;

different concentrations

same cation or anion;

different counter ion;

same concentration

no common ions,

and/or one common

ion; different concs

782

Type 1

Type 2

(α)

(β)

Type 3

we use ti values, which are based on kinetic transport to determine


783

Type 1

Type 2

(α)

(β)

Type 3

… just the ratio of the

conductivity attributable

to the dissimilar ions

(with a few assumptions, pg. 72)



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Type 1

Type 2… just the ratio of the



(α)

(β)

… sign depends on the charge of the dissimilar ion:

(+) when cations are dissimilar, and (–) when anions are dissimilar

Type 3




785

Type 1

Type 2

(α)

(β)

Type 3

the Henderson Eq. (with a few assumptions, pg. 72)









786

Type 1

Type 2

(α)

(β)

Type 3

the Henderson Eq. (with a few assumptions, pg. 72)

… as written, these equations calculate Ej at β vs α









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787example: B&F Problem 2.14d

Calculate Ej for NaNO3 (0.10 M) / NaOH (0.10M)

1. What Type?



1. What Type? Type 2

2. Polarity?

789

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Calculate Ej for NaNO3 (0.10 M) | NaOH (0.10M)


2. Polarity? Polarity should be clear… compare mobilities;

OH– is higher so NaNO3 side will be (–).

3. Calculate it:

… (–) due to anions moving-




2. Polarity? Polarity should be clear… compare mobilities;

OH– is higher so NaNO3 side will be (–).

3. Calculate it:

… (–) due to anions moving-

𝐸𝑗 = −0.05916 logµNO3−

µOH−

𝐸𝑗 = −0.05916 log7.404 x 10−4

25.05 x 10−4= 0.02617 V = +26.17 mV

as predicted, a (+) LJ potential correlates with the compartment in the denominator, β, vs α

… a rather large Ej (at β vs α)!

α | β

792

… so, why do trained

electrochemists prefer

to use saturated KCl (or

KNO3) as the salt to fill

reference electrodes?

… similar mobilities

and thus, similar ti’s

and thus,…

… vanishingly small LJ

potentials!

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793

Type 3:

Type “2”:

Last point: clarifying sign conventions in B&F so that this is crystal clear…

where Λ (the equivalent conductivity) is defined as follows:

794

Type 3:

Type “2”:

where Λ (the equivalent conductivity) is defined as follows:

… and where Ceq is the concentration of positive or negative charges

associated with a particular salt in solution, and so rearranging 2.3.40…

… and by comparing this to the form of our Type 2 equation, one sees…

, where


795

Type 3:

Type “2”:

Note: We switched α and β…Type 2:


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796

Type 3:

Type “2”:

… off by a factor of (-1)… let’s look at that pre-factor for a specific example:

0.1M HCl (α) | 0.1M KCl (β)


797

Type 3:

… off by a factor of (-1)… let’s look at that pre-factor for a specific example:

0.1M HCl (α) | 0.1M KCl (β)

… so this really means that the Lewis–

Sargent relation should have a (–) in

front of it when net cations diffuse… so

switch the sign, as ∓, or use ours.


798

Type 3:

Type “2”:

… since we know that the β side will be (+) in the previous case,

this really means the Lewis–Sargent Eq. should have a (–) sign

in front of it when net cations diffuse…

… if we’re sticking to our convention that the potential is the β

(product/red. phase) versus the α (reactant/ox. phase)…


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… anyway…

800

a cell

Donnan potential: A special liquid-junction potential due to fixed charges

… here are two systems in which Donnan potentials play a prominent role:

semipermeable

membrane

membrane impermeable to

charged macromolecules

an ionomer film

http://www.futuremorf.com/

Nafion

http://www.williamsclass.com/

801

a film of poly(styrene sulfonate)

Na+

CNaCl

cNa+ cCl-cNa+ cCl-

m m s s

R–

… consider this model which applies to both scenarios…

http://www.futuremorf.com

http://www.williamsclass.com

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802

m s

𝜇𝑖𝑜,𝑚 + 𝑅𝑇 ln γ𝑖

𝑚 + 𝑅𝑇 ln 𝑐𝑖𝑚 + 𝑧𝑖𝐹𝜙

𝑚 = 𝜇𝑖𝑜,𝑠 +𝑅𝑇 ln γ𝑖

𝑠 + 𝑅𝑇 ln 𝑐𝑖𝑠 + 𝑧𝑖𝐹𝜙

𝑠

Because differences in electrochemical potential ( 𝜇𝑖𝑜) – think free

energy – drive net mass transport (of unstirred solutions), mobile

Na+ and Cl– partition between the membrane and the solution in

compliance with their electrochemical potentials:

803

(for ion "i"… its electrochemical potential in the membrane… is the same as in

solution… this is the definition of something that has equilibrated!)





𝑠





804Because differences in electrochemical potential ( 𝜇𝑖𝑜) – think free








𝑠

… Assuming that standard state chemical potentials (𝜇𝑖𝑜) are the same

inside and outside of the membrane, we can easily solve for the "Galvani" /

inner electric potential difference, ϕm – ϕs

… which is exactly what was required to calculate liquid-junction potentials!

𝜙𝑚 − 𝜙𝑠 =𝑅𝑇

𝑧𝑖𝐹ln

γ𝑖𝑠𝑐𝑖

𝑠

γ𝑖𝑚𝑐𝑖

𝑚 = 𝐸Donnan

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805

… so we can express EDonnan, an equilibrium electric potential difference,

in terms of any ion that has access to both the membrane and the solution:

𝐸Donnan =𝑅𝑇

1 𝐹ln

𝑎Na+𝑠

𝑎Na+𝑚 =

𝑅𝑇

−1 𝐹ln

𝑎Cl−𝑠

𝑎Cl−𝑚

… Assuming that standard state chemical potentials (𝜇𝑖𝑜) are the same

inside and outside of the membrane, we can easily solve for the "Galvani" /

inner electric potential difference, ϕm – ϕs

… which is exactly what was required to calculate liquid-junction potentials!

𝜙𝑚 − 𝜙𝑠 =𝑅𝑇

𝑧𝑖𝐹ln

γ𝑖𝑠𝑐𝑖

𝑠

γ𝑖𝑚𝑐𝑖

𝑚 = 𝐸Donnan









𝑠

806

Aside #1: Recall Type 1 case of LJ potential… but now with t– = 0…

(α)

(β)

… with β being the membrane


𝐹ln

𝑎Na+𝑠

𝑎Na+𝑚 = −

𝑅𝑇

𝐹ln

𝑎Cl−𝑠

𝑎Cl−𝑚

𝐸j =𝑅𝑇

𝐹ln

𝑎1 α

𝑎1 β= 𝐸Donnan Na+

807

Aside #1: Recall Type 1 case of LJ potential… but now with t– = 0…

(α)

(β)

𝐸j =𝑅𝑇

𝐹ln

𝑎1 α

𝑎1 β= 𝐸Donnan Na+

… with β being the membrane

Aside #2: This is what B&F write for this (Donnan) potential…

… Check!

Eqn. (2.4.2)


𝐹ln

𝑎Na+𝑠

𝑎Na+𝑚 = −

𝑅𝑇

𝐹ln

𝑎Cl−𝑠

𝑎Cl−𝑚

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808

Anyway… now divide both sides by RT/F and invert the argument of

the “ln()” on the right to eliminate the negative sign, and we have...


𝐹ln

𝑎Na+𝑠

𝑎Na+𝑚 = −

𝑅𝑇

𝐹ln

𝑎Cl−𝑠

𝑎Cl−𝑚

809

… or…

𝑎Na+s 𝑎Cl−

s = 𝑎Na+m 𝑎Cl−

m

Anyway… now divide both sides by RT/F and invert the argument of

the “ln()” on the right to eliminate the negative sign, and we have...


𝐹ln

𝑎Na+𝑠

𝑎Na+𝑚 = −

𝑅𝑇

𝐹ln

𝑎Cl−𝑠

𝑎Cl−𝑚

810

a film of poly(styrene sulfonate)

Na+

CNaCl

cNa+ cCl-cNa+ cCl-

m m s s

R–

… recall the scenario we are analyzing (with R– representing the fixed charges) …

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811

now, there is an additional constraint: the bulk of the solution and the

bulk of the membrane must be electrically neutral:

… an equation quadratic in cmCl- is obtained as follows…

… if these are dilute electrolytes, we can neglect activity coefficients…

𝑐Na+s 𝑐Cl−

s = 𝑐Na+m 𝑐Cl−

m

𝑐Na+m = 𝑐Cl−

m + 𝑐R−m𝑐Na+

s = 𝑐Cl−s

𝑎Na+s 𝑎Cl−

s = 𝑎Na+m 𝑎Cl−

m

𝑐Na+s = 𝑐Cl−

s

𝑐Na+s 𝑐Cl−


m

812

because in the solution, cNa+ = cCl-

for goodness sakes!


𝑐Cl−s 2

= 𝑐Cl−m 2

+ 𝑐Cl−m 𝑐R−

m


s

𝑐Na+s 𝑐Cl−


m


m + 𝑐R−m

… an equation quadratic in cmCl- is obtained as follows… 813

𝑐Cl−s 2

= 𝑐Cl−m 2

+ 𝑐R−m 𝑐Cl−

m

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814

… use the quadratic formula to solve for cmCl- and one gets…


𝑐Cl−s 2

= 𝑐Cl−m 2


m

0 = 𝑐Cl−m 2


m − 𝑐Cl−s 2

𝑐Cl−𝑚 =

−𝑐R−𝑚 + 𝑐R−

𝑚 2+ 4 𝑐Cl−

𝑠 2

2=𝑐R−𝑚

21 + 4

𝑐Cl−𝑠

𝑐Cl−𝑅

2

− 1


s

𝑐Na+s 𝑐Cl−


m


m + 𝑐R−m

815

if (which is the typical case of interest), then…

𝑐Cl−𝑚 =


𝑚 2+ 4 𝑐Cl−

𝑠 2

2=𝑐R−𝑚

21 + 4

𝑐Cl−𝑠

𝑐R−𝑚

2

− 1

𝑐Cl−𝑠 ≪ 𝑐R−

𝑚

816


𝑐Cl−𝑚 =


𝑚 2+ 4 𝑐Cl−

𝑠 2

2=𝑐R−𝑚

21 + 4

𝑐Cl−𝑠

𝑐R−𝑚

2

− 1


𝑚

1 + 4𝑐Cl−𝑠

𝑐R−𝑚

2

≈ 1 + 2𝑐Cl−𝑠

𝑐R−𝑚

2

(Taylor/Maclaurin series expansion to the first 3 (or 4) terms)

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817

… fixed charge sites are responsible for the electrostatic exclusion of mobile

“like” charges (co-ions) from a membrane, cell, etc. This is Donnan Exclusion.


𝑐Cl−𝑚 =


𝑚 2+ 4 𝑐Cl−

𝑠 2

2=𝑐R−𝑚

21 + 4

𝑐Cl−𝑠

𝑐R−𝑚

2

− 1


𝑚

1 + 4𝑐Cl−𝑠

𝑐R−𝑚

2

≈ 1 + 2𝑐Cl−𝑠

𝑐R−𝑚

2

𝑐Cl−𝑚 =

𝑐R−𝑚

21 + 2

𝑐Cl−𝑠

𝑐R−𝑚

2

− 1 =

… the larger CR-m, the samller CCl-

m

818

… is a reasonable assumption? What is CR-m?

… well, for Nafion 117, the sulfonate concentration is 1.13 M…

… for CR61 AZL from Ionics, the sulfonate concentration is 1.6 M…

so, as an example, if CCl-s = 0.1 M…

… so how excluded is excluded?

… an order of magnitude lower

than CCl-s… rather excluded!

Source: Torben Smith Sørensen, Surface Chemistry and Electrochemistry of

Membranes, CRC Press, 1999 ISBN 0824719220, 9780824719227

=0.1 2

1.0= 0.01 M


𝑚

819… so how excluded is excluded?

… an order of magnitude lower

than CCl-s… rather excluded!

… but what if CCl-s is also large (e.g. 1 M)? … No more Donnan exclusion!

… is a reasonable assumption? What is CR-m?

… well, for Nafion 117, the sulfonate concentration is 1.13 M…

… for CR61 AZL from Ionics, the sulfonate concentration is 1.6 M…

so, as an example, if CCl-s = 0.1 M…

=0.1 2

1.0= 0.01 M


𝑚

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820… but Donnan exclusion is “amplified” in Nafion

and some other polymers… how?

Nafion phase separates into a hydrophobic phase, concentrated in

-(CF2)- backbone, and hydrophilic clusters of SO3– solvated by water…

Mauritz & Moore, Chem. Rev., 2004, 104, 4535

821… but Donnan exclusion is “amplified” in Nafion


Nafion phase separates into a hydrophobic phase, concentrated in

-(CF2)- backbone, and hydrophilic clusters of SO3– solvated by water…

… and, FYI, SO3– clusters are interconnected by channels that percolate

through the membrane, imparting a percolation network for ionic conduction

822

http://www.ocamac.ox.ac.uk/

http://www.msm.cam.ac.uk/doitpoms/tlplib/fuel-cells/figures/dow.png

http://www.chemie.uni-osnabrueck.de/

… but Donnan exclusion is “amplified” in Nafion


http://www.ocamac.ox.ac.uk

http://www.chemie.uni-osnabrueck.de

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823

So in Nafion, "amplifying" effects operate in parallel…

1) The aqueous volume accessible to ions of EITHER

charge is a small fraction of the polymer’s overall

volume, and…

2) The local concentration of SO3– (CR-

m) is much

larger than calculated based on the polymer's density

and equivalent weight (i.e. the molecular weight per

sulfonic acid moiety).

824the Donnan potential can be used to concentrate metal ions...

cation-exchange membrane

(e.g. PSS)

[H3O+]R = 10 mM

[Cu2+]R = ?? ppm

start

[SO42-]R =

5 mM

[H3O+]L = 1 mM

[Cu2+]L = 30 ppm

[SO42-]L =

0.5 mM

Wallace, I&EC Process Design Development, 1967, 6, 423 from Wiki

P-Chemist

Frederick George Donnan

(1870–1956)


… there are actually two Donnan

potentials… one at each membrane–

solution interface…

… but that is fine because they are not

equal and opposite… their sum is the

net Donnan potential between the two

solutions wetting the membrane… we

do not need to know each Donnan

potential separately, and in fact, it is

nearly impossible to measure each

individually…

… as long as there is

decent Donnan

Exclusion, the system

will be rather stable

and work as desired

[H3O+]R = 10 mM

[Cu2+]R = ?? ppm

start

[SO42-]R =

5 mM

[H3O+]L = 1 mM

[Cu2+]L =

30 ppm

[SO42-]L =

0.5 mM


P-Chemist


(1870–1956)


(e.g. PSS)

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[H3O+]R = 10 mM

[Cu2+]R = ?? ppm

start

[SO42-]R =

5 mM

[H3O+]L = 1 mM

[Cu2+]L = 30 ppm

[SO42-]L =

0.5 mM

– ––


P-Chemist


(1870–1956)


– ––

= 3000 ppm

––

2+


equilibrium

[H3O+]R = 10 mM

[Cu2+]R = 3000 ppm

[SO42-]R =

5 mM

[H3O+]L = 1 mM

[Cu2+]L =

30 ppm

[SO42-]L =

0.5 mM

… one net Donnan potential for the entire system that satisfies

equilibrium for each ion at each interface… Cool!


(e.g. Nafion or PSS)

[H3O+]R = 10 mM

[Cu2+]R = ?? ppm

start

[SO42-]R =

5 mM

[H3O+]L = 1 mM

[Cu2+]L =

30 ppm

[SO42-]L =

0.5 mM

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829… the less-frequently discussed four-electrode measurement!

Slade, …, Walsh, J. Electrochem. Soc., 2002, 149, A1556

CE/CA1 (S3/P2)WE/CA2 (P1)

WE'/CA2'

(S1)

RE/ref2

(S2)(EC-Lab diagram, from Bio-Logic)

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