lecture #8 of 17 - uci department of chemistryardo/echem/uci-chem248... · … again, at...
TRANSCRIPT
529
Lecture #8 of 17
530
Q: What’s in this set of lectures?
A: B&F Chapters 1, 15 & 4 main concepts:
● Section 1.1: Redox reactions
● Chapter 15: Electrochemical instrumentation
● Section 1.2: Charging interfaces
● Section 1.3: Overview of electrochemical experiments
● Section 1.4: Mass transfer and Semi-empirical treatment of
electrochemical observations
● Chapter 4: Mass transfer
531
Looking forward… Section 1.4 and Chapter 4
● Mass transfer
● Diffusion
● Migration / Drift
● Convection
● Semi-empirical models
● Conductivity
● Transport (Transference) number
● Balance sheets
532… again, at steady-state, ionic migration current must equal electronic current (due to KCL!)
This is a negative current…
}6e– need 6 “+”s,
whose ratios are ti
0.001 M Cu2+/Cu+
… and so is this!
B&F Example #2 (of 3)
migration
533
-5(2) + 7(1) = -3
This is a negative current…
= 3(1) = +3
… neutral!
}6e– need 6 “+”s,
whose ratios are ti
0.001 M Cu2+/Cu+
… and so is this!
… again, at steady-state, counterions move to maintain electroneutrality in diffusion layers
B&F Example #2 (of 3)
migration
534
5(2) + -7(1) = +3 -5(2) + 7(1) = -3
… and this is a positive current (at that electrode)…
= -3(1) = -3
… also neutral!
}
0.001 M Cu2+/Cu+
… and so is this!
6e– need 6 “+”s,
whose ratios are ti
= 3(1) = +3
… neutral!
… again, at steady-state, counterions move to maintain electroneutrality in diffusion layers
B&F Example #2 (of 3)
migration
535… RECALL FROM LAST TIME… adding a supporting electrolyte
reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
}Na+ClO4
– dominates
because it carries
the most current; its
contribution to σ is
large, and thus the
components have
the largest value of t
B&F Example #3 (of 3)
migration
536… adding a supporting electrolyte reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
But this is actually
much more complex
because of the…
… diffusion effects
here
}Na+ClO4
– dominates
because it carries
the most current; its
contribution to σ is
large, and thus the
components have
the largest value of t
B&F Example #3 (of 3)
migration
537… adding a supporting electrolyte reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
But this is actually
much more complex
because of the…
… diffusion effects
here }Na+, ClO4
–, and Cl–
should share charge
balance responsibilities
}Na+ClO4
– dominates
because it carries
the most current; its
contribution to σ is
large, and thus the
components have
the largest value of t
B&F Example #3 (of 3)
migration
538… adding a supporting electrolyte reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
But this is actually
much more complex
because of the…
… diffusion effects
here }Na+, ClO4
–, and Cl–
should share charge
balance responsibilities
}Na+ClO4
– dominates
because it carries
the most current; its
contribution to σ is
large, and thus the
components have
the largest value of t
CHECK: -5.97(2) + 6.029(1) = -5.911…
migration
539… adding a supporting electrolyte reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
But this is actually
much more complex
because of the…
… diffusion effects
here }Na+, ClO4
–, and Cl–
should share charge
balance responsibilities
}Na+ClO4
– dominates
because it carries
the most current; its
contribution to σ is
large, and thus the
components have
the largest value of t
CHECK: -5.97(2) + 6.029(1) = -5.911… +2.92(1) + 2.92(1) = 5.84…
migration
540… adding a supporting electrolyte reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
But this is actually
much more complex
because of the…
… diffusion effects
here }Na+, ClO4
–, and Cl–
should share charge
balance responsibilities
}
CHECK: -5.97(2) + 6.029(1) = -5.911… +2.92(1) + 2.92(1) = 5.84… so, missing +0.071 for Cl– toward the cathode
… and -0.071 for Cl– away from the anode
Na+ClO4– dominates
because it carries
the most current; its
contribution to σ is
large, and thus the
components have
the largest value of t
migration
541… adding a supporting electrolyte reduces the original im values
0.001 M Cu2+/Cu+
0.1 M NaClO4
PRECISION: -5.9709(2) + 6.0291(1) = -5.9127… +2.91(1) + 2.91(1) = 5.82… so, +0.0927 for Cl– toward the cathode
2.91
2.91 2.91
2.91
5.9709 5.9709
6.0291 6.0291
0.0927, so sum is 6
* Balance sheet
now balances
after corrections
here, which are
“underlined”
… and -0.0927 for Cl– away from the anode
* This is a very
challenging sign
convention, so
don’t worry
about it. Just
use absolute
numbers!
migration
542
FeIII(CN)63- + 1e- ⇌ FeII(CN)6
4-
cathode anode
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
?
? ?
* anions move toward the anode
* cations move toward the cathode
Let’s end this discussion of balance sheets with our own example…
Consider a 0.10 M equimolar solution of K3Fe(CN)6 and K4Fe(CN)6:
543
1) Calculate all t’s (for now, assume mobilities are equal, and thus cancel).
1.0
+
2) Convert t’s to migrational ion displacements…
𝒕𝐅𝐞(𝐂𝐍)𝟔
𝟑− =𝟑 𝟎. 𝟏 𝐱 𝟏𝟎−𝟑 𝐦𝐨𝐥/𝐜𝐦𝟑
𝟑 𝟎. 𝟏 𝐱 𝟏𝟎−𝟑 𝐦𝐨𝐥/𝐜𝐦𝟑 + 𝟒 𝟎. 𝟏 𝐱 𝟏𝟎−𝟑 𝐦𝐨𝐥/𝐜𝐦𝟑 + 𝟏 𝟎. 𝟕 𝐱 𝟏𝟎−𝟑 𝐦𝐨𝐥/𝐜𝐦𝟑
𝒕𝐅𝐞(𝐂𝐍)𝟔
𝟑− =𝟑 𝟎. 𝟏
𝟑 𝟎. 𝟏 + 𝟒 𝟎. 𝟏 + 𝟏 𝟎. 𝟕= 𝟎. 𝟐𝟏𝟒 ≈ 𝟎. 𝟐
𝒕𝐅𝐞(𝐂𝐍)𝟔
𝟒− =𝟒 𝟎. 𝟏
𝟑 𝟎. 𝟏 + 𝟒 𝟎. 𝟏 + 𝟏 𝟎. 𝟕= 𝟎. 𝟐𝟖𝟔 ≈ 𝟎. 𝟑
𝒕𝐊+ =𝟏 𝟎. 𝟕
𝟑 𝟎. 𝟏 + 𝟒 𝟎. 𝟏 + 𝟏 𝟎. 𝟕≈ 𝟎. 𝟓
5442) Convert t’s to migrational ion displacements...
3) Calculate diffusional ion displacements by difference:
10 x 0.5 x 1 = 5 K+
10 x 0.2 x 1/3 = 2/3 Fe(CN)63-
10 x 0.3 x 1/4 = 3/4 Fe(CN)64-
545
? ?
10 x 0.5 x 1 = 5 K+
10 x 0.2 x 1/3 = 2/3 Fe(CN)63-
10 x 0.3 x 1/4 = 3/4 Fe(CN)64-
FeIII(CN)63- + 1e- ⇌ FeII(CN)6
4-
cathode anode
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
Let’s end this discussion of balance sheets with our own example…
Consider a 0.10 M equimolar solution of K3Fe(CN)6 and K4Fe(CN)6:
5462) Convert t’s to migrational ion displacements...
10 x 0.5 x 1 = 5 K+
10 x 0.2 x 1/3 = 2/3 Fe(CN)63-
10 x 0.3 x 1/4 = 3/4 Fe(CN)64-
3) Calculate diffusional ion displacements by difference:
cathode10 Fe(CN)6
3- (needed) = -2/3 Fe(CN)63- (mig) + 10 2/3 Fe(CN)6
3- (diff)
-10 Fe(CN)64- (needed) = -3/4 Fe(CN)6
4- (mig) – 9 1/4 Fe(CN)64- (diff)
anode-10 Fe(CN)6
3- (needed) = 2/3 Fe(CN)63- (mig) – 10 2/3 Fe(CN)6
3- (diff)
10 Fe(CN)64- (needed) = 3/4 Fe(CN)6
4- (mig) + 9 1/4 Fe(CN)64- (diff)
547
10 x 0.5 x 1 = 5 K+
10 x 0.2 x 1/3 = 2/3 Fe(CN)63-
10 x 0.3 x 1/4 = 3/4 Fe(CN)64-
FeIII(CN)63- + 1e- ⇌ FeII(CN)6
4-
cathode anode
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
Let’s end this discussion of balance sheets with our own example…
Consider a 0.10 M equimolar solution of K3Fe(CN)6 and K4Fe(CN)6:
- 9 1/4 Fe(CN)64-
10 2/3 Fe(CN)63- - 10 2/3 Fe(CN)6
3-
9 1/4 Fe(CN)64-
548
4) Counter ions move to maintain electroneutrality in diffusion layers:
cathode(+10 2/3 Fe(CN)6
3-)(3) + (-9 1/4 Fe(CN)64-)(4)
32 – 37 = -5 (means +5 counterion diffusion… away from the electrode)
anode(+9 1/4 Fe(CN)6
4-)(4) + (-10 2/3 Fe(CN)63-)(3)
37 – 32 = +5 (means -5 counterion diffusion… toward the electrode)
3) Calculate diffusional ion displacements by difference:
cathode10 Fe(CN)6
3- (needed) = -2/3 Fe(CN)63- (mig) + 10 2/3 Fe(CN)6
3- (diff)
-10 Fe(CN)64- (needed) = -3/4 Fe(CN)6
4- (mig) – 9 1/4 Fe(CN)64- (diff)
anode-10 Fe(CN)6
3- (needed) = 2/3 Fe(CN)63- (mig) – 10 2/3 Fe(CN)6
3- (diff)
10 Fe(CN)64- (needed) = 3/4 Fe(CN)6
4- (mig) + 9 1/4 Fe(CN)64- (diff)
549
10 x 0.5 x 1 = 5 K+
10 x 0.2 x 1/3 = 2/3 Fe(CN)63-
10 x 0.3 x 1/4 = 3/4 Fe(CN)64-
FeIII(CN)63- + 1e- ⇌ FeII(CN)6
4-
cathode anode
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
Let’s end this discussion of balance sheets with our own example…
Consider a 0.10 M equimolar solution of K3Fe(CN)6 and K4Fe(CN)6:
5 K+ - 5 K+
- 9 1/4 Fe(CN)64-
10 2/3 Fe(CN)63- - 10 2/3 Fe(CN)6
3-
9 1/4 Fe(CN)64-
550
so, for the reaction of Fe(CN)63- at the cathode:
5. Finally, we could tally im and id:
𝑖𝑚 =− 2 3
− 2 3 + 10 2 3
= −0.066 (–) means migration will
"reduce" the total current
551
so, for the reaction of Fe(CN)63- at the cathode:
and, for the reaction of Fe(CN)64- at the anode:
5. Finally, we could tally im and id:
(–) means migration will
"reduce" the total current
(+) … the diffusional current
should always be positive
𝑖𝑚 =− 2 3
− 2 3 + 10 2 3
= −0.066
𝑖𝑑 =10 2 3
− 2 3 + 10 2 3
= +1.066
552
10 x 0.5 x 1 = 5 K+
10 x 0.2 x 1/3 = 2/3 Fe(CN)63-
10 x 0.3 x 1/4 = 3/4 Fe(CN)64-
FeIII(CN)63- + 1e- ⇌ FeII(CN)6
4-
cathode anode
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
10 e-
10 Fe(CN)64-
10 Fe(CN)63-
Let’s end this discussion of balance sheets with our own example…
Consider a 0.10 M equimolar solution of K3Fe(CN)6 and K4Fe(CN)6:
5 K+ - 5 K+
- 9 1/4 Fe(CN)64-
10 2/3 Fe(CN)63- - 10 2/3 Fe(CN)6
3-
9 1/4 Fe(CN)64-
553
so, for the reaction of Fe(CN)63- at the cathode:
and, for the reaction of Fe(CN)64- at the anode:
5. Finally, we could tally im and id:
Id will be larger by 6.6% as
compared to its value in the
absence of migration
Id will be smaller by 7.5% as
compared to its value in the
absence of migration
𝑖𝑚 =− 2 3
− 2 3 + 10 2 3
= −0.066
𝑖𝑑 =10 2 3
− 2 3 + 10 2 3
= +1.066
𝑖𝑚 =+ 3 4
+ 3 4 + 9 1 4
= +0.075
𝑖𝑑 =9 1 4
+ 3 4 + 9 1 4
= +0.925
(–) means migration will
"reduce" the total current
(+) means migration will
“add to" the total current
554
so, for the reaction of Fe(CN)63- at the cathode:
and, for the reaction of Fe(CN)64- at the anode:
5. Finally, we could tally im and id: … Boom!… Tough problem analyzed!
𝑖𝑚 =− 2 3
− 2 3 + 10 2 3
= −0.066
𝑖𝑑 =10 2 3
− 2 3 + 10 2 3
= +1.066
𝑖𝑚 =+ 3 4
+ 3 4 + 9 1 4
= +0.075
𝑖𝑑 =9 1 4
+ 3 4 + 9 1 4
= +0.925
Id will be larger by 6.6% as
compared to its value in the
absence of migration
Id will be smaller by 7.5% as
compared to its value in the
absence of migration
(–) means migration will
"reduce" the total current
(+) means migration will
“add to" the total current
555… so supporting electrolyte removes (most) drift for redox species
of interest… but it also removes iRu drop from data (2-for-1!)…
… let’s just look at some figures: 1, 4, 5, 7, 14 ,16
556… so supporting electrolyte removes (most) drift for redox species
of interest… but it also removes iRu drop from data (2-for-1!)…
Japp = constant
Problems
557… so supporting electrolyte removes (most) drift for redox species
of interest… but it also removes iRu drop from data (2-for-1!)…
Japp = constant
Problems Solutions
558… so supporting electrolyte removes (most) drift for redox species
of interest… but it also removes iRu drop from data (2-for-1!)…
Japp = constant
during CV
Problems Solutions
559Day-long 1st exam assigned “tomorrow” and due ≤ Friday
For HW: Box in answers! For B&F 2.4, do not include
junction potentials; for B&F 2.11, Hittorf method is
discussed in JCE 2001 paper; Problem #2 is cumulative
from the class (and does not cover Fick’s 2nd Law)…
Uncompensated resistance:
κ = σ Planar WE
… current is limited by largest resistor in series… which is at WE
… and finally, iRu drop does not always have a typical “R”…
x
Units: Ohm-1 cm-1
560Day-long 1st exam assigned “tomorrow” and due ≤ Friday
For HW: Box in answers! For B&F 2.4, do not include
junction potentials; for B&F 2.11, Hittorf method is
discussed in JCE 2001 paper; Problem #2 is cumulative
from the class (and does not cover Fick’s 2nd Law)…
Uncompensated resistance:
κ = σ Planar WE Point WE see B&F,pp. 27–28,and pp. 216–218 for details
Němec, J. Electroanal. Chem., 1964, 8, 166
… current is limited by largest resistor in series… which is at WE
What happens when x ∞?
… and finally, iRu drop does not always have a typical “R”…
x
561Day-long 1st exam assigned “tomorrow” and due ≤ Friday
For HW: Box in answers! For B&F 2.4, do not include
junction potentials; for B&F 2.11, Hittorf method is
discussed in JCE 2001 paper; Problem #2 is cumulative
from the class (and does not cover Fick’s 2nd Law)…
Uncompensated resistance:
κ = σ Planar WE Point WE see B&F,pp. 27–28,and pp. 216–218 for details
Němec, J. Electroanal. Chem., 1964, 8, 166
… current is limited by largest resistor in series… which is at WE
What happens when x ∞?
… and finally, iRu drop does not always have a typical “R”…
x
Just this term... with 1/circumference in place of x/A
562
A review of Section 1.4 (and Chapter 4)
● Mass transfer
● Diffusion
● Migration / Drift
● Convection
● Semi-empirical models
● Conductivity
● Transport (Transference) number
● Balance sheets
563
Q: What was in this set of lectures?
A: B&F Chapters 1, 15 & 4 main concepts:
● Section 1.1: Redox reactions
● Chapter 15: Electrochemical instrumentation
● Section 1.2: Charging interfaces
● Section 1.3: Overview of electrochemical experiments
● Section 1.4: Mass transfer and Semi-empirical treatment of
electrochemical observations
● Chapter 4: Mass transfer
564
Time-Dependence in
Electrochemistry
Chapters 4 and 5
565
Q: What’s in this set of lectures?
A: B&F Chapters 4 & 5 main concepts:
● Section 4.4.2: Fick’s Second Law of Diffusion
● Section 5.1: Overview of step experiments
● Section 5.2: Potential step under diffusion controlled
● Sections 5.3 & 5.9: Ultramicroelectrodes
● Sections 5.7 – 5.8: Chronoamperometry/Chronocoulometry
566
Fick’s 1st Law of Diffusion:Adolph Eugen Fick
B&F, pg. 149
… derive it (approximately) in a similar fashion as the diffusion coefficient…get your favorite beverage!
we use both of Fick’s laws of diffusion to derive equations for
transport-controlled electrochemistry...
567
Fick’s 1st Law of Diffusion:Adolph Eugen Fick
B&F, pg. 149
This is the net flux (correct dimensions)…… with half moving right and half moving left
we use both of Fick’s laws of diffusion to derive equations for
transport-controlled electrochemistry...
568
Fick’s 1st Law of Diffusion:Adolph Eugen Fick
B&F, pg. 149
we use both of Fick’s laws of diffusion to derive equations for
transport-controlled electrochemistry...
569
Fick’s 1st Law of Diffusion:Adolph Eugen Fick
B&F, pg. 149
… derived!Recall…
we use both of Fick’s laws of diffusion to derive equations for
transport-controlled electrochemistry...
570
Fick’s 1st Law of Diffusion:
Fick’s 2nd Law of Diffusion:
Adolph Eugen Fick
we use both of Fick’s laws of diffusion to derive equations for
transport-controlled electrochemistry...
B&F, pg. 149
… derive it (approximately) in asimilar fashion as Fick’s first law…
571
B&F, pp. 149–150
… derivation is not so bad…
FOR YOUR REFERENCE
572
B&F, pp. 149–150
… derivation is not so bad…
FOR YOUR REFERENCE
573
(First Law)
… derived!
B&F, pp. 149–150
… derivation is not so bad…
FOR YOUR REFERENCE
574The experiment we will model is a potential step experiment…
key points: at E1: no reaction (CO(x, 0) = CO*)
at E2: diffusion-controlled reaction (CO(0, t) = 0)
Eeq
> 200 mV
> 200 mV
How to derive expressions for diffusion-controlled current vs. time: 575
1. Solve Fick’s Second Law to get CO(x, t), and in the
process of doing this, you will use boundary conditions
that “customize” the solution for the particular
experiment of interest:
How to derive expressions for diffusion-controlled current vs. time: 576
1. Solve Fick’s Second Law to get CO(x, t), and in the
process of doing this, you will use boundary conditions
that “customize” the solution for the particular
experiment of interest:
2. Use Fick’s First Law to calculate JO(0, t) from CO(x, t):
How to derive expressions for diffusion-controlled current vs. time: 577
1. Solve Fick’s Second Law to get CO(x, t), and in the
process of doing this, you will use boundary conditions
that “customize” the solution for the particular
experiment of interest:
3. Calculate the time-dependent diffusion-limited current:
i = nFAJO(0, t)
2. Use Fick’s First Law to calculate JO(0, t) from CO(x, t):
How to derive expressions for diffusion-controlled current vs. time: 578
1. Solve Fick’s Second Law to get CO(x, t), and in the
process of doing this, you will use boundary conditions
that “customize” the solution for the particular
experiment of interest:
3. Calculate the time-dependent diffusion-limited current:
i = nFAJO(0, t)
2. Use Fick’s First Law to calculate JO(0, t) from CO(x, t):
… using the… Laplace transform, integration by parts, L’Hôpital’s rule, Schrödinger
equation, complementary error function, Leibniz rule, chain rule… Cool!
579Step 1 is the kicker… we’ll use the Laplace Transform to solve the
linear partial differential equation
The Laplace transform of any function F(t) is:
580
how about F(t) = 1?
Step 1 is the kicker… we’ll use the Laplace Transform to solve the
linear partial differential equation
The Laplace transform of any function F(t) is:
𝐿 1 =
0
∞
𝑒−𝑠𝑡 1 𝑑𝑡 = 𝑒−𝑠𝑡
−𝑠0
∞
= 0 −1
−𝑠=
1
𝑠
581
how about F(t) = 1?
Step 1 is the kicker… we’ll use the Laplace Transform to solve the
linear partial differential equation
The Laplace transform of any function F(t) is:
𝐿 1 =
0
∞
𝑒−𝑠𝑡 1 𝑑𝑡 = 𝑒−𝑠𝑡
−𝑠0
∞
= 0 −1
−𝑠=
1
𝑠
𝐿 𝑘𝑡 =
0
∞
𝑒−𝑠𝑡 𝑘𝑡 𝑑𝑡 = 𝑘
0
∞
𝑡𝑒−𝑠𝑡𝑑𝑡 = 𝑘𝑒−𝑠𝑡
𝑠2−𝑠𝑡 − 1
0
∞
how about F(t) = kt?
582Integrated by partshow about F(t) = kt?
𝐿 𝑘𝑡 =
0
∞
𝑒−𝑠𝑡 𝑘𝑡 𝑑𝑡 = 𝑘
0
∞
𝑡𝑒−𝑠𝑡𝑑𝑡 = 𝑘𝑒−𝑠𝑡
𝑠2−𝑠𝑡 − 1
0
∞
583Integrated by parts
Used L’Hôpital’s rule
how about F(t) = kt?
𝐿 𝑘𝑡 =
0
∞
𝑒−𝑠𝑡 𝑘𝑡 𝑑𝑡 = 𝑘
0
∞
𝑡𝑒−𝑠𝑡𝑑𝑡 = 𝑘𝑒−𝑠𝑡
𝑠2−𝑠𝑡 − 1
0
∞
= 𝑘 0 −1
𝑠2−1 =
𝑘
𝑠2
584
how about F(t) = e–at?
Integrated by parts
Used L’Hôpital’s rule
how about F(t) = kt?
𝐿 𝑘𝑡 =
0
∞
𝑒−𝑠𝑡 𝑘𝑡 𝑑𝑡 = 𝑘
0
∞
𝑡𝑒−𝑠𝑡𝑑𝑡 = 𝑘𝑒−𝑠𝑡
𝑠2−𝑠𝑡 − 1
0
∞
= 𝑘 0 −1
𝑠2−1 =
𝑘
𝑠2
𝐿 𝑒−𝑎𝑡 =
0
∞
𝑒−𝑠𝑡𝑒−𝑎𝑡𝑑𝑡 =
0
∞
𝑒− 𝑠+𝑎 𝑡𝑑𝑡 = 𝑒− 𝑠+𝑎 𝑡
− 𝑠 + 𝑎0
∞
585
how about F(t) = e–at?
how about F(t) = kt? Integrated by parts
Used L’Hôpital’s rule
𝐿 𝑘𝑡 =
0
∞
𝑒−𝑠𝑡 𝑘𝑡 𝑑𝑡 = 𝑘
0
∞
𝑡𝑒−𝑠𝑡𝑑𝑡 = 𝑘𝑒−𝑠𝑡
𝑠2−𝑠𝑡 − 1
0
∞
= 𝑘 0 −1
𝑠2−1 =
𝑘
𝑠2
𝐿 𝑒−𝑎𝑡 =
0
∞
𝑒−𝑠𝑡𝑒−𝑎𝑡𝑑𝑡 =
0
∞
𝑒− 𝑠+𝑎 𝑡𝑑𝑡 = 𝑒− 𝑠+𝑎 𝑡
− 𝑠 + 𝑎0
∞
= 0 −1
− 𝑠 + 𝑎=
1
𝑠 + 𝑎
586
OK, now for our case:
Recall, Second Law:
–F(t) =
587
OK, now for our case: –F(t) =
–L{ }= ?
588
well, wait a second, this
term is not so bad…
OK, now for our case: –F(t) =
–L{ }= ?
0
∞
𝑒−𝑠𝑡𝐷𝜕2𝐶 𝑥, 𝑡
𝜕𝑥2𝑑𝑡 = 𝐷
𝜕2
𝜕𝑥2
0
∞
𝑒−𝑠𝑡𝐶 𝑥, 𝑡 𝑑𝑡 = 𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
589
the Laplace transform
of C(x, t)? … Isn’t this
cheating?
OK, now for our case:
well, wait a second, this
term is not so bad…
–F(t) =
–L{ }= ?
0
∞
𝑒−𝑠𝑡𝐷𝜕2𝐶 𝑥, 𝑡
𝜕𝑥2𝑑𝑡 = 𝐷
𝜕2
𝜕𝑥2
0
∞
𝑒−𝑠𝑡𝐶 𝑥, 𝑡 𝑑𝑡 = 𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
590
the Laplace transform
of C(x, t)? … Isn’t this
cheating? Well, ahem,
no!
OK, now for our case:
well, wait a second, this
term is not so bad…
–F(t) =
–L{ }= ?
0
∞
𝑒−𝑠𝑡𝐷𝜕2𝐶 𝑥, 𝑡
𝜕𝑥2𝑑𝑡 = 𝐷
𝜕2
𝜕𝑥2
0
∞
𝑒−𝑠𝑡𝐶 𝑥, 𝑡 𝑑𝑡 = 𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
591
?
OK, now for our case:
not so lucky with
this term…
–F(t) =
–L{ }= ?
−𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
592
g(x) f '(x)
Integration, by parts, again!
0
∞
𝑒−𝑠𝑡𝜕𝐶𝑂 𝑥, 𝑡
𝜕𝑡𝑑𝑡
𝑎
𝑏
𝑔 𝑥 𝑓′ 𝑥 𝑑𝑥 = 𝑔 𝑥 𝑓 𝑥𝑎
𝑏−
𝑎
𝑏
𝑓 𝑥 𝑔′ 𝑥 𝑑𝑥
593
… and at time = 0, what is the value of C, anywhere?
Integration, by parts, again!
g(x) f '(x)
0
∞
𝑒−𝑠𝑡𝜕𝐶𝑂 𝑥, 𝑡
𝜕𝑡𝑑𝑡
𝑎
𝑏
𝑔 𝑥 𝑓′ 𝑥 𝑑𝑥 = 𝑔 𝑥 𝑓 𝑥𝑎
𝑏−
𝑎
𝑏
𝑓 𝑥 𝑔′ 𝑥 𝑑𝑥
= 𝑒−𝑠𝑡𝐶 𝑥, 𝑡0
∞−
0
∞
𝐶 𝑥, 𝑡 −𝑠𝑒−𝑠𝑡 𝑑𝑥
= 0 − 𝐶 𝑥, 0 + 𝑠 𝐶 𝑥, 𝑠
594
… and at time = 0, what is the value of C, anywhere?
… just C*!
Integration, by parts, again!
g(x) f '(x)
0
∞
𝑒−𝑠𝑡𝜕𝐶𝑂 𝑥, 𝑡
𝜕𝑡𝑑𝑡
𝑎
𝑏
𝑔 𝑥 𝑓′ 𝑥 𝑑𝑥 = 𝑔 𝑥 𝑓 𝑥𝑎
𝑏−
𝑎
𝑏
𝑓 𝑥 𝑔′ 𝑥 𝑑𝑥
= 𝑒−𝑠𝑡𝐶 𝑥, 𝑡0
∞−
0
∞
𝐶 𝑥, 𝑡 −𝑠𝑒−𝑠𝑡 𝑑𝑥
= 0 − 𝐶 𝑥, 0 + 𝑠 𝐶 𝑥, 𝑠
595L.T. of Fick’s 2nd Law…
now is turns out that the
L.T. of this…
see B&F,pg. 775,for details
is this…
–F(t) =
–L{ }
𝑠 𝐶 𝑥, 𝑠 − 𝐶∗ − 𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
596L.T. of Fick’s 2nd Law…
now is turns out that the
L.T. of this…
is this…
our equation:
Now what? Well, recall these terms are equal to each other (= 0), then rearrange…
… and what does it look like?
–F(t) =
–L{ }
𝑠 𝐶 𝑥, 𝑠 − 𝐶∗ − 𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
597L.T. of Fick’s 2nd Law…
now is turns out that the
L.T. of this…
is this…
our equation:
Now what? Well, recall these terms are equal to each other (= 0), then rearrange…
… and what does it look like?
–F(t) =
–L{ }
𝑠 𝐶 𝑥, 𝑠 − 𝐶∗ − 𝐷𝜕2
𝜕𝑥2 𝐶 𝑥, 𝑠
the time-independent
Schrödinger Eq.
in 1D…
𝑑2
𝑑𝑥2𝜓 𝑥 −
2𝑚
ℏ2𝐸 − 𝑉 𝑥 𝜓 𝑥 = 0
598
the solution of the
Schrödinger Eq. is:
our equation:
the time-independent
Schrödinger Eq.
in 1D…
𝑑2
𝑑𝑥2𝜓 𝑥 −
2𝑚
ℏ2𝐸 − 𝑉 𝑥 𝜓 𝑥 = 0
𝜓 𝑥 = 𝐴′exp− 2𝑚 𝐸 − 𝑉 𝑥
ℏ𝑥 + 𝐵′exp
2𝑚 𝐸 − 𝑉 𝑥
ℏ𝑥
599
… and by analogy, the solution of our equation is:
the solution of the
Schrödinger Eq. is:
our equation:
the time-independent
Schrödinger Eq.
in 1D…
𝑑2
𝑑𝑥2𝜓 𝑥 −
2𝑚
ℏ2𝐸 − 𝑉 𝑥 𝜓 𝑥 = 0
𝜓 𝑥 = 𝐴′exp− 2𝑚 𝐸 − 𝑉 𝑥
ℏ𝑥 + 𝐵′exp
2𝑚 𝐸 − 𝑉 𝑥
ℏ𝑥
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥 + 𝐵′ 𝑠 exp
𝑠
𝐷𝑥
600
Now, what are A' and B' and how do we get rid of the “s”?
… we need some boundary conditions!
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥 + 𝐵′ 𝑠 exp
𝑠
𝐷𝑥
601
called semi-infinite linear
(because of x) diffusion
Now, what are A' and B' and how do we get rid of the “s”?
… we need some boundary conditions!
1.
L.T.
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥 + 𝐵′ 𝑠 exp
𝑠
𝐷𝑥
lim𝑥→∞
𝐶 𝑥, 𝑠 =𝐶∗
𝑠
602
called semi-infinite linear
(because of x) diffusion
Now, what are A' and B' and how do we get rid of the “s”?
… we need some boundary conditions!
1.
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥 + 𝐵′ 𝑠 exp
𝑠
𝐷𝑥
lim𝑥→∞
𝐶 𝑥, 𝑠 =𝐶∗
𝑠
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥 + 𝐵′ 𝑠 exp
𝑠
𝐷𝑥
What does this do for us?
0… and so B' must be equal to 0
∞
L.T.
603
some more boundary conditions…
2.
L.T.
𝐶 0, 𝑠 = 0
𝐶 0, 𝑡 = 0
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥
604
some more boundary conditions…
What does this do for us?
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥
2.
L.T.
𝐶 0, 𝑠 = 0
𝐶 0, 𝑡 = 0
1
𝐴′ 𝑠 = −𝐶∗
𝑠… and so
𝐶 𝑥, 𝑠 =𝐶∗
𝑠+ 𝐴′ 𝑠 exp −
𝑠
𝐷𝑥
605now our solution is fully constrained… and we need “t” back!!
inverse L.T. using Table A.1.1 in B&F
=
606What’s efrc?… Well, first of all, what’s the error function: erf?
607Now… what’s efrc?
608Now… what’s efrc?
Gaussian distribution,
with mean = 0 and std.
dev. = 1/sqrt(2)
609
… well, for large x, erfc = 0 (erf = 1) and so C(x, t) = C* … Check!
… and for x = 0, erfc = 1 (erf = 0) and so C(x, t) = 0 … Check!
… Let’s plot it!
Does this make sense?
610
C* = 1 x 10-6 M
D = 1 x 10-5 cm2 s-1
t = 1s
0.1s0.01s0.0001s
Hey, these look completely reasonable!
(100 µm)
611How large is the diffusion layer? Recall the rms displacement…
root mean square
(rms) displacement
(standard deviation) Δ = 2𝑑 𝐷𝑡, where d is the dimension
D Δ*=
1D
2D
3D
*the rms displacement
… plane
… wire, line, tube
… point, sphere, disk
a characteristic
"diffusion length"
2𝐷𝑡
4𝐷𝑡
6𝐷𝑡
Δ = 2𝑑 𝐷𝑡 =cm2
ss = cm
From a…
612
4 µm
14 µm 40 µm
t = 1s
0.1s
Hey, these look completely reasonable!
0.01s0.0001s
… use the geometric area for calculations
C* = 1 x 10-6 M
D = 1 x 10-5 cm2 s-1
2𝐷𝑡 =(100 µm)
OK… that’s Step #1… 613
1. Solve Fick’s Second Law to get CO(x, t), and in the
process of doing this, you will use boundary conditions
that “customize” the solution for the particular
experiment of interest:
3. Calculate the time-dependent diffusion-limited current:
i = nFAJO(0, t)
2. Use Fick’s First Law to calculate JO(0, t) from CO(x, t):
614
… but we just derived CO(x, t):
… now Step #2…
(Fick’s First Law)
… and so we need to evaluate:
−𝐽𝑂 𝑥, 𝑡 = 𝐷𝑂
𝜕
𝜕𝑥𝐶∗erf
𝑥
2 𝐷𝑂𝑡
615… now Step #2…
… we use the Leibniz rule, to get d/dx(erf(x)) as follows:
see B&F,pg. 780,for details
−𝐽𝑂 𝑥, 𝑡 = 𝐷𝑂
𝜕
𝜕𝑥𝐶∗erf
𝑥
2 𝐷𝑡
616… now Step #2…
… we use the Leibniz rule, to get d/dx(erf(x)) as follows:
see B&F,pg. 780,for details
… and using this in conjunction with the chain rule, we get:
−𝐽𝑂 𝑥, 𝑡 = 𝐷𝑂
𝜕
𝜕𝑥𝐶∗erf
𝑥
2 𝐷𝑡
−𝐽𝑂 𝑥, 𝑡 = 𝐷𝑂𝐶∗1
2 𝐷𝑂𝑡
2
𝜋exp
−𝑥2
4𝐷𝑂𝑡
617
… we use the Leibniz rule, to get d/dx(erf(x)) as follows:
… now Step #2…
see B&F,pg. 780,for details
… and using this in conjunction with the chain rule, we get:
… and when x = 0 (at the electrode), we have what we wanted…
−𝐽𝑂 𝑥, 𝑡 = 𝐷𝑂
𝜕
𝜕𝑥𝐶∗erf
𝑥
2 𝐷𝑡
−𝐽𝑂 0, 𝑡 = 𝐶∗𝐷𝑂
𝜋𝑡
−𝐽𝑂 𝑥, 𝑡 = 𝐷𝑂𝐶∗1
2 𝐷𝑂𝑡
2
𝜋exp
−𝑥2
4𝐷𝑂𝑡
618
1. Solve Fick’s Second Law to get CO(x, t). In the
process of doing this, you will use boundary conditions
that “customize” the solution for the particular
experiment of interest.
3. Calculate the time-dependent diffusion-limited current:
i = nFAJ(0, t)
2. Use Fick’s First Law to calculate J0(0, t) from C(x, t):
OK… that’s Steps #1 and 2…
619… and finally, Step #3 using Step #2…
−𝐽𝑂 0, 𝑡 = 𝐶∗𝐷𝑂
𝜋𝑡
the Cottrell Equation
… and with i = nFAJ(0, t)…
620… and finally, Step #3 using Step #2…
the Cottrell Equation
Frederick Gardner Cottrell, in 1920b. January 10, 1877, Oakland, California, U.S.A.
d. November 16, 1948, Berkeley, California, U.S.A.
… established Research Corp. in 1912
… initial funding from profits on patents for
the electrostatic precipitator, used to clear
smokestacks of charged soot particles
… and with i = nFAJ(0, t)…
−𝐽𝑂 0, 𝑡 = 𝐶∗𝐷𝑂
𝜋𝑡
621
http://en.wikipedia.org/wiki/Electrostatic_precipitatorhttp://en.wikipedia.org/wiki/Corona_discharge
> 1 kVneutral particles
corona dischargecapture plates
charged particles
Cottrell, then at UC Berkeley, invented the electrostatic precipitator
used to clear smokestacks of charged soot particles…
622
623
D = 1.5 x 10-6 cm2 s-1
D = 1 x 10-5 cm2 s-1
… OK, so what does it predict?
the Cottrell
Equation
(10 ms)
624
D = 1.5 x 10-6 cm2 s-1
D = 1 x 10-5 cm2 s-1
… OK, so what does it predict?
the Cottrell
Equation
625… OK, so what does it predict?
the Cottrell
Equation
slope =
nFAπ -1/2D1/2C*
626
slope =
nFAπ -1/2D1/2C*
… OK, so what does it predict?
?????
the Cottrell
Equation
627
1. Huge initial currents… compliance current!
2. Noise.
3. Roughness factor adds to current at early times.
4. RC time limitations yield a negative deviation at short times.
5. Adsorbed (electrolyzable) junk adds to current at short times.
6. Convection, and “edge effects,” impose a “long” time limit on
an experiment.
… use the Cottrell Equation to measure D!
… but what are the problems with this approach?
the Cottrell
Equation
628… use the Cottrell Equation to measure D!
… but what are the problems with this approach?
… solution: Integrate it, with respect to time:
the integrated
Cottrell Equation
1. Huge initial currents… compliance current!
2. Noise.
3. Roughness factor adds to current at early times.
4. RC time limitations yield a negative deviation at short times.
5. Adsorbed (electrolyzable) junk adds to current at short times.
6. Convection, and “edge effects,” impose a “long” time limit on
an experiment.
the Cottrell
Equation