470 Lecture #7 of 18

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1 Lecture #7 of

2 471 Q: What s in this set of lectures? A: Introduction, Review, and B&F Chapter 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

3 472 Looking forward Section 1.4 and Chapter 4 Mass transfer Diffusion Migration / Drift Convection Semi-empirical models Balance sheets

4 now, a simplified NernstPlanck equation 473 From before, for one species the total flux in one-dimension is N = Dc RT d μ dx cv, {several math steps from before} N = D dc zfd dφ c cv dx RT dx which can also be written using the ES equation N = D dc flux dx J i x = D i C i x x μc dφ dx cv z if RT D φ x ic i x μ i = z i FD i RT D i = RTμ i z i F B&F, C iv x

5 there is some value in thinking semi-quantitatively about mass transport and B&F develop a formalism for this (pp. 2935): 474 Why? The goal: Derive an expression for the current as a function of the applied potential in our electrochemical cell.

6 there is some value in thinking semi-quantitatively about mass transport and B&F develop a formalism for this (pp. 2935): 475 first, let s eliminate contributions to transport from migration/drift and convection SO WE CAN FOCUS ON DIFFUSION EFFECTS J i x = D i C i x x z if RT D φ x ic i x add supporting electrolyte do not stir C iv x Now, imagine a scenario where this is not only true, but where dc/dx is time invariant, and thus you should see no hysteresis. Ironically, this situation is encountered when the solution is stirred just not too vigorously J i x = D i C i x x

7 there is some value in thinking semi-quantitatively about mass transport and B&F develop a formalism for this (pp. 2935): 476 first, let s eliminate contributions to transport from migration/drift and convection SO WE CAN FOCUS ON DIFFUSION EFFECTS J i x = D i C i x x J i x = D i C i x x z if RT D φ x ic i x add supporting electrolyte do not stir C iv x now, consider specifically the reduction of some molecule O : O ne R (example: Fe(CN)6 3-1e Fe(CN)6 4- )

8 now, let s approximate the concentration gradient near the WE as a linear function: C O C O x = 0 J O x = D O where C O * is the bulk concentration of O, δ is the Nernst diffusion layer thickness δ O 477 Bard & Faulkner, 2nd Ed., Figure stirring causes δ to become welldefined, time-invariant, and a short distance to encounter bulk solution conditions

9 now, let s approximate the concentration gradient near the WE as a linear function: C O C O x = 0 J O x = D O where C O * is the bulk concentration of O, δ is the Nernst diffusion layer thickness δ O 478 an unstirred solution will have δ cm (50 µm) after ~1 sec (Bockris, Reddy, and Gamboa-Aldeco, Modern EChem, Vol. 2A, 2002, pg. 1098) Bard & Faulkner, 2nd Ed., Figure stirring causes δ to become welldefined, time-invariant, and a short distance to encounter bulk solution conditions

10 now, let s approximate the concentration gradient near the WE as a linear function: C O C O x = 0 J O x = D O where C O * is the bulk concentration of O, δ is the Nernst diffusion layer thickness the linear approximation δ O 479 the TRUE concentration gradient Bard & Faulkner, 2nd Ed., Figure 1.4.1

11 because it will be convenient later, group the diffusion coefficient with the diffusion layer thickness: m O = D O δ O where m O is the mass transfer coefficient (units: cm s -1 ) * Note, dimensionally we have (cm 2 s -1 )/cm 480

12 because it will be convenient later, group the diffusion coefficient with the diffusion layer thickness: 481 m O = D O δ O where m O is the mass transfer coefficient (units: cm s -1 ) substituting J O x = m O C O C O x = 0 * Note, dimensionally we have (cm 2 s -1 )/cm moles s -1 cm -2 cm s -1 moles cm -3 * Note: we also dropped the () sign

13 because it will be convenient later, group the diffusion coefficient with the diffusion layer thickness: m O = D O δ O where m O is the mass transfer coefficient (units: cm s -1 ) 482 substituting J O x = m O C O C O x = 0 * Note, dimensionally we have (cm 2 s -1 )/cm writing the flux (i.e. rate) in terms of the current i nfa = m O C O C O x = 0 (1.4.6) we can also express the flux in terms of the transport of "R": i nfa = m R C R x = 0 C R

14 i nfa = m R C R x = 0 C R and if no R is presently initially then C R* = 0 483

15 i nfa = m R C R x = 0 C R and if no R is presently initially then C R* = 0 i nfa = m O C O C O x = 0 and fastest rate, i l, occurs when C O (x = 0) = 0 484

16 i nfa = m R C R x = 0 C R and if no R is presently initially then C R* = 0 i nfa = m O C O C O x = 0 and fastest rate, i l, occurs when C O (x = 0) = to obtain the potential dependence of the current, we make two substitutions into the Nernst Equation. We assume that electrontransfer from/to the electrode to/from both O and R is rapid enough that equilibrium concentrations are maintained at the electrode surface. i nfa = m RC R x = 0 C O x = 0 = i l i nfam O E = E 0 RT nf ln C O x = 0 C R x = 0 i l nfa = m OC O

17 486 E = E 0 RT nf ln m O m R RT nf ln i l i i formal potential algebra, algebra i nfa = m RC R x = 0 C O x = 0 = i l i nfam O E = E 0 RT nf ln C O x = 0 C R x = 0 i l nfa = m OC O

18 487 E = E 0 RT nf ln m O m R RT nf ln i l i i formal potential and this equation is further simplified by using the definition for the half-wave potential: E = E 1/2 when i = i l /2. E 1/2 = E 0 RT nf ln m O m R RT nf ln i i l l 2 i l 2 E 1/2 = E 0 RT nf ln m O m R

19 488 E = E 0 RT nf ln m O m R RT nf ln i l i i formal potential and this equation is further simplified by using the definition for the half-wave potential: E = E 1/2 when i = i l /2. E 1/2 = E 0 RT nf ln m O m R RT nf ln i i l l 2 i l 2 E 1/2 = E 0 RT nf ln m O m R

20 E = E 1/2 RT nf ln i l i i 489 What happens when i i l? What happens when i 0?

21 and when C R * E = E 0 RT nf ln m O m R RT nf ln i l,c i i i l,a What happens when i i l? What happens when i 0?

22 and when C R * E = E 0 RT nf ln m O m R Looks linear Could define a resistance for this using Ohm s law R mt RT nf ln i l,c i i i l,a What happens when i i l? What happens when i 0?

23 and when C R * E = E 0 RT nf ln m O m R Looks linear Could define a resistance for this using Ohm s law R mt RT nf ln i l,c i i i l,a Could also define a linearized overpotential formula here η conc What happens when i i l? What happens when i 0?

24 and when C R * E = E 0 RT nf ln m O m R Looks linear Could define a resistance for this using Ohm s law R mt RT nf ln i l,c i i i l,a Could also define a linearized overpotential formula here η conc Recall that this is all due to mass transport by diffusion What if we now also included mass transport by migration / drift?

but first, the Drude model for conduction of electrons / Ions A potential bias creates an electric field that drives electrons preferentially in one direction.

(e.g. other electrons) which slow the transport of the electrons and, as before, they are said to have a resulting average velocity, v d (see below).

25 but first, the Drude model for conduction of electrons / Ions A potential bias creates an electric field that drives electrons preferentially in one direction. These electrons bounce off of stationary crystal ions (and other things (e.g. other electrons) which slow the transport of the electrons and, as before, they are said to have a resulting average velocity, v d (see below). With each bounce, they lose kinetic energy to heat (inelastic collision) and this removes some of their driving force (potential). Thus, at this point, they have less (free) energy to move ( potential drop ) or, they are stopped and they start again due to the potential at this new starting location. 494 R Ω

26 Equivalent circuit representations 495 The potential across an electrochemical cell must be dropped across the intervening media (e.g. electrodes, electrolytes, interfaces). Thus, one can model an electrochemical cell as the sum of the voltage drops because they are electrically in series. Each of these voltage drops is a function of the steady-state current, because the current through each is the same (I = I 1 = I 2 = I 3 ), written as E(I) = E 1 (I) E 2 (I) E 3 (I). Randles equivalent circuit Why is this a common equivalent circuit?

27 Electrical Engineering Physics Given a resistor with resistance, R, look what you can calculate

28 Electrical Engineering Physics Given a resistor with resistance, R, look what you can calculate ρ = RA which is the resistivity (Ω cm) l the resistivity is great because it is independent of size!

29 Electrical Engineering Physics Given a resistor with resistance, R, look what you can calculate ρ = RA which is the resistivity (Ω cm) l the resistivity is great because it is independent of size! σ = 1 ρ = l RA = Gl A which is the conductivity (Ω-1 cm -1 = mho cm -1 (< 1971) = S cm -1 ) and where S stands for siemen the conductivity is equally as great because it is also independent of size!

Electrical Engineering Physics 101 499 Given a resistor with resistance, R, look what you can calculate ρ = RA which is the resistivity (Ω cm) l the resistivity is great because it is independent of

30 Electrical Engineering Physics Given a resistor with resistance, R, look what you can calculate ρ = RA which is the resistivity (Ω cm) l the resistivity is great because it is independent of size! σ = 1 ρ = l RA = Gl A which is the conductivity (Ω-1 cm -1 = mho cm -1 (< 1971) = S cm -1 ) and where S stands for siemen the conductivity is equally as great because it is also independent of size! Well, the conductivity of an electrolyte can be determined based on the properties of its constituents

31 recall, the simplified NernstPlanck equation (with no mixing ) for one species and as a current density J = zfn = zfd dc dx z Fμc dφ dx 500

32 recall, the simplified NernstPlanck equation (with no mixing ) for one species and as a current density J = zfn = zfd dc dx z Fμc dφ dx when current is passing, it is assumed that the bulk concentration of species does not change (much) and so J diff is small (~0) thus, J = i J migr,i only and recall Ohm s law 501 J = i z i Fμ i c i dφ dx = E RA = EG A, where G is the conductance (S)

33 recall, the simplified NernstPlanck equation (with no mixing ) for one species and as a current density J = zfn = zfd dc dx z Fμc dφ dx when current is passing, it is assumed that the bulk concentration of species does not change (much) and so J diff is small (~0) thus, J = i J migr,i only and recall Ohm s law 502 J = i z i Fμ i c i dφ dx = E RA = EG A, where G is the conductance (S) G = AF E dφ dx i z i μ i c i = A l σ, where σ is the conductivity (S cm-1 )

34 recall, the simplified NernstPlanck equation (with no mixing ) for one species and as a current density J = zfn = zfd dc dx z Fμc dφ dx when current is passing, it is assumed that the bulk concentration of species does not change (much) and so J diff is small (~0) thus, J = i J migr,i only and recall Ohm s law 503 J = i z i Fμ i c i dφ dx = E RA = EG A, where G is the conductance (S) G = AF E dφ dx i z i μ i c i = A l σ, where σ is the conductivity (S cm-1 ) and because the concentration of species does not change, the change in electric potential (dφ) is the potential difference (E) and assuming linear slope in potential over space, dx = l, and so G = AF E E l i z i μ i c i = A l σ

35 recall, the simplified NernstPlanck equation (with no mixing ) for one species and as a current density J = zfn = zfd dc dx z Fμc dφ dx when current is passing, it is assumed that the bulk concentration of species does not change (much) and so J diff is small (~0) thus, J = i J migr,i only and recall Ohm s law 504 J = i z i Fμ i c i dφ dx = E RA = EG A, where G is the conductance (S) G = AF E dφ dx i z i μ i c i = A l σ, where σ is the conductivity (S cm-1 ) and because the concentration of species does not change, the change in electric potential (dφ) is the potential difference (E) and assuming linear slope in potential over space, dx = l, and so G = AF E E l i z i μ i c i = A l σ σ = F i z i μ i c i

36 so, who can tell me what the transport (transference) number is? 505 σ = F i z i μ i c i

37 so, who can tell me what the transport (transference) number is? 506 It is the fraction of the conductivity that each ion carries, and thus, the t i must sum to one! σ = F i z i μ i c i

38 so, who can tell me what the transport (transference) number is? 507 It is the fraction of the conductivity that each ion carries, and thus, the t i must sum to one! It is the fraction of the current (density) that each ion carries J = i z i Fμ i c i dφ dx = E RA = EG A = Eσ l, where G is the conductance (S) σ = F i z i μ i c i

39 again, the NernstPlanck equation 508 J i x = D i C i x x diffusion z if RT D φ x ic i x migration C iv x convection let s continue to suppose that we do not stir J i x = D i C i x x diffusion z if RT D φ x ic i x migration C iv x convection

40 again, the NernstPlanck equation 509 J i x = D i C i x x diffusion z if RT D φ x ic i x migration C iv x convection let s continue to suppose that we do not stir J i x = D i C i x x diffusion z if RT D φ x ic i x migration C iv x convection and, as an aside, this is the equation used in the drift diffusion model for transport and equilibration in semiconductors.

41 again, the NernstPlanck equation 510 J i x = D i C i x x diffusion z if RT D φ x ic i x migration C iv x convection let s continue to suppose that we do not stir J i x = D i C i x x diffusion z if RT D φ x ic i x migration C iv x convection Question: How can we analyze an electrochemical experiment in order to ascertain whether migration has been reduced to the point where it can be completely neglected? flux current i = i d(iffusion) i m(igration) = nfaj x

42 So, clearly, i m(igration) can reinforce i d(iffusion), or oppose it 511

43 So, clearly, i m(igration) can reinforce i d(iffusion), or oppose it in an electrolytic cell A 512 anode cathode

44 So, clearly, i m(igration) can reinforce i d(iffusion), or oppose it in an electrolytic cell A 513 anode cathode at anode, i m adds to the current; at cathode, i d subtracts at anode, i d subtracts from the current; at cathode, i m adds

45 So, clearly, i m(igration) can reinforce i d(iffusion), or oppose it in an electrolytic cell 514 cathodes source negative charge; anodes sink it so, migration drives anions from the cathode to the anode, and cations from the anode to the cathode neutral species are not affected Huh?

46 So, clearly, i m(igration) can reinforce i d(iffusion), or oppose it in an electrolytic cell 515 cathodes source negative charge; anodes sink it so, migration drives anions from the cathode to the anode, and cations from the anode to the cathode neutral species are not affected Huh? example:

47 B&F s balance sheets help us understand the role of migration in steady-state electrolytic mass transport. Let s break them down: M HCl

48 B&F s balance sheets help us understand the role of migration in steady-state electrolytic mass transport. Let s break them down: all ions in the bulk of the cell contribute to migration by an amount proportional to their transport number M HCl

49 B&F s balance sheets help us understand the role of migration in steady-state electrolytic mass transport. Let s break them down: all ions in the bulk of the cell contribute to migration by an amount proportional to their transport number M HCl t H = t Cl = x 10 3 cm 2 s 1 V x 10 3 mol/cm x 10 3 cm 2 s 1 V x 10 3 mol/cm x 10 4 cm 2 s 1 V x 10 3 = mol/cm3 2 ~8

50 B&F s balance sheets help us understand the role of migration in steady-state electrolytic mass transport. Let s break them down: diffusion occurs only in regions where a concentration gradient exists near the electrodes M HCl these coefficients are the values necessary to satisfy the steady-state condition of i = i m i d for example, at the cathode (left): 10H (needed) = 8H (migration) 2H (diffusion) 2Cl (diffusion) for charge balance

51 B&F s balance sheets help us understand the role of migration in steady-state electrolytic mass transport. Let s break them down: M HCl and at the anode (right): 10Cl (needed) = 2Cl (migration) 8Cl (diffusion) 8H (diffusion) for charge balance

52 B&F s balance sheets help us understand the role of migration in steady-state electrolytic mass transport. Let s break them down: 521 If we focus on the cathode H current and anode Cl current 0.1 M HCl we can estimate the relative i d and i m at each electrode: cathode (for H ) : i = 10; i d = 2, i m = 8 (80% of the H current is migration) anode (for Cl ) : i = 10; i d = 8, i m = 2 (20% of the Cl current is migration)

529 Lecture #8 of 18

529 Lecture #8 of 18 Lecture #8 of 18 529 530 Q: What s in this set of lectures? A: Introduction, Review, and B&F Chapter 1, 15 & 4 main concepts: Section 1.1: Redox reactions Chapter 15: Electrochemical instrumentation Section