A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 69
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The drop remains in place until a mechanical, solenoid-driven drop knocker dislodges it upon receiving another electronic signal. The SMDE can serve as a hangingmercury drop electrode (HMDE) or, in a repetitive mode, as a replacement for the DME.In the latter role, it retains all of the important advantages of the DME (Section 7.1.3), andit has the added feature that the area does not change at the time of measurement. Mostcontemporary polarographic work is carried out with SMDEs.In the sections below, we will discuss polarographic concepts first in the context ofthe DME, then with reference to the SMDE.Additional fine points about behavior at the DME are discussed in Section 5.3 of the first edition.7.1 Behavior at Polarographic Electrodes263Hg reservoirDrop knockerElectronicallycontrolled valveOpenClosedLarge-bore capillaryПTimeCounter electrodeFigure 7.1.2 Schematic diagram of a static mercury drop electrode. The typical unit includes acell stand, as shown here, and facilities for stirring and deaeration of the solution by bubbling withan inert gas.
These functions are normally controlled electronically by an automated potentiostat,which also manages the issuance and dislodgment of the mercury drops and applies the potentialprogram that they experience while they are active as working electrodes. When a new drop isneeded, the old one is dislodged by a command to the drop knocker, then the electronicallycontrolled valve is opened for 30-100 ms (see graph on left). The drop is formed during this briefperiod (see graph on right), then it remains indefinitely stable in size as it is employed as theworking electrode.7.1.2Diffusion-Limited Responses at the DME and SMDE(a) The Ilkovic EquationLet us consider the current that flows during a single drop's lifetime when a DME is heldat a potential in the mass-transfer-controlled region for electrolysis.
That is, we seek thediffusion-limited current, essentially as we did in Section 5.2 for stationary planar andspherical electrodes. The problem was solved by Koutecky (9, 10), but the treatment requires consideration of the relative convective movement between the electrode and solution during drop growth. The mathematics are rather complicated and give littleintuitive feel for the effects involved in the problem. The treatment we will follow, originally due to Lingane and Loveridge (11), makes no pretense to rigor. It is only an outlineto the problem, but it highlights the differences between a DME and a stationary electrode (3, 11-14).The typical values of drop lifetime and drop diameter at maturity ensure that lineardiffusion holds at a DME to a good approximation [Section 5.2.2(c)]. Thus, we beginby invoking the Cottrell relation, (5.2.11), while remembering that for the moment weare considering electrolysis only at potentials on the diffusion-limited portion of thevoltammetric response curve.
Since the drop area is a function of time, we must determine A(t) explicitly. If the rate of mercury flow from the DME capillary (mass/time) ism and the density of mercury is с1щ, then the weight of the drop at time t ismt = &ridHg(7.1.1)264 • Chapter 7. Polarography and Pulse VoltammetryThe drop's radius and its area are then given bySubstitution into the Cottrell relation givesjyH2W'3tm(7-1.4)In addition to the effect of the changing area, which progressively enlarges the diffusion field, there is a second consideration that we might call the "stretching effect." Thatis, at any time t, expansion of the drop causes the existing diffusion layer to stretch over astill larger sphere, much like the membrane of an expanding balloon.
This has the effectof making the layer thinner than it otherwise would be, so that the concentration gradientat the electrode surface is enhanced and larger currents flow. It turns out that the result isthe same as if the effective diffusion coefficient were (7/3)DQ; hence (7.1.4) requires multiplication by (7/3)1/2:Evaluating the constant in brackets, we havelA=(7.1.6)where i& is in amperes, DQ in cm2/s, CQ in mol/cm3, m in mg/s, and t in seconds. Alternatively, /d can be taken in /xA, and CQ in mM.Ilkovic was first to derive (7.1.6); hence this famous relation bears his name (12-16).His approach was much more exact than ours has been, as was that of MacGillavry andRideal (17), who provided an alternative derivation a few years afterward.
Actually theLingane-Loveridge approach is not independent of these more rigorous treatments, forthey arrived at the (7/3)1/2 stretching coefficient merely by comparing the bracketed factorin (7.1.4) with the factor 708 given by Ilkovic and by MacGillavry and Rideal. All threetreatments are based on linear diffusion.Figure 7.1.3 is an illustration of the current-time curves for several drops as predicted by the Ilkovic equation.
Immediately apparent is that the current is a monotonically increasing function of time, in direct contrast to the Cottrell decay found at astationary planar electrode. Thus, the effects of drop expansion (increasing area andstretching of the diffusion layer) more than counteract depletion of the electroactive substance near the electrode. Two important consequences of the increasing current-timefunction are that the current is greatest and its rate of change is lowest just at the end ofthe drop's life. As we will see, these aspects are helpful for applications of the DME insampled-current voltammetric experiments.Dc polarograms, as obtained in historic practice, are records of the current flow at aDME as the potential is scanned linearly with time, but sufficiently slowly (1-3 mV/s)that the potential remains essentially constant during the lifetime of each drop.
This constancy of potential is the basis for the descriptor "dc" in the name of the method. In moremodern practice, the potential is applied as a staircase function, such that there is a smallshift in potential (normally 1-10 mV) at the birth of each drop, but the potential otherwise7.1 Behavior at Polarographic Electrodes *4 265- Drop fallFigure 7.1.3 Currentgrowth during threesuccessive drops of a~~t DME.2-4 sremains constant as a drop grows through its lifetime.
The current oscillations arisingfrom the growth and fall of the individual drops are ordinarily quite apparent if the currentis recorded continuously. A typical case is shown in Figure 7.1.4. The most easily measured current is that which flows just before drop fall, and within the linear approximationit is given by(7.1.7)where £max is the lifetime of a drop (usually called the drop time and often symbolizedmerely as t).2'3(b) Transient Behavior at the SMDEIn most respects, the SMDE presents a much simpler situation than the classical DME,because the drop is not growing during most of its life.
In parallel with our discussionof diffusion-controlled currents at the DME, we confine our view now to the situationwhere the SMDE is held constantly at a potential in the mass-transfer controlled region. In the earliest stages of a drop's life (on the order of 50 ms), when the valve controlling mercury flow is open and the drop is growing, the system is convective. Masstransfer and current flow are not described simply. After the valve closes, and the dropstops growing, the current becomes controlled by the spherical diffusion of electroactive species.2Much of the older polarographic literature involves measurements of the average current, /d, flowing during adrop's lifetime.
This practice grew up when recording was typically carried out by damped galvanometers thatresponded to the average current. From the Ilkovic equation, one can readily find id to be six-sevenths of themaximum current:*d = 607 nDo ComtmaxThe first edition has more on average currents, including a discussion of Koutecky's treatment of the effects ofsphericity (pp.
150-152).3Sometimes polarographic current-potential curves show peaks, called polarographic maxima, whichcan greatly exceed the limiting currents due to diffusion. These excess currents arise at the DME fromconvection around the growing mercury drop. The convection apparently comes about (a) because differencesin current density at different points on the drop (e.g., at the shielded top versus the accessible bottom) causevariations in surface tension across the interface, or (b) because the inflow of mercury causes disturbances of thesurface. Surfactants, such as gelatin or Triton X-100, termed maximum suppressors, have been foundexperimentally to eliminate these maxima and are routinely added in small quantities to test solutions when themaxima themselves are not of primary interest. Convective maxima are rarely important in polarography at anSMDE.266Chapter 7.
Polarography and Pulse Voltammetry-0.8-1.0-1.2-1.4-1.6E, Vvs. SCEFigure 7.1.4 Polarogram for 1 mM CrO^" in deaerated 0.1 M NaOH, recorded at a DME. TheIlkovic equation describes current flow in the plateau region, at potentials more negative than about-1.3 V. The lower curve is the residual current observed in the absence of СЮ4". The recorderwas fast enough to follow the current oscillations through most of each drop's life, but not at themoment of drop fall, as one can see by the fact that the trace does not reach the zero-current linebefore starting a fresh rise with the new drop.This system is similar to that treated in Section 5.2.2, but the parallel is not exact because of the drop growth and convection in the early part of the experiment.
After thedrop becomes static, the current declines with time toward an asymptote. This behaviordiffers markedly from that at the DME, where the current rises with time because of continuous expansion of the drop. If the current at the SMDE is sampled electronically at atime т after the birth of the drop, then the current sample is given approximately from(5.2.18) as,(7.1.8)where r$ is the radius of the mercury droplet.
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