B1 Closed Form Theoretical Models

We can generalize from the preceding discussion that a nonlinear regression model must contain a mathematical description of the signal resulting from the sample in addition to a description of the background, unless the background can be accurately subtracted from the raw data. Theoretical equations describing the instrumental response are excellent starting points for such regression models. When used with the appropriate background terms, computations are usually fast, and physically significant parameters are obtained. In this section, we present another example of such a model. The first such example was the exponential decay system presented in the previous chapter.

Response curves of current (i) in an electrolytic cell containing a planar working electrode can be obtained when the potential (E) is varied linearly with time. Under experimental conditions, where a steady state exists between the rate of electrolysis and the rate of mass transport to the electrode, sigmoid-shaped i-E curves (Figure 3.3) result. These conditions can be

Figure 3.3 Sigmoid-shaped steady state voltammogram with background.

Figure 3.3 Sigmoid-shaped steady state voltammogram with background.

achieved by stirring the solution, by rotating the electrode, or by using very tiny electrodes. These sigmoid curve shapes are therefore found in the electrochemical techniques of polarography, normal pulse voltammetry, rotating disk voltammetry, and slow scan voltammetry using disk "ultrami-croelectrodes" with radii less than about 20 /xm. The results of such experiments are called steady state voltammograms.

The simplest electrode process giving rise to a sigmoid-shaped steady state response is a single-electron transfer unaccompanied by chemical reactions. Thus, O the oxidant, and R the reductant, are stable on the experimental time scale.

Equation (3.4) represents a fast, reversible transfer of one electron to an electroactive species O from an electrode. The resulting voltammetric current-potential (i vs. E) curve centers around the formal potential E0' (Figure 3.3). The half-wave potential (EU2) is the value of E when the current is one half of the current at the plateau between about -70 to -90 mV. This plateau current is called the limiting or diffusion current (id) and is proportional to the concentration of O in the solution. E0' is the same as the half-wave potential if the diffusion coefficients of O and R are equal.

Assuming that the solution contains only O initially in the solution, the equation describing the steady state faradaic i vs. E response for eq. (3.4) is where d = cxp[(£ - E°')(F/RT)], R is the universal gas constant, F is Faraday's constant, and T is the temperature in Kelvins. The use of this model for nonlinear regression provides estimates of E0' and i\. A typical example for data without background shows excellent results (Table 3.5) when evaluated by our usual criteria.

Table 3.5 Results of the Model in Eq. (3.5) Fit onto Steady State Voltammetry Data

Parameter/statistic

Initial value

True value'

1.01 36

1.003 38.72 -0.2000 0.0028 Random

Deviation plot

" Data generated by using eq. (3.5) with absolute normally distributed noise at 0.5% of the maximum y.

For a well-resolved experimental steady state reduction curve with signal to noise ratio of 200/1 or better in the limiting current region, an appropriate model [1] for a regression analysis combines eq. (3.5) with a linearly varying background current:

The b!/(I + d) term on the right-hand side of eq. (3.6) describes the faradaic current from electron transfer in eq. (3.4) and also contains the parameters b2 = F/RT, and b3 = E0'. The value of b2 is known and fixed by the value of T. It is included as an adjustable parameter in the model as an additional check for correctness. Its value returned by the regression program can be checked against the theoretical value.

The b4x + b5 terms on the right-hand side of eq. (3.6) accounts for the background. The parameter b4 is the slope of the background current, in ju,A/V, for example, and b5 is the current intercept in ¡xA at zero volts (x = 0). A correction factor accounting for observed nonparallel plateau and baseline is sometimes included in the model but may not always be necessary [1], The quantities b4 and b5 may be used as regression parameters, or one or both of them can be kept fixed. In the latter case, their values must be measured from data obtained in a range of potentials before the rise of the sigmoid wave and then provided to the program.

As a second example of the importance of background considerations, we generated a steady state voltammogram by using eq. (3.6) and fit these data to eq. (3.5), which has no background terms. With a rather small linear background having a slope of 0.2 /¿A/V, errors of about 10% are found in parameters fr, and b2 (Table 3.6). The value of SD is about fivefold larger than that found for data without background (Table 3.5), and SD > ey, where ev here is the amount of absolute normally distributed noise added to the data, 0.005. The data points fall away from the regression line, and the deviation plot is clearly nonrandom (Figure 3.4).

If you have read the preceding pages of this book, you already know y = b,/( 1 + 0) + b4x + b5.

Table 3.6 Results of the Model in Eq. (3.5) without Background Term Fit onto Steady State Voltammetry Data with Background

Parameter/statistic

Initial value

True value"

Final value

Error

1.01

1.000

1.084

8.4%

b2

36.0

38.92

35.03

10.0%

b,

-0.21

-0.200

-0.198

1%

b4

Not used

-0.200

bs

Not used

0.005

SD

(ev = 0.005)

0.0107

Deviation plot

Nonrandom

" Data generated by using eq. (3.5) with absolute normally distributed noise at 0.5% of the maximum y.

how to correct the errors in this example. We can simply fit the data to eq. (3.6), which contains the linear background expression. The results of such an analysis (Table 3.7) indicate a good fit of the model to these data. The errors in the parameters and SD are four- to fivefold smaller than in

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