Polarography

Let the k-th reactant M^{z+} is the depolarizer (M^{z+} + z e^{−} = M), a_{k}^{∗} (c_{k}^{∗}) and a_{M}^{∗} (c_{M}^{∗}) are activities (free concentrations) of the depolarizer and metal, respectively, at the mercury drop electrode surface (i.e. at the interface between solution and amalgam), c_{k} is the free concentration of the depolarizer in the solution. At each point of the polarographic curve¶ I ∼ E

E = E^{0}(M^{z+}/M) + \frac{RT}{zF} ln \frac{a_{k}^{∗}}{a_{M}^{∗}} = E^{0}(M^{z+}/M) + \frac{RT}{zF} ln \frac{γ_{k}}{γ_{M}} + \frac{RT}{zF} ln \frac{c_{k}^{∗}}{c_{M}^{∗}} = E_{0} + \frac{RT}{zF} ln \frac{c_{k}^{∗}}{c_{M}^{∗}} ,

where E_{0} is constant, if both ionic strength of the solution and current density (i.e. also γ_{k} and γ_{M}) are constant.

If there is only the depolarizer† in the solution, then

I = κ_{k}(c_{k} − c_{k}^{∗}) = κ_{M}c_{M}^{∗} ,

c_{k}^{∗} = \frac{κ_{k}c_{k} − I}{κ_{k}} = \frac{I_{ℓ} − I}{κ_{k}} ,

c_{M}^{∗} = \frac{I}{κ_{M}} ,

E = E_{0} + \frac{RT}{zF} ln \frac{κ_{M}}{κ_{k}} + \frac{RT}{zF} ln \frac{I_{ℓ} − I}{I} ,

I = \frac{I_{ℓ}}{2} ⇒ E = E_{1/2} ,

E_{1/2} ≡ E_{1/2k} = E_{0} + \frac{RT}{zF} ln \frac{κ_{M}}{κ_{k}} .

If the solution contains (in addition to the depolarizer) also other reactant(s) (and their complexes), then

I = κ_{k} (c_{k} − c_{k}^{∗}) + ∑_{i=1}^{m} κ_{n+i} (c_{n+i} − c_{n+i}^{∗}) = κ_{M}c_{M}^{∗} ,

c_{n+i} = β_{i} ∏_{j=1}^{n} c_{j}^{q_{ij}} .

Let

c_{n+i}^{∗} = β_{i} ∏_{j=1}^{n} c_{j}^{∗}^{q_{ij}} = β_{i}c_{k}^{∗}^{q_{ik}} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ,

i.e. we assume that “fast exchange” is in progress in the solution (all complexes are kinetically labile so that chemical equilibrium is being attained at the mercury drop electrode surface during the electrolysis, i.e. diffusion rate is negligible compared with the rate of all chemical reactions) and that free concentrations of all components, which do not contain the depolarizer§, are \underline{the same} at the electrode surface as in the solution‡, and let

q_{ik} = 0 ∨ q_{ik} = 1 , i = 1, …, m ,

¶ dependence of the current I passing through the solution on the potential E of the mercury drop electrode

† probably as an aqua-complex

§ including inert ions

‡ this assumption can be accepted e.g. if analytical concentration of the ligand is much greater than analytical concentration of the metal

i.e. all complexes containing the depolarizer are mononuclear. Then^{1} (∀i : q_{ik} ≠ 0)

c_{n+i} = β_{i}c_{k} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ,

c_{n+i}^{∗} = β_{i}c_{k}^{∗} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ,

C_{k} = c_{k} + ∑_{{i=1 \atop q_{ik}≠0}}^{m} c_{n+i} = c_{k} (1 + ∑_{{i=1 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) ,

C_{k}^{∗} = c_{k}^{∗} + ∑_{{i=1 \atop q_{ik}≠0}}^{m} c_{n+i}^{∗} = c_{k}^{∗} (1 + ∑_{{i=1 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) = \frac{C_{k}c_{k}^{∗}}{c_{k}} ,

C_{k} and C_{k}^{∗} are analytical concentrations of the depolarizer in the solution and at the mercury drop electrode surface, respectively.

I = κ [(c_{k} − c_{k}^{∗}) + ∑_{{i=1 \atop q_{ik}≠0}}^{m} (c_{n+i} − c_{n+i}^{∗})] = κ(C_{k} − C_{k}^{∗}) ,

κ ≡ κ(c, n, V, c’, q, β) = [κ_{k} (c_{k} − c_{k}^{∗}) + ∑_{{i=1 \atop q_{ik}≠0}}^{m} κ_{n+i} (c_{n+i} − c_{n+i}^{∗})] /(C_{k} − C_{k}^{∗}) =

= (κ_{k} + ∑_{{i=1 \atop q_{ik}≠0}}^{m} κ_{n+i}β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) / (1 + ∑_{{i=1 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) = (κ_{k}c_{k} + ∑_{{i=1 \atop q_{ik}≠0}}^{m} κ_{n+i}c_{n+i}) /C_{k} ,

I = κ (C_{k} − \frac{C_{k}c_{k}^{∗}}{c_{k}} ) = κC_{k} − κ\frac{C_{k}c_{k}^{∗}}{c_{k}} ,

c_{k}^{∗} = \frac{κC_{k} − I}{κ} \frac{c_{k}}{C_{k}} = \frac{I_{ℓ} − I}{κ} \frac{c_{k}}{C_{k}} ,

c_{M}^{∗} = \frac{I}{κ_{M}} ,

E = E_{0} + \frac{RT}{zF} ln \frac{κ_{M}}{κ} + \frac{RT}{zF} ln \frac{c_{k}}{C_{k}} + \frac{RT}{zF} ln \frac{I_{ℓ} − I}{I} ,

I = \frac{I_{ℓ}}{2} ⇒ E = E_{1/2} ,

E_{1/2} = E_{0} + \frac{RT}{zF} ln \frac{κ_{M}}{κ} + \frac{RT}{zF} ln \frac{c_{k}}{C_{k}} .

From the difference of half-wave potentials of the complexed and “free” depolarizer (at the same ionic strength and current density) it is possible to express analytically¶ the ratio of the free and analytical concentration of the k-th reactant \frac{c_{k}}{C_{k}}

E_{1/2} − E_{1/2k} = \frac{RT}{zF} ln \frac{κ_{k}}{κ} + \frac{RT}{zF} ln \frac{c_{k}}{C_{k}} ,

¶ we must not forget any of the approximations made above

\frac{c_{k}}{C_{k}} = \frac{κ}{κ_{k}} e^{^{\frac{zF}{RT}} (E_{1/2}−E_{1/2k})} .

I would rather say it once more. The given simple dependence of the ratio of the free and analytical concentration of the k-th reactant \frac{c_{k}}{C_{k}} on the quantities κ, κ_{k}, E_{1/2}, E_{1/2k} was derived on three pivotal assumptions: all complexes are kinetically labile, free concentrations of all components, which do not contain the depolarizer, are the same at the electrode surface as in the solution and all complexes containing the depolarizer are mononuclear. Violation of the first assumption causes the occurrence of several polarographic waves instead of the “average” one. Violation of any of the rest two assumptions lead to much more complicated dependence than the derived one.

For illustration, let us consider that one of the complexes containing the depolarizer is binuclear, e.g.

q_{1k} = 2 , q_{ik} = 0 ∨ q_{ik} = 1 , i = 2, …, m .

Then (the first and second assumption keep valid)

c_{n+1} = β_{1}c_{k}^{2} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{1j}} ,

c_{n+1}^{∗} = β_{1}c_{k}^{∗}^{2} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{1j}} ,

C_{k} = c_{k} (1 + β_{1}c_{k} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} + ∑_{{i=2 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) ,

C_{k}^{∗} = c_{k}^{∗} (1 + β_{1}c_{k}^{∗} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} + ∑_{{i=2 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) = c_{k}^{∗} (β_{1}c_{k}^{∗} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} + \frac{C_{k}}{c_{k}} − β_{1}c_{k} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) ,

I = κ [C_{k} − c_{k}^{∗} (1 + β_{1}c_{k}^{∗} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} + ∑_{{i=2 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} )]

and c_{k}^{∗} is evidently the solution of the quadratic equation

κβ_{1}c_{k}^{∗}^{2} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} + κc_{k}^{∗} (1 + ∑_{{i=2 \atop q_{ik}≠0}}^{m} β_{i} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) + I − κC_{k} = 0 ,

κβ_{1}c_{k}^{∗}^{2} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} + κc_{k}^{∗} (\frac{C_{k}}{c_{k}} − β_{1}c_{k} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) + I − κC_{k} = 0 ,

c_{k}^{∗} = \{−κ (\frac{C_{k}}{c_{k}} − β_{1}c_{k} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ) + [κ^{2} (\frac{C_{k}}{c_{k}} − β_{1}c_{k} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} )^{2} − 4κβ_{1} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}}(I − κC_{k})]^{\frac{1}{2}} \} /2κβ_{1} ∏_{{j=1 \atop j≠k}}^{n} c_{j}^{q_{ij}} ,

c_{M}^{∗} = \frac{I}{κ_{M}} .

It would be fond to believe that after substitution for c_{k}^{∗} and c_{M}^{∗} in the relation for the potential of the mercury drop electrode E, expression of E_{1/2} and subtraction of E_{1/2k} we get something, that is in whatever way suggestive of the dependence of the ratio of the free and analytical concentration of the k-th reactant \frac{c_{k}}{C_{k}} on the quantities κ, κ_{k}, E_{1/2}, E_{1/2k} (in the case of all complexes mononuclear) derived before.

If we neglect the dependence of the diffusion current constant κ ≡ κ(c, n, V, c’, q, β) on the values of the chemical parameters (variation of diffusion current caused by complexation of the depolarizer is usually small), we can use calibration functions of types −5 and −6 to describe the polarographic data. However, the measured quantity¶ y = \frac{c_{k}}{C_{k}} (or y = \frac{a_{k}c_{k}}{C_{k}}) does not have constant variance, if the measured quantities E_{1/2} and E_{1/2k} have constant variances (variances of the measured quantities κ and κ_{k} can be neglected considering the slight influence of the values κ and κ_{k} on the value y). Such a variance must be corrected with non-trivial variance-covariance matrix V. The logarithmic transformation of the measured quantity y = − ln \frac{c_{k}}{C_{k}} = − ln c_{k} + ln C_{k} (or y = − ln a_{k} − ln c_{k} + ln C_{k}), i.e. the use of calibration functions of types −5ln and −6ln, is another (and probably better) solution:

1.

y = \frac{zF}{RT} (E_{1/2k} − E_{1/2}) + ln \frac{κ_{k}}{κ} †

2.

y = \frac{zF}{RT} (E_{1/2k} − E_{1/2}) − ln κ , ln a_{k} = ln κ_{k}

3.

y = −\frac{zF}{RT} E_{1/2} + ln \frac{κ_{k}}{κ} , ln a_{k} = \frac{zF}{RT} E_{1/2k}

4.

y = −\frac{zF}{RT} E_{1/2} − ln κ , ln a_{k} = \frac{zF}{RT} E_{1/2k} + ln κ_{k}

If we mark the parameter a_{k} unknown in any of the given cases, its value must be common to several measured values y§. Otherwise there would be more unknown parameters than points.

1. DeFord D. D., Hume D. N., J. Am. Chem. Soc. 73 (1951) 5321

ln a_{k} = \frac{zF}{RT} ΔE_{1/2k} + Δ ln κ_{k} ,

¶ random variable

† If a calibration function of type −5ln is used, then ln a_{k} = 0. If we mark the parameter a_{k} unknown (with its value common to all measured values y), it can be interpreted by the relation where ΔE_{1/2k} and Δ ln κ_{k} are “corrections” to the measured values E_{1/2k} and ln κ_{k}.

§ e.g. to all measured values y for the given depolarizer M^{z+}