Example of Using Nernst Equation

Nernst Equation

The Nernst equation is an important relation which is used to determine reaction equilibrium constants and concentration potentials as well as to calculate the minimum energy required in electrodialysis as will be shown later.

From: Membrane Science and Technology , 2004

An introduction to electrochemical methods for the functional analysis of metalloproteins

Vincent Fourmond , Christophe Léger , in Practical Approaches to Biological Inorganic Chemistry (Second Edition), 2020

Derivation of Eq. (9.9)

We write the Nernst equation first for the alkaline couple Ox/Red, and then for both forms (protonated and deprotonated) of the redox couple:

(9.36a) $E=E_{\mathrm{alk}}^0+\frac{RT}{nF}\ln \left(\frac{[\mathrm{O}\mathrm{x}]}{[\mathrm{R}\mathrm{e}\mathrm{d}]}\right)$

(9.36b) $E=E^{0\prime }([H^{+}])+\frac{RT}{nF}\ln \left(\frac{[\mathrm{O}\mathrm{x}]+[\mathrm{O}\mathrm{x}\mathrm{H}]}{[\mathrm{R}\mathrm{e}\mathrm{d}]+[\mathrm{R}\mathrm{e}\mathrm{d}\mathrm{H}]}\right)$

We rewrite Eq. (9.36b) as follows:

(9.37a) $E=E^{0\prime }([H^{+}])+\frac{RT}{nF}\ln \frac{[\mathrm{O}\mathrm{x}]\left(1+\left([H^{+}]/K_{\mathrm{Ox}}\right)\right)}{[\mathrm{R}\mathrm{e}\mathrm{d}]\left(1+\left([H^{+}]/K_{\mathrm{Red}}\right)\right)}$

(9.37b) $E=E^{0\prime }([H^{+}])+\frac{RT}{nF}\ln \frac{[\mathrm{O}\mathrm{x}]}{[\mathrm{R}\mathrm{e}\mathrm{d}]}+\frac{RT}{nF}\ln \frac{1+\left([H^{+}]/K_{\mathrm{Ox}}\right)}{1+\left([H^{+}]/K_{\mathrm{Red}}\right)}$

Equating Eqs. (9.37b) and (9.36a) gives Eq. (9.9):

(9.38) $E^{0\prime }([H^{+}])=E_{\mathrm{alk}}^0+\frac{2.3RT}{nF}\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}\left(\frac{1+\left([H^{+}]/K_{\mathrm{Red}}\right)}{1+\left([H^{+}]/K_{\mathrm{Ox}}\right)}\right)$

Check that E ^0′ ([H⁺]) tends to $E_{\mathrm{alk}}^0$ when [H⁺] is small.

Using

(9.39) $E=E_{\mathrm{acid}}^0+\frac{RT}{nF}\ln \frac{[\mathrm{O}\mathrm{x}\mathrm{H}]}{[\mathrm{R}\mathrm{e}\mathrm{d}\mathrm{H}]}$

instead of Eq. (9.36a) gives

(9.40) $E^{0\prime }([{\mathrm{H}}^{+}])=E_{\mathrm{acid}}^0+\frac{2.3RT}{nF}\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}\left(\frac{1+\left(K_{\mathrm{Red}}/[H^{+}]\right)}{1+\left(K_{\mathrm{Ox}}/[H^{+}]\right)}\right)$

Check that $E^{0\prime }([H^{+}])$ tends to $E_{\mathrm{acid}}^0$ when [H⁺] is large.

The relation between $E_{\mathrm{alk}}^0$ and $E_{\mathrm{acid}}^0$ is simply obtained by equating Eqs. (9.37a) and (9.40):

(9.41) $E_{\mathrm{acid}}^0=E_{\mathrm{alk}}^0+\frac{RT}{nF}\ln \frac{K_{\mathrm{Ox}}}{K_{\mathrm{Red}}}$

Check that with pK _Ox<pK _Red, $E_{\mathrm{acid}}^0>E_{\mathrm{alk}}^0$ .

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Electrochemical methods

P. WESTBROEK , in Analytical Electrochemistry in Textiles, 2005

2.2.2 Fundamentals of potentiometry

Potentiometric measurements are based on the Nernst equation, which was developed from thermodynamic relationships and is therefore valid only under equilibrium (read thermodynamic) conditions. As mentioned above, the Nernst equation relates potential to the concentration of electroactive species. For electroanalytical purposes, it is most appropriate to consider the redox process that occurs at a single electrode, although two electrodes are always essential for an electrochemical cell. However, by considering each electrode individually, the two-electrode processes are easily combined to obtain the entire cell process. Half reactions of electrode processes should be written in a consistent manner. Here, they are always written as reduction processes, with the oxidised species, O, reduced by n electrons to give a reduced species, R:

[2.1] $\text{O}+n{\text{e}}^{-}\leftrightarrows \text{R}$

For such a half reaction the free energy is given by the relation:

[2.2] $\Delta G=\Delta G^0+\text{R}T\text{ln}\frac{a_{\text{R}}}{a_{\text{o}}}$

or:

[2.3] $-\Delta G=-\Delta G^0+\text{R}T\text{ln}\frac{a_{\text{o}}}{a_{\text{R}}}$

where –ΔG is the electromotive force indicating the tendency for the reaction to go to the right; R is the gas constant (8.317 J mol⁻¹ K⁻¹), T is the temperature (K), and the quantity ΔG ⁰ is the free energy of the half reaction when the activities of the reactant and product have values of unity and is directly proportional to the standard half-cell potential for the reaction as written. The electromotoric force, also called the free energy of this half reaction, is related to the electrode potential, E, by the expressions:

[2.4] $-\Delta G=r\text{F}E$

[2.5] $-\Delta G^0=r\text{F}{E}^0$

It also is a measure of the equilibrium constant for the half reaction assuming the activity of electrons is unity and under this condition the following equation is valid:

[2.6] $-\Delta G^0=\text{R}T\text{ln K}$

Beware of the fact that E ⁰ values are valid only under equilibrium conditions and for a _O=a _R=1. In practice, this condition is, in most cases, not fulfilled.

When Equations 2.2 and 2.4 are combined, the resulting equation relates the half-cell potential to the effective activity of the redox couple:

[2.7] $E=E^0-\frac{\text{R}T}{n\text{F}}\text{ln}\frac{a_{\text{R}}}{a_{\text{o}}}=E^0+\frac{\text{R}T}{n\text{F}}\text{ln}\frac{a_{\text{o}}}{a_{\text{R}}}$

a relation better known as the Nernst equation. This quantity is equal to the concentration of the species times a mean activity coefficient:

[2.8] $a_{{\text{M}}^{y+}}={\gamma }_{\pm {\text{C}}_{{\text{M}}^{y+}}}$

Although there is no straightforward and convenient method for evaluating activity coefficients for individual ions, the Debye–Hückel relationship permits an evaluation of the mean activity coefficient (γ_±), for ions at low concentrations (usually < 0.01 moll⁻¹):

[2.9] $\text{log}{\gamma }_{\pm }=-0.509z^2\frac{\sqrt{I}}{1+\sqrt{I}}$

where z is the charge on the ion and I is the ionic strength given by:

[2.10] $I=0.5\Sigma c_i{z}_i^2$

The reaction of an electrochemical cell always involves a combination of two redox half reactions such that one species oxidises a second species to give the respective redox products. Thus, the overall cell reaction can be expressed by a balanced chemical equation:

[2.11] $a{\text{O}}_1+{n\text{e}}^{-}\leftrightarrows c{R}_1{E}_1$

[2.12] $\underset{¯}{b{\text{R}}_2\leftrightarrows d{\text{O}}_2+n{\text{e}}^{-}{E}_2}$

[2.13] $a{\text{O}}_1+b{\text{R}}_2\leftrightarrows c{\text{R}}_1+d{\text{O}}_{2}K$

However, electrochemical cells are most conveniently considered as two individual half reactions, whereby each is written as a reduction in the form indicated by Equations 2.11 and 2.12. When this is done and values of the appropriate quantities are inserted, a potential can be calculated for each half cell of the electrode system. Then the reaction corresponding to the half cell with the more positive potential will be the positive terminal in a galvanic cell, and the electromotive force of that cell will be represented by the algebraic difference between the potential of the more-positive half cell and the potential of the less-positive half cell:

[2.14] $E_{\text{cell}}=E_1-E_2(E_{\text{cell}}\text{being positive})$

Insertion of the appropriate forms of Equation 2.7 gives an overall expression for the cell potential:

[2.15] $E_{\text{cell}}=E_1^0-E_2^0+\frac{\text{R}T}{n\text{F}}\text{ln}\frac{{\text{a}}_{{\text{O}}_1}^{\text{a}}{\text{a}}_{{\text{R}}_2}^{\text{b}}}{{\text{a}}_{{\text{O}}_1}^{\text{a}}{\text{a}}_{{\text{R}}_2}^{\text{b}}}$

The equilibrium constant for the chemical reaction expressed by Equation 2.15 is related to the difference of the standard half-cell potentials by the relation:

[2.16] $\text{ln}K=\frac{n\text{F}}{\text{R}T}(E_1^0-E_2^0)$

Equation 2.16 shows that potentiometry is a valuable method for the determination of equilibrium constants. However, it should be borne in mind that the system should be in equilibrium. Some other conditions, which are described below, also need to be fulfilled for use of potentiometry in any application. The basic measurement system must include an indicator electrode that is capable of monitoring the activity of the species of interest, and a reference electrode that gives a constant, known half-cell potential to which the measured indicator electrode potential can be referred. The voltage resulting from the combination of these two electrodes must be measured in a manner that minimises the amount of current drawn by the measuring system. This condition includes that the impedance of the measuring device should be much higher than that of the electrode.

For low-impedance electrode systems, a conventional potentiometer is satisfactory. However, electrochemical measurements with high-impedance electrode systems, and in particular the glass-membrane electrode, require the use of an exceedingly high-input impedance-measuring instrument (usually an electrometer amplifier with a current drain of less than 1 pA). Because of the logarithmic nature of the Nernst equation, the measuring instrumentation must have considerable sensitivity. Another important condition is that the potential response is directly dependent on the temperature of the measuring system. Thus, if the correct temperature is not used in the Nernst expression, large absolute errors can be introduced in the measurement of the activity for an electroactive species. In addition, temperature indirectly has an influence through the activity coefficients, ionic strength and dilution of the solution.

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Redox Cell Biology and Genetics Part B

Wei Feng , Isaac N. Pessah , in Methods in Enzymology, 2002

Experimental Design to Obtain Redox Potential

The glutathione redox potential is calculated by the Nernst equation from the GSH : GSSG ratios and the total glutathione concentration according to

$E_{\text{h}}=E_0+2.303\times RT/Fn\times \log \frac{\left[\text{GSSG}\right]}{{\left[{\text{GS}}^{-}\right]}^2}$

where E _h is the redox potential referred to the normal hydrogen electrode, V; E ₀ is the standard potential of glutathione, −   0.24 V; R is the gas constant, 8.31 J/deg · mol; T is the absolute temperature (K); n is the number of electrons transferred, n = 2 for SH-SS exchanges; F is the Faraday constant, 96,406 J/V; and [GSSG] and [GSH] are the molar concentrations of oxidized and reduced glutathione, respectively.

At the typical temperature for bilayer measurements (22   °),

$\begin{array}{lll}\hfill E_h& =& -0.24+29.28\times log\frac{[\text{GSSG}]}{{{[\text{GS}}^{-}]}^2}\\ \hfill [{\text{GS}}^{-}]& =& {([\text{GSSG}]/{10}^{(E_h+0.24)/29.28})}^{1/2}\\ \hfill [\text{GSSG}]& =& {{[\text{GS}}^{-}]}^2\times {10}^{(E_h+0.24)/29.28}\end{array}$

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POLAROGRAPHY | Overview

V. Cerdà , in Encyclopedia of Analytical Science (Second Edition), 2005

Current–Potential Curves for a Reversible Electrode Reaction

Along the rising part of the polarographic wave the Nernst equation is valid:

[9] $E=E^0+\frac{RT}{nF}\ln \left(\frac{\left[{\mathrm{Red}}_0\right]}{\left[{\mathrm{Ox}}_0\right]}\right)$

where E ⁰ is the standard redox potential, n is the number of exchanged electrons, and the subscript 0 denotes the concentrations at the electrode surface. Three different cases should be taken into account depending on the species that are present in the bulk of the solution:

1.

only the oxidized form is reduced;

2.

only the reduced form is oxidized (also in the case of amalgams); and

3.

both the oxidized and the reduced forms are present in the solution.

The resulting curves are denoted as cathodic, anodic, and cathodic–anodic curves, respectively. At the potentials where the surface concentration is nonzero the product must diffuse away from the electrode due to the resulting concentration gradient. The net current is expressed by the concentration of the oxidized species

[10] $i=-\kappa \left(\left[\mathrm{Ox}\right]-{\left[\mathrm{Ox}\right]}_0\right)$

or of the reduced species

[11] $i=-{\kappa }^{\prime }([\mathrm{Red}]-{\left[\mathrm{Red}\right]}_0)$

The parameters are denoted as

[12] $\kappa =0.627nF{D}_{\mathrm{ox}}^{1/2}{m}^{2/3}{t}^{1/6}$

[13] ${\kappa }^{\prime }=0.627nF{D}_{\mathrm{red}}^{1/2}{m}^{2/3}{t}^{1/6}$

and are often denoted as the Ilkovič constants. Simple arithmetic operations lead to the equation of a reversible polarographic wave:

[14] $E=E_0\mp \frac{RT}{nF}ln\sqrt{\frac{D_{\text{ox}}}{D_{\text{red}}}}\mp \frac{RT}{nF}ln\left(\frac{\overline{i}}{{\overline{i}}_{\text{d}}-i}\right)$

where minus and plus signs denote cathodic and anodic waves, respectively, and i _d is again the limiting diffusion current. For the cathodic–anodic wave

[15] $E=E_0-\frac{RT}{nF}ln\sqrt{\frac{D_{\text{ox}}}{D_{\text{red}}}}-\frac{RT}{nF}ln\left(\frac{\overline{i}-{\overline{i}}_{\text{da}}}{{\overline{i}}_{\text{dc}}-i}\right)$

where i _da and i _dc are anodic and cathodic limiting currents, respectively. The potential at which the current is i=1/2i _d is denoted as the half-wave potential E _1/2 and, assuming D _ox=D _red for a reversible system is equal to E ⁰:

[16] $E=E_{1/2}-\frac{RT}{nF}ln\left(\frac{\overline{i}-{\overline{i}}_{\text{da}}}{{\overline{i}}_{\text{dc}}-i}\right)\approx E_{1/2}-\frac{RT}{nF}ln\left(\frac{\overline{i}-{\overline{i}}_{\text{da}}}{{\overline{i}}_{\text{dc}}-i}\right)$

The properties of diffusion controlled currents are as follows:

1.

Current at a given potential is linearly proportional to the bulk concentration of the redox form undergoing the electrode process.

2.

Current is proportional to the square root of the diffusion coefficient, which may be thus evaluated.

3.

Mean limiting diffusion current is proportional to the square root of the mercury height h ^1/2.

4.

The shape of the i–E curve determines the dependence of ln{i/(i _d−i)} versus E, which should be linear with a slope of ±RT/nF. (This so-called log-plot analysis is often used for testing the reversibility of a reaction or for estimating the number of electrons (n) involved in a given electrode process.)

5.

The half-wave potential E _1/2 is independent of the concentration and for D _ox=D _red equals the standard redox potential E ⁰ of a corresponding redox couple.

Finite Charge-Transfer Rate in Electrode Processes (Irreversible or Slow Waves)

If the electron-transfer rate is not high enough it becomes the controlling factor and the observed electrode processes are called irreversible or slow. The Nernst equation is not obeyed in this situation.

Reversibility is not an absolute property in electrochemistry. It depends on the mutual ratio of the electrochemical time constant and the charge-transfer reaction rate. In direct current polarography the method's time constant is given by the drop time, the time available for the establishment of equilibrium, which can be varied only over a limited interval. Electrode reactions with charge-transfer control

[17] $\text{Ox}\stackrel{k_{+\text{e}}}{\underset{k_{-\text{e}}}{\leftrightharpoons }}\text{Red}$

are described by the rate constants k _+e and k _−e for reduction and oxidation, respectively. Both constants are potential dependent:

[18] $\begin{array}{cc}{k}_{-\text{e}}=& k^{0}exp\left(-\frac{(1-\alpha )nF}{RT}(E-E^0)\right)\\ k_{+\text{e}}=& k^{0}exp\left(-\frac{\alpha nF}{RT}(E-E^0)\right)\end{array}$

where k ⁰ is the standard heterogeneous rate constant corresponding to the rate at E ⁰ and is expressed in centimeters per second and α is the transfer coefficient. At potentials far from E ⁰ the charge-transfer rate becomes so high that the diffusion rate assumes control and a limiting diffusion current is observed.

The half-wave potential is as follows:

[19] $E_{1/2}=E^0+\frac{2.3RT}{\alpha nF}\log \left(0.886k^0\sqrt{\frac{t_1}{D}}\right)$

The following features of irreversible polarographic waves are relevant:

1.

The half-wave potential differs from E ⁰, depends on t ₁, and the anodic and cathodic waves of the same redox couple have different E _1/2 values.

2.

The irreversible polarographic wave is more protracted when compared with a reversible wave.

3.

The log-plot analysis does not yield the approximate n value because the slope also includes the α transfer coefficient.

4.

The upper limit of the heterogeneous charge-transfer rate, distinguishable from the diffusion control, is about $k^0\ll 0.02{\text{cm s}}^{-1}$ . Larger values of k ⁰ must be measured by faster techniques (AC or pulse polarography).

5.

Irreversible electrode reactions with the participation of complex ions or of organic compounds cannot be analyzed by means of expressions for reversible compounds: more complicated expressions need to be applied.

6.

The E _1/2 value may also depend on the concentration and on the type of the indifferent electrolyte.

Oxygen Behavior

Usually it is necessary to work in the absence of oxygen, since in the potential working zone of the mercury it is active, the following reduction reactions taking place:

$\begin{array}{l}{\text{O}}_2+2{\text{H}}^{+}+2\text{e}\to {\text{H}}_2{\text{O}}_2\\ {\text{H}}_2{\text{O}}_2+2{\text{H}}^{+}+2\text{e}\to 2{\text{H}}_2\text{O}\end{array}$

which give two reduction waves for potentials ∼−0.1 and −0.9   V (versus SCE), which may interfere with the polarographic waves of the analytes to be determined. The nonremoved oxygen traces (a nitrogen flow of sulfite is used) defines, together with the residual current, the detection limit of the technique. These reduction waves of oxygen are, on the other hand, the basis of the electrochemical Clark probe for dissolved oxygen in waters.

Polarographic Maxima

Certain anomalies related to polarographic curves are known by this name and consist in sharp intensity increases on the rising zone of the curve or on that in which the diffusion limit is attained. Depending on both their morphology and origin they are classified as first, second, and third type maxima. Besides, those corresponding to the first kind can be positive and negative (Figure 3).

Figure 3. Polarographic maxima: A, positive first kind maxima; B, negative first kind maxima; C, second kind maxima; D, third kind maxima.

The high increase of intensity regarding the different types of maxima is always due to the internal movements of the mercury drop, which in turn induces movements in the adjacent zone of the solution. Consequently, the flow rate of the electroactive species toward the electrode is higher than that which would take place by diffusion.

Polarographic maxima may be eliminated by addition to the solution of tensioactive substances. At present, Triton X-100 is the most widely used compound owing to the higher stability of its solutions with time, together with the fact that since it is a nonionic tensioactive agent, it is, thus, absorbed within a wide range of potentials and is unlikely to chemically interact with other compounds of the solution.

The use of the static mercury electrode enhances the elimination of polarographic maxima, since measurements are carried out with the mercury not flowing.

Quantification

Regardless of the degree of reversibility of the electrode process, both reversible and irreversible processes can be used for qualitative identification (using E _1/2), and the quantitative determination based on the Ilkovič equation. If the classical DC polarographic method is applied the lower limit of determination is in the region of 2×l0⁻⁵  mol   l⁻¹. This sensitivity is not sufficient in comparison with the more advanced analytical methods. The evaluation is carried out by calibration curves or by standard addition.

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Polarography

DONALD J. PIETRZYK , CLYDE W. FRANK , in Analytical Chemistry, 1979

Half-Wave Potential

In an electrochemically reversible reaction the wave is governed by the Nernst equation. Thus, the midpoint of the polarographic wave should be independent of the concentration. This point is defined as the half-wave potential ( E _1/2) and is characteristic of the species undergoing reduction (or oxidation) for a specified set of experimental conditions. As an approximation, E _1/2 values correspond to the standard reduction potential for the reaction being investigated.

The E _1/2 is related to the potential at the DME by the Heyrovsky–Ilkovic equation

(29-6) $E_{\text{DME}}=E_{1/2}+\frac{0.0592}{n}\log \frac{i_{\text{d}}-i}{i}$

where i is the current for the potential at the E _DME. This equation, which applies only to reversible systems, is useful for several reasons. For example, a plot of E _DME (choose potentials along the rising portion of the wave) vs log (i _d − i)/i gives a straight line whose slope is 0.059/n and intercept is the E _1/2 (when i = i _d/2, log (i _d − i)/i = 0, and E _DME = E _1/2). Thus, the half-wave potential and the number of electrons participating in the electrochemical reaction can be determined.

The half-wave potential can also be determined by a geometrical treatment of the polarogram. This is shown in Fig. 29-4.

Fig. 29-4. Geometric treatment of the polarographic wave.

The half-wave potentials for many inorganic ions have been determined under a variety of experimental conditions. Table 29-1 contains a partial listing of half-wave potentials. In general, if the half-wave potentials differ by at least 0.3 V a well-defined polarographic wave for each species is observed. This is illustrated in Fig. 29-5.

Table 29-1. Examples of E _1/2 Values for Inorganic Substances

Substance	Supporting electroyte	E 1,2 vs SCE (V)
Al(III)	0.1 F KCl	−1.70
As(III)	2 F Acetic acid, 2 F NH4C2H3O2	−0.92
Br−	0.1 F KNO3	+0.12
BrO3 −	0.1 F H2SO4	−0.41
Cd(II)	0.1 F KCl	−0.60
Cr(II)	0.1 F KCl	−0.34
Cu(II)	1 F NaOH	−0.41
Fe(II)	0.1 F KCl	−1.3
In (III)	0.1 F KCl	−0.561
Mn(II)	1.0 F KCl	−1.364
Ni(II)	0.1 F KCl	−1.1
Pb(II)	0.1 F KCl	−0.40
Sn(II)	0.1 F HCl	−0.83
ssU(VI)	0.1 F KCl	−0.185
Zn(II)	0.1 F KCl	−0.995

Fig. 29-5. Polarograms of (a) approximately 0.1 mM each of silver(I), thallium(I), cadmium(II), nickel(II), and zinc(II) listed in the order in which their waves appear, in 1 F ammonia–1 F ammonium chloride containing 0.002% Triton X-100; (b) the supporting electrolyte alone. (From L. Meites, "Polarographic Techniques," Interscience, New York, 1965.)

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Apoptosis

Naoufal Zamzami , ... Guido Kroemer , in Methods in Enzymology, 2000

Quantification of Mitochondrial Transmembrane Potential in Cells

Lipophilic cations accumulate in the mitochondrial matrix, driven by the electrochemical gradient following the Nernst equation, according to which every 61.5 mV increase in membrane potential (usually 120–170 mV) corresponds to a 10-fold increase in cation concentration in mitochondria. Therefore, the concentration of such cations is 2 to 3 logs higher in the mitochondrial matrix than in the cytosol. Several different cationic fluorochromes can be employed to measure mitochondrial transmembrane potentials. These markers include 3,3′-dihexyloxacarbocyanine iodide [DiOC ₆(3)] (fluorescence in green), ⁵ chloromethyl-X-rosamine (CMXRos) (fluorescence in red) ^{6 , 7} and 5,5′,6,6′-tetrachloro-1,1′,3,3′-tetraethylbenzim-idazolcarbocyanine iodide (JC-1) (fluorescence in red and green). ⁸ As compared with rhodamine 123 (Rh123), which we do not recommend for cytofluorometric analyses, ⁹ DiOC₆(3) offers the important advantage that it does not show major quenching effects. JC-1 incorporates into mitochondria, where it either forms monomers (fluorescence in green, 527 nm) or, at high transmembrane potentials, aggregates (fluorescence in red, 590 nm). ⁸ Thus, the quotient between green and red JC-1 fluorescence provides an estimate of ΔΨ_m that is (relatively) independent of mitochondrial mass.

Materials

Stock solutions of fluorochromes: DiOC₆(3) should be diluted to 40 μM in dimethyl sulfoxide (DMSO), CMXRos to 1 mM in DMSO, and JC-1 to 10 mM in DMSO. All three fluorochromes can be purchased from Molecular Probes (Eugene, OR) and should be stored, once diluted, at −20° in the dark

Working solutions: Dilute DiOC₆(3) to 400 nM [10 μ1 of stock solution plus 1 ml of phosphate-buffered saline (PBS)], CMXRos to 10 μM nM (10 μ1 of stock solution plus 1 ml of PBS or mitochondrial resuspension buffer), and JC-1 to 20 μM (2 μ1 of stock solution plus 1 ml of PBS). These solutions should be prepared fresh for each series of stainings

Carbonyl cyanide m-chlorophenylhydrazone (CCCP) diluted in ethanol (stock at 20 mM): A protonophore required for control purposes (ΔΨ_m disruption)

Cytofluorometer with appropriate filters

Staining Protocol

1.

Cells (5–10 × 10⁶ in 0.5 ml of PBS) should be kept on ice until staining. If necessary, cells can be labeled with specific antibodies conjugated to compatible fluorochromes [e.g., phycoerythrin for DiOC₆(3), fluorescein isothiocyanate for CMXRos] before determination of mitochondrial potential.

2.

For staining, add the following amounts of working solutions to 0.5 ml of cell suspension: 25 μl of DiOC₆(3) (final concentration, 20 nM), 25 μl of CMXRos (final concentration, 100 nM), or 25 μl of JC-1 (final concentration, 1 μM) and transfer the tubes to a water bath kept at 37°. After 15–20 min of incubation, return the cells to ice. Do not wash the cells. As a negative control, in each experiment aliquots of cells should be labeled in the presence of the protonophore CCCP (100 μM).

3.

Perform cytofluorometric analysis within 10 min, while gating the forward and sideward scatters on viable, normal-sized cells. When large series of tubes are to be analyzed (>10 tubes), the interval between labeling and cytofluorometric analysis should be kept constant. When using an Epics Profile cytofluorometer (Coulter, Hialeah, FL), DiOC₆(3) should be monitored in FL1 (Fig. 1), CMXRos in FL3 (excitation, 488 nm; emission, 599 nm), and JC-1 in FL1 versus FL3 (excitation, 488 nm; emission, 527 and 590 nm). The following compensations are recommended for JC-1: 10% of FL2 in FL1, and 21% of FL1 in FL2 (indicative values).

Fig. 1. Representative examples for ΔΨ_m measurements. (A) ΔΨ_m measurement in Jurkat cells that were either left untreated (control) or were treated for 3 hr with a CD95 cross-linking antibody (CH11; 1 μg/ml) for a period of 3 hr, followed by staining with DiOC₆(3) in the presence (solid line) or absence of CCCP (dashed line). (B) ΔΨ_m measurement in isolated mouse liver mitochondria treated with 5 mM atractyloside, followed by staining with CMXRos in the presence or absence of CCCP and cytofluorometric evaluation of the CMXRos-dependent fluorescence. Note that higher fluorescence values imply a higher ΔΨ_m. (C) ΔΨ_m measurement using Rh123. The same samples as in (A) were stained with Rh123 and evaluated in a fluorometer. Emission spectra for an excitation of 488 nm are shown. Note the inverse correlation between ΔΨ_m and Rh123-dependent fluorescence.

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ION-SELECTIVE ELECTRODES | Solid-State

V. Horváth , G. Horvai , in Encyclopedia of Analytical Science (Second Edition), 2005

The Nernst Equation

The relationship between an e.m.f. (E ) and primary ion activity is ideally given by the Nernst equation:

[1] $E=E°+\frac{RT}{F}\text{ln}{a}_{\mathrm{I}}$

where E° (standard e.m.f.) is a constant, R, T, and F have their usual meaning, and a _I is the activity of the primary ion I⁺ in the sample. This equation may be derived by observing that the e.m.f. adds up from several potential drops in the cell but sample composition should influence only the potential drop across the membrane and at the sample/reference electrode liquid junction (in cells with transference). Neglecting the changes of the latter we need to pay attention only to the potential drop between sample and internal reference solution, i.e., across the membrane. In the absence of interferences this potential difference can be calculated from thermodynamic considerations:

[2] $\Delta {\phi }_{1,2}=\frac{RT}{F}\text{ln}\frac{a_{\text{I},1}}{a_{\text{I},2}}=\frac{RT}{F}\text{ln}\frac{a_{\text{A},2}}{a_{\text{A},1}}$

where a _I,1 is the activity of the primary cation in the sample, a _I,2 is the same in the internal reference solution, and the a _As are the respective quantities for the primary anion. The composition of the internal reference solution is usually independent of the sample, so that eqn [1] or a similar equation for the primary ion is valid.

When writing eqn [2] it has been assumed that the internal reference solution is a soluble salt of I⁺ when this ion is measured, and a soluble salt of A⁻ when this is the analyte. In practice, there is no need to change the internal electrolyte according to the measured ion. If the internal solution is an I⁺ salt but the analyte is an A⁻ salt the Nernst equation is still valid albeit with a different E°. Figure 1 shows typical calibration lines with an unchanged internal solution. Apparent deviations from the Nernstian straight line are observed at low analyte concentration due to membrane solubility (see the section on detection limit).

Figure 1. Typical calibration lines of a silver halide, AgX electrode for Ag⁺ and X⁻.

The I⁺A⁻ membrane may be contacted on the inner side with a solid, electronically conducting material to form an all-solid-state electrode. This is generally regarded to be a sound practice only if the two contacting solid phases can equilibrate, i.e., there is at least one charged species (ion or electron) that can easily transgress the phase boundary in both directions. One example is metallic silver (or silver containing adhesive) in contact with a silver salt membrane. Thermodynamic reasoning yields in this case again the Nernst equation. Solid contacts that do not appear to fulfill the above requirement have occasionally been used successfully.

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FUEL CELLS – MOLTEN CARBONATE FUEL CELLS | Cells and Stacks

A.L. Dicks , in Encyclopedia of Electrochemical Power Sources, 2009

Operating Conditions

There are good reasons for wanting to operate the MCFC at elevated pressure. In line with the Nernst equation, the efficiency of an MCFC increases significantly as the operating pressure increases. The greatest gains are found by increasing the cell pressure from 1 to 5  bar. Net gains in efficiency in operating at over 10   bar are negated by the additional parasitic losses associated with pressurizing the gases. High operating pressures also have the disadvantage of promoting increased nickel dissolution at the MCFC cathode, as described in FUEL CELLS – MOLTEN CARBONATE FUEL CELLS: Overview. This can seriously curtail the stack lifetime and has prompted much research to develop alternative cathode materials to the conventional nickel oxide.

Another reason for operating the MCFC above atmospheric pressure is to enable the stack to be closely coupled to a gas turbine, making use of exhaust heat from the stack to generate additional electrical power. This has been tested to a limited extent by some developers. To operate at elevated pressures, an MCFC cell or stack has to be enclosed within a pressure vessel so that the differential pressures between anode and cathode compartments and between the inside and the outside of the stack can be controlled within a few millibars. The TWINSTACK® system devised by Ansaldo Fuel Cells spa (AFCo) (see below) is an example of a system that is amenable to operation at elevated pressures.

The operating temperature of the MCFC must also be controlled within fairly narrow limits. Too low a temperature and the molten carbonate will not have sufficient ionic conductivity and too high a temperature will result in loss of carbonate from the cell by evaporation. Typically, the inlet of the cell is maintained at around 600   °C and the maximum outlet temperature is ∼650   °C.

The matrix itself needs to have a high degree of mechanical stability to withstand temperature excursions that occur during normal operation as fuel and oxidant flows change in response to electrical demand. More importantly, if the cell temperature is allowed to cool below the melting point of the electrolyte, solidification within the pores of the matrix can cause cracking of the ceramic material especially along the edges of the cell, resulting in gas crossover with subsequent loss of cell performance and shortening of stack lifetime. The mechanical strength of the matrix is influenced by the method of its preparation. This means fewer layers of matrix and electrolyte are desirable, as are faster drying rates, especially if water-based systems are to be employed for fabrication on a large scale (organic binder is currently used in some of the tape cast materials). Alumina fibers may be added to the matrix to improve its mechanical stability; some other basic components may be added such as barium and/or strontium salts. Some research is being carried out to move away from lithium aluminate altogether to materials such as strontium titanate (SrTiO₃). Additives are also incorporated into the matrix material to reduce cathode dissolution, as described in FUEL CELLS – MOLTEN CARBONATE FUEL CELLS: Cathodes.

Electrolyte migration from cathode to anode, with the flow of the carbonate ions (known as ion-pumping), can give rise to loss of electrolyte at the extremities of a stack. This loss is usually through the gaskets that seal the manifolds to the cell stack. With internal manifolding of stacks, this issue is virtually eliminated.

Further specific information on the cell materials is given in FUEL CELLS – MOLTEN CARBONATE FUEL CELLS: Anodes and FUEL CELLS – MOLTEN CARBONATE FUEL CELLS: Cathodes and a chronology of materials development is given in Fuel Cell Handbook published by the US Department of Energy. This handbook also discusses the influence of operating conditions on the performance of the MCFC. In recent years, many researchers in North America, Europe, Japan, and Korea have been involved in improving the materials to (1) increase cell performance, (2) increase cell lifetime, and (3) reduce costs.

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Membrane Proteins—Production and Functional Characterization

Adriana Rycovska-Blume , ... Oliver Einsle , in Methods in Enzymology, 2015

4.1.4 Conclusions

The reconstitution of NirC into PLBs was straightforward and yielded macroscopic currents that reversed at the potentials predicted by Nernst's equation (Fig. 2A ). In line with the findings from the three-dimensional structure, where the N-terminal helices of each protomer were adhering tightly to the protein rather than showing significant flexibility (Lü, Schwarzer, et al., 2012), macroscopic currents for NirC did not show a pH-dependent gating behavior (Fig. 2B). Contrary to the complex dual role of FocA as an export channel and import transporter (Lü et al., 2013), NirC is thought to act as a bidirectional permease for monovalent anions at all times, so that a regulation by (external) pH is not required. However, currents measured at different pH values lead to a characteristic shift in reversal potential, indicating that protons are involved in NirC function. This was assembled into a functional model, wherein a proton resides within the membrane and helps the anionic cargo to cross a hydrophobic barrier through transient protonation (Lü et al., 2013). Also, this finding is relevant in light of the results from SSM electrophysiology and fluorescence dequenching (see Sections 4.2 and 4.3).

Figure 2. Electrophysiological characterization of StNirC in a PLB. Solubilized protein was reconstituted in a planar bilayer of 200   μm diameter. (A) Current–voltage diagram of macroscopic current measurements at two different nitrite gradients. The observed reversal potentials match the expectations from Nernst's equation. (B) Voltage-clamp studies of NirC at pH 7.9 (red) (gray in the print version) and pH 4.0 (black) show that StNirC is not gated in a pH-dependent manner. However, the curves show a shift in reversal potential. (C) The channel exhibits bursts of fast gating, as well as a slower gating process. In the transport model, the fast process could correspond to functional opening and closing of the constrictions in the substrate channel, while a longer closed phase is entered upon loss of a proton that is required to transiently neutralize the anionic cargo.

Using lower protein:lipid ratios during the formation of proteoliposomes, planar lipid membranes with only one or two NirC pentamers could be obtained that were suitable for single-channel recordings (Fig. 2C). Here, the observed conductance for single channels was in the low pA range, providing evidence that the protein acted as a fast ion channel. Gating of currents was observed on two levels, with bursts of fast gating, interrupted by slower gating events on a longer timescale (Fig. 2C). This was interpreted within the existing mechanistic model in that the fast gating reflects the transient opening events of the selectivity filter that must be short in order to avoid the passage of water or the uncoupling of existing ion gradients. In contrast, the slow gating was suggested to represent the loss of a proton to either side of the membrane that then needs to be replenished before anions can again pass the protein (Lü et al., 2013; Lü, Schwarzer, et al., 2012).

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New Approaches for Flavin Catalysis

Shelbi L. Christgen , ... Donald F. Becker , in Methods in Enzymology, 2019

2 Spectroelectrochemistry method

The reduction potential (E _m ) value for a redox center in a protein is determined using the Nernst equation (1), which relates the ratio of oxidized and reduced species with E _meas, which is

(1) $E_{\text{meas}}=E_{\mathrm{m}}+\left(2.\text{3RT}/n\mathrm{F}\right)log\left[\mathrm{ox}\right]/\left[\mathrm{red}\right]$

the redox potential measured at equilibrium during a potentiometric titration. In the Nernst equation E _m is the standard reduction potential of the redox center, R is the gas constant (8.314   J   K^{−   1}  mol^{−   1}), F is the Faraday constant (96,485   J   V^{−   1}  mol^{−   1}), and n is the number of electrons transferred. In a potentiometric titration, E _meas is recorded at different concentrations of the oxidized [ox] and reduced [red] species (Stankovich, 1980; Wilson, 1978). A plot of E _meas versus [ox]/[red], called a Nernst plot, is then used to determine E _m, which is equal to E _meas when [ox]/[red]   =   1. A Nernst plot also reveals the number of electrons transferred during the titration. A slope of 0.0295   V (25   °C) indicates a two-electron reduction process whereas a slope of 0.059   V (25   °C) reflects one-electron transfer. Observation of a flavin semiquinone during the potentiometric titration would be consistent with one-electron transfer.

The potentiometric method for determining the E _m of redox centers involves measuring the redox potential (E _meas) of the protein solution in the cell at different concentrations of [ox] and [red] species. The [ox]/[red] ratio is determined by monitoring a unique feature of the redox center, such as a change in the UV–visible spectrum of the protein-bound flavin. Examples of other techniques used for monitoring [ox] and [red] concentrations are electron paramagnetic resonance (Becker, Leartsakulpanich, Surerus, Ferry, & Ragsdale, 1998; Harder, Feinberg, & Ragsdale, 1989) and limited proteolysis (Zhu & Becker, 2003). Potentiometric cells can thus be modified accordingly for the type of measurement used for monitoring [ox] and [red] species. Regardless of the modifications made, a potentiometric cell houses three electrodes, which includes a working electrode (e.g., gold or platinum), an auxiliary electrode (Ag/AgCl), and a reference electrode (Ag/AgCl). A spectroelectrochemical cell with a three-electrode system was designed by Stankovich (1980) and has been the archetype for potentiometric measurements of flavoproteins (Pellet & Stankovich, 2002). Miniaturized spectroelectrochemical cells can now also be designed to reduce the amount of protein needed for an experiment.

During a potentiometric titration, the contents of the cell are reduced by bulk electrolysis using a potentiostat with an applied negative potential (e.g., −   600   mV). Reduction of the cell contents occurs at the working electrode while oxidation reactions occur at the auxiliary electrode. In theory, it would seem that the redox center of a protein could be reduced directly at the working electrode. Electron transfer between a protein and an electrode, however, can be very slow due to the buried nature of the redox center, which is typical of flavoproteins. Often in enzymes, the flavin cofactor is buried and less accessible to a solid electrode surface, thus making direct electron transfer generally difficult with flavoenzymes (Pellet & Stankovich, 2002). Additionally, proteins sometimes adsorb onto the electrode surface causing irreversible electron transfer behavior. To circumvent these problems, a redox mediator dye such as benzyl viologen or methyl viologen is added to the protein solution to facilitate reduction of the flavin. Benzyl viologen and methyl viologen are particularly useful because they have redox potentials that are more negative (−   350   mV and −   446   mV, respectively) than most protein-bound flavins and they have no visible spectrum in the oxidized state (Clark, 1960; Massey, 1991; Pellet & Stankovich, 2002). Additionally, benzyl viologen and methyl viologen remain positively charged upon one-electron reduction (methyl viologen⁺⁺  +   1e⁻  =   methyl viologen ⁺) thereby reducing kinetic barriers for reducing flavins in proteins. The other type of dye used in a potentiometric titration experiment is a reference or indicator dye. These dyes link the equilibrium of the protein-dye mixture with the working electrode when the potential of the cell is recorded. Once equilibrium is achieved, the ratio of the oxidized and reduced dye in solution reports on the ratio of oxidized and reduced flavin in the enzyme, thus the measured potential reflects the redox potential of the protein-bound flavin. The indicator dyes used for a potentiometric titration need to be chosen carefully and should have well defined optical and electrochemical properties and, as a general guideline, should have a redox potential that is ±   30   mV (n  =   2) from the E _m of the protein-bound flavin (Pellet & Stankovich, 2002).

Potentiometric experiments need to be performed under anaerobic conditions using a sealed anaerobic electrochemical cell or under a nitrogen atmosphere in an anaerobic chamber. To make a cell anaerobic, the cell containing the protein-dye mixture is flushed with nitrogen or argon and residual oxygen is removed by repeated cycles of vacuum and inert gas using a gas/vacuum line. Because oxygen reacts readily with redox mediators and indicators, oxygen must be excluded from potentiometric experiments and maintained at <   5   ppm. A potentiometric titration is performed by stepwise reduction of the cell contents by cycling between "On," when a negative potential is applied, and "Off," when no potential is applied (Fig. 2A). In the "Off" state, the potential at the working electrode is monitored as the protein-dye mixture is given time to equilibrate (~   30   min–2   h). The protein solution is considered to be at equilibrium when there is minimal drifting (drift is less than 1   mV in 5   min) of the measured redox potential (E _meas) and no further change in the UV–visible spectrum. Once the measured potential is stable, the potential between the reference (e.g., Ag/AgCl) and working electrode is recorded along with the UV–visible spectrum of the dye-protein mixture. The titration resumes upon applying a negative potential again to the system. The protein-dye solution is thus reduced in a step-wise fashion from the fully oxidized (quinone) to the fully reduced (hydroquinone) state with the redox potential and UV–visible spectrum being recorded at each step of reduction. The measured potentials are reported in reference to the normal hydrogen electrode by calibrating the reference electrode using a potassium ferricyanide/ferrocyanide standard solution (Stankovich, 1980). The concentration of oxidized and reduced species is determined at each point in the titration from the UV–visible spectra of the flavin. To accurately quantify the oxidized and reduced forms of the protein-bound flavin, the indicator dye spectra are subtracted from the spectra of the dye-protein mixture at each corresponding measured potential. Spectra of the dye alone must thus be acquired at similarly measured redox potentials as for the protein-dye mixture.

The advantage of using a potentiostat is the ability to manipulate the redox environment of the cell in the reductive and oxidative directions. The E _m value determined from a potentiometric titration ideally should not depend on whether equilibrium was approached from the reductive or oxidative direction. To check this, an oxidative potentiometric titration is performed in which the protein is first fully reduced and then a positive potential is applied in a stepwise fashion to oxidize the redox center. For an oxidative titration, ferrocyanide or ferrocene (+   422   mV) are good choices as a mediator dye for readily oxidizing flavin proteins upon applying a positive potential to the protein-dye mixture (Pellet & Stankovich, 2002; Vinod, Bellur, & Becker, 2002). In practice, it is best to apply an oxidative potential to the protein-dye mixture periodically throughout the reductive titration to ensure that the protein-dye mixture is truly in equilibrium.

In lieu of a potentiostat, step-wise reduction of a protein can also be performed with chemical reducing agents as outlined by Dutton (1978) using sodium dithionite or by photoreduction using 5-deaza-flavin (Klopprogge & Schmitz, 1999; McIver et al., 1998; Ravasio, Curti, & Vanoni, 2001). Similarly to that described above, mediator and indicator dyes are included in the cell solution and after each addition of reducing equivalent, the redox potential is recorded after the protein-dye mixture has fully equilibrated. To test the reversible behavior of a redox center the reduced protein can be titrated with a chemical oxidizing agent such as potassium ferricyanide.

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Example of Using Nernst Equation

Source: https://www.sciencedirect.com/topics/chemistry/nernst-equation

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