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theory:electrochemistry:electrochemistry

An Introduction to Electrochemistry

When connecting electrodes to liquids, tissue or biological materials, we create an interface between electronics and chemistry. In the electrodes and electronic equipment, charge is conducted by electrons, while it is conducted by ions in the biomaterials and liquids. The field of science about the interface of electronics and liquids is called electrochemistry. This page descibes the basic knowledge to understand the phenomena of electrochemical methods.

The simplest electrochemical systems is created by two or three metal electrodes in a liquid. We can supply an electrical current (or potential) to the electrodes, while measuring the potential difference (or current) that results from it. The basic physical and chemical principles describing the relation between potential and current normally originate from three locations in the system:

  • The bulk of the liquid with dissolved ions and dipoles will have a resistance because the charged particles need effort to move under the applied potential difference over the electrodes
  • The dissolved charged particles will be transported from and towards electrodes as a result of the applied potential. This transport will result in a potential difference over the electrode-liquid interface and is referred to as electrode polarization
  • Next, the ions in the solution may give or get an electron from the electrode, and so charge is exchanged between the electrode and the liquid. This exchange invloves a chemical transition and the crossing of a barrier, also resulting into an electrode-liquid potential difference

In more complex systems like the Ion Sensitive Transistor (ISFET), liquid conductors and glass electrodes, all three phenomane will take place.

In almost all cases, the aim of an electrochemical experiment is the determination of information about a solution. This information can be pH (acidity), the electrical conductivity, or a certain specific concentration. In electrochemical cells, however, we only have electrical potentials and currents as inputs and outputs. By picking the right input (frequency, amplitude, pulse shape, etc.), and interpreting the response appropriately, we can isolate one of the phenomena and learn something specific about the solution1). For example, in the kHz range, the phenomena at the electrodes are too slow to have a significant influence, hence only the resistance of the bulk solution determines the behaviour of the I-V relation. In that case, we are measuring the electrical conductivity of a solution without being affected by electrode polarization. When using a DC current, however, the third phenomenon will dominate the behaviour.

This tutorial starts with phenomena at the electrode surface. Next, it will be explained how these phenomena are using in an electrochemical half-cell with a reference electrode as an example. Next, methods that are using these half cells are discussed, with the ISFET in some more detail as an electronic component. After the elektrode interface phenomena, the transport in the bulk of a solution will be discussed which is the basis for electrical conductivity measurements. The page ends with some examples that illustrate the application of all theory.

Faradeic and non-faradeic processes

As described in the introduction, there are two processes by which a potential difference accross the electrode-liquid interface can be created. The difference between these two processes is in the flow of electrons:

  • When electrons are exchanged with the electrode, we speak of a Faradeic process. The energy step needed to cross the interface is the origin of the Faradeic electrode potential, and is described by the Nernst equation.
  • When there is just accumulation of charge at the electrode boundary, we speak of a non-Faradeic process. In that case, the behaviour is described by capacitor like models.

The Nernst equation

Consider an electrode in a liquid. The electrode can take or give electrons to a particle in that liquid. As a result, the state and charge of that particle will change. When electrons are taken from electrodes, we speak of reduction, when the liquid gives electrons to the electrode we speak of oxidation. An electrode at which reduction occurs is a cathode, an electrode at wich oxidation takes place is an anode. So with a cathodic current, electrons flow from the electrode to the liquid, with an anodic current in the opposite direction. See figure 1 for a graphical representation.

image21.jpg Fig. 1: Reduction and oxidation

For the particle in the liquid, we use the following terminology: with a reduction reaction, we call the original $Ox$, while the result is $Red$. This has to do with applications where there is a reduction reaction without the intervention of a cathode where electrons are pulled out of another couple. The other couple is then oxidized and we can call $Ox$ the “oxidator”. So:

  • An oxidator particle can oxidize another particle by which it is reduced itself, and
  • A reductor particle can reduce another particle by which it is oxidized itself.

The equilibrium between the oxidized phase and the reduced phase can be described by:

\begin{equation} Ox+n\cdot e^{-}\leftrightarrow Red. \label{eq:eqRedOx} \end{equation}

Here is $Ox$ the oxidized phase and Red the reduced phase. The number of electrons needed in the equilibrium is $n$. In a reversible system, the relation between electrode potential and the ratio in the equilibrium is given by the Nernst equation:

\begin{equation} E = E^{0}+\frac{RT}{nF}\ln \frac{a_{Ox}}{a_{Red}} \label{eq:Nernst} \end{equation}

with $R$ the molar gas constant, $T$ the temeprature, $F$ Faraday's constant (see table 3) and $a_{Ox}$ and $a_{Red}$ the activities of $Ox$ and $Red$. Faraday's constant is nothing else than the charge of one mole of electrons. In semiconductor models we see the factor of $kT/q$ which is about $25 mV (mJ/C)$, in electrochemistry this factor is replaced by $RT/F$, so

\begin{equation} \frac{kT}{q} = \frac{RT}{F} \approx 25 mV. \end{equation}

The activities are related to concentrations by means of the activity coefficient $g$. For practical (low) concentrations, the activity coefficienrts are equal to one, so that the Nernst equation can also be writtten in terms of ionic concentrations.

In a reversible system, the equilibrium can always return to the original position. In an irreversible system this can not, for example because there is a solid precipitation or because gas diffuses out of the solution.

The electrode potential with a current through the electrode

The Nernst equation describes the potential difference accross the metal-liquid interface, but can not be used to determine the complete current-potential relation. An electrode interface is called polarized when there is a potential difference accroess the interface as a result of the current. Deviation form that potential is called overpotential:

\begin{equation} \eta = E - E_{eq}. \end{equation}

We distinguish between polarization as a result of charge transfer (Nernstian) and polarization as a result of diffusion (concentration polarization).

A Tafelplot is the curve of the electrode current as a function of the overpotential, and contains information about the transfer coefficient and the exchange current $i_{0}$2).

The double layer capacitance

With Faradeic processes, as described before, charge is exchanged between the electrode and the liquid. Elements in the liquid will be modified chemically. A second process is the accumulation of charge at the electrode surface in the liquid, with an equivalent amount of charge in the electrode. This accumulation does not result into an immediate chemical modification of components. In Faradeic processes we have already seen this as concentration polarzation, but also in non-Faradeic processes this polarization will take place.

In the metal, the charge accumulation consists of electrons, while in the liquid the charge comprises ions. In case of charge in the form of ions, we should distinguish two types. First, ions can adsorb on the interface. We call this specific adsorption because the properties of the ions themselves determines the behaviour. These ions will be the closest to the interface. The layer comprising these ions is called the Helmholtz layer, Sternlayer, or Inner Helmholtz Plane (IHP). The second accumulation is with ions in the solution. Ions in the solution have a water shield consisting of the dipoles of the water molecules. Their chemical properties are invisible for the metal surface. Nevertheless, there can be an electrostatic interaction with the electrode. This results into a second layer: the diffusion layer. The diffusion layer will have a larger distance to the electrode surface as the Sternlayer.

The total accumulatiomn of charge results into a dubbellayer capacitance because charge is accumulated as a result of an electric field, just like in a normal capacitor. The value of this capacitor is normally in the range of $10$ tot $40 \mu F/cm^{2}$, but depends strongly on the concentrations and potential difference accross the interface. The simplification to an ideal capacitor can be used in most cases, but to explain some phenomena, we have to use frequency dependent components in this capacitor (the Warburg impedance).

The electrochemical cell

In practical configurations, we can never have a single metal-liquid interface. To make a closed electronic circuit, we need at least two interfaces. In the total electronic loop, there will be voltage drops due to the two interfaces plus voltage drops due to bulk electrolyte resistances and electrical resistances of the wires. The whole system is referred to as an electrochemical cell.

Principle

In a galvanic cell, electrochemical reactions occur autonomously, without the need for external energy. We know this as a battery. The anode, which has per definition the oxidation, is then at negative potential. The cathode is at positive potential. In an electrolytic cell, on the other hand, electrochemical reactions are activated by external energy (a potential difference). In that case, the oxidizing anode is positive and the cathode negative.

The cathode/anode definitions of figure 1 are still valid as illustrated in figure 2.

cel.jpg Fig. 2: Electrochemical cell (left hand) and galvanic cell (right hand)

The standard reference

The potential difference accross the two electrodes is determined by two electrochemical semi-cells. Because we want to investigate the electrochemical phenomena at a single electrode, we have to create a sitation where the other interface is “known” and “defined”. We can make the phenomena at an electrode constant and known by defining the concentrations in the effective vicinity of the electrode. The Nernst equation \eqref{eq:Nernst} then states that the potential difference accross the electrode will stay constant. Such a well defined electrode is called a reference electrode. A reference electrode must never carry a current, because then the local concentrations are altered and the potential is not guaranteed anymore.

In fact, “the” potential of an electrolyte is always unknown because there are always two interfaces in the loop. Even the potential as given by the Nernst equation is defined with respect to a chosen reference. This chosen reference is the Normal Hydrogen Electrode (NHE) which is defined as zero Volt.

Electrochemical methods

The potentiostat/galvanostat

As described in the previous section, it is not allowed to pull a current through a reference electrode, because it will no longer define a reliable reference potential. If we want to study an electrode on how it passes a current, we will need a third electrode that sources or drains the current.

The basic scheme for a potentiostat is sketched in figure 3a. In a potentiostat the applied potential difference is defined over the working electrode and the reference electrode, while the current is measured through the working electrode which flows to or from the couter electrode. When we are interested in the electrode potential as a result of an imposed current, we speak of a galvanostat as sketched in figure 3b. Potentiostat/galvalostats are for example available from Princeton Applied Research, where the PAR173 and PAR273 are quite commonly used.

image24.jpg Fig. 3: A potentiostat (a) and a galvanostat (b)

Cyclic voltammetry

cv2.jpg Fig. 4: A typical voltammogram of water: 1 mm^2 Pt, 100 mV/sec, 100 mM KNO3.

Amperometry

Chrono amperometry

Consider an involved reaction for reduction at a working electrode

\begin{equation} Ox+n\cdot e^{-}\rightarrow Red. \end{equation}

which is evoked after applying a negative potential step to the working electrode. In case of a potential step which is high enough to deplete the reacting species $Ox$ at the electrode surface completely during this step, the resulting current response for a planar electrode is given by the Cottrell equation3)

\begin{equation} i\left ( t \right ) = nFAC_{Ox} \sqrt{\frac{D_{Ox}}{\pi t}} \end{equation}

with $F$ the Faraday constant, $n$ the number of electrons transferred, $A$ the working electrode size, $C_{Ox}$ the bulk $Ox$ concentration and $D_{Ox}$ the diffusion coefficient of $Ox$. So, after applying a voltage step, the monitored current response is proportional to the $Ox$ concentration.

Sampled current voltammetry

Coulometry

Potentiometry

The Ion Sensitive Field Effect Transistor (ISFET)

The ISFET resembles a MOSFET, but with an ISFET the metal gate is replaced by an electrolyte. So the electrolyte is in direct contact with the gate oxide, the gate contact consists of a reference electrode in the electrolyte (figure 5).

image26.jpg Fig. 5: Schematic representation of an ISFET

The threshold potential of the ISFET is the result of the summation of a few physical and chemical potentials, of which one, the electrostatic potential $\psi_{0}$, appears to be pH dependent according to4):

\begin{equation} \frac{\partial \psi _{0}}{\partial \text {pH} _{B}} = -2.3 \frac{kT}{q} \alpha \end{equation}

with $pH_{B}$ the bulk pH and $\alpha$ a sensitivity parameter between $0$ and $1$. In practice this results into a pH sensitivity of the theshold voltage of $59 mV/pH$. This pH dependency is the result of the buffering of $H^{+}$ ions at the oxide interface.

Electronically seen, the ISFET is a MOSFET with a pH dependent threshold voltage $V_{T}$. For a p-type substrate we find the relations of table 1

Mode Condition Draincurrent Conduction in channnel
Saturated $V_{DS} > V_{GS} - V_{T}$ $I_{D} = \mu C_{Ox} \tfrac{1}{2} \frac{W}{L} \cdot \left ( V_{GS} - V_{T} \right )^{2}$ Holes
Unsaturated $V_{DS} < V_{GS} - V_{T}$ $I_{D} = \mu C_{Ox} \frac{W}{L} \cdot \left [ \left ( V_{GS} - V_{T} \right ) V_{DS}- \tfrac{1}{2}{}V_{DS}^{2} \right ] $ Electrons (inversion)
Tab. 1: Operational regions in a Field Effect Transistor

The modi of the transistor curves are also sketched in figure 6.

image28.jpg Fig. 6: Transistor curves

Because the $V_{T}$ is pH dependent, the drain current will also be pH dependent (regardless the saturation mode). However, it is convenient to have a linear dependency between $I_{D}$ and pH. Two methods are quite commonly used5).

Method 1: Constant drain-source voltage

Take the unsaturated mode and keep $V_{DS}$ constant. The draincurrent becomes:

\begin{equation} I_{D} = C_{1} + V_{T}C_{2} \end{equation}

Take care that $V_{DS}<V_{GS}-V_{T}$, which means we need a depletion type transistor ($V_{T}$ negative. Orion uses this method in their ISFET based pH meter “pHuture”.

Method 2: Constant drain-source voltage and drain current

Keep $I_{D}$ constant and $V_{DS}$ as well. The equations for the drain current show that this requires a $V_{GS}$ that is equal to $V_{T}$, regardless the saturation mode. Nevertheless, the unsaturated mode is preferred because then $I_{D}$ and $V_{DS}$ are lower and so is the dissipated power $I_{D} \cdot V_{DS}$.

A concvenient circuit for method 2 is given in figure 7. The operational amplifier keeps the potential of the drain equal to the potential defined by the voltage divider $R_{1} : R_{2}$. The drain-source potential difference is then equal to:

\begin{equation} V_{DS} = \frac{R_{2}}{R_{1} + R_{2}} V_{Ref} \end{equation}

which is normally kept at $0.5 Volt$. The drain current is in this circuit equal to the current through $R_{3}$ and so

\begin{equation} I_{D} = \frac{R_{1}}{ \left ( R_{1} + R_{2} \right ) R_{3}} V_{Ref}. \end{equation}

An $I_{D}$ of $100 \mu A$ is a quite commonly used value.

image32.jpg Fig. 7: An ISFET in constant VDS and constant ID mode

Ions in an electrolyte

Particles in a solution can move due to three different mechanisms. This division is made based on the driving force causing the motion:

  • Convection or migration: when the liquid moves or is stirred
  • Drift: a charged particle in an applied electric field
  • Diffusion: in a concentration gradient

The operational principle fo electrolyte conductivity sensors is based on the characteristic drift of charged particles, while potentiometric and amperometric methods are normally diffusion limited. Convection of the medium is normally an undesirable effect and will not be discussed here.

Diffusion

Diffusion is motion of particles due to a concentration gradient. The basis is the second law of Fick which states that the change of the amount of particles in a volume is proportional to the flux of particles through the boundary of the enclosed volume:

\begin{equation} \frac{\mathrm{d} c}{\mathrm{d} t} = D \nabla^{2}c \end{equation}

with $D$ the diffuscion coefficient $[m^{2}/sec]$ and $c$ the concentration $[mol/l]$.

The same differential equation is true for the diffusion of heat in solids6). This means we can re-use the generic solution. For an instantaneous pointsource at $t = 0 sec$ at location $r_{0}$, measured at location $r$, we find the solution:

\begin{equation} c_{pointsource} \left ( r, t \right ) = \frac{Q}{\left( 4 \pi D t \right )^{\frac{3}{2}} } \cdot e^{- \frac{\left( x-x_{0} \right )^{2}}{4Dt}} \end{equation}

with which, by integration over the total source surface, the response on the pointsource can be found:

\begin{equation} c_{source} \left ( r, t \right ) = \int_{source}^{ } c_{pointsource} \left ( r, t \right ) dr_{0} \end{equation}

By convolution with the source signal (actuator current or power of the heater) we can find the concentration in the total system:

\begin{equation} c \left ( r, t \right ) = \int_{0}^{t} i \left ( \tau \right ) \cdot c_{source} \left ( r, t-\tau \right ) d\tau \end{equation}

The solution contains

  • an exponential term - as a result of diffusion, and
  • some error-functions - as a result of the actuator geometry.

The used analogy between the thermal and chemical domain is also valid for electrical transmission lines. table 2 summarizes the mathematics and meaning of the state valiables for the three domains.

Heat Ions Electrical transmission lines
Flow through cross section A $\frac{\mathrm{d}Q}{\mathrm{d}t} = \lambda A \frac{\mathrm{d}T}{\mathrm{d}x}$ $I = \frac{dq}{dt} = D \cdot A \cdot F \frac{\mathrm{d}c}{\mathrm{d}x}$ $I = \frac{\mathrm{d}q}{\mathrm{d}t} = A \cdot \sigma E = \frac{1}{R} \frac{\mathrm{d}U}{\mathrm{d}x}$
Through slice with thickness dl $\left. \frac{\mathrm{d}Q}{\mathrm{d}t} \right| _{\mathrm{d}l} = \lambda A \frac{\mathrm{d}^{2}T}{\mathrm{d}x^{2}} \mathrm{d}l$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right| _{V} = D \cdot A \cdot F \frac{\mathrm{d}^{2}c}{\mathrm{d}x^{2}} \mathrm{d}l$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right | _{\mathrm{d}l}= \frac{1}{R} \frac{\mathrm{d}^{2}U}{\mathrm{d}x^{2}} \mathrm{d}l$
Storage in disc $\left. \frac{\mathrm{d}Q}{\mathrm{d}t} \right| _{\mathrm{d}l} = c_{m} \rho A \frac{\mathrm{d}T}{\mathrm{d}t} \mathrm{d}l$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right| _{V} = A \cdot F \frac{\mathrm{d}c}{\mathrm{d}t} dl$ $I = \left. \frac{\mathrm{d}q}{\mathrm{d}t} \right | _{\mathrm{d}l}= C \frac{\mathrm{d}U}{\mathrm{d}t} \mathrm{d}l$
Differential equation $\frac{\mathrm{d}T}{\mathrm{d}t} = \frac{\lambda}{c_{m}\rho}\nabla^{2}T$ $\frac{\mathrm{d}c}{\mathrm{d}t} = D \nabla^{2}c$ $\frac{\mathrm{d}U}{\mathrm{d}t} = \frac{1}{RC} \nabla^{2}U$
$Q$ : heat $q$: charge $q$ : charge
$\partial Q/\partial t$ : heat flow $I$: electric current $I$: electric current
$T$ : temperature $c$: concentration $U$ : potential
$Adl$ : slice $Adl$ : slice $Adl$ : slice
$\lambda$ : thermal conductivity $D$ : diffusion coefficient $R$ : resistance per $dl$
$\rho$ : density $F$: Faraday constant $C$ : capacity per $dl$
$c_{m}$ : specific heat $\sigma$ : conductivity
Tab. 2: Analogies in the thermal, chemical and electrical domain

Drift

Drift is the movement of charged particles in an electric field. In an electrolyte, the drift of ions determines the electric current. The contribution of an ion i to the total current is expressed by the transport number $t_{i}$ where the sum of all transport numbers of all charged particles in the electrolyte equals one:

\begin{equation} \sum_{i} t_{i} = 1 \end{equation}

The electrolyte conductivity κ can be expressed in terms of the molar conductivity $\Lambda$:

\begin{equation} \kappa=c \cdot \Lambda \end{equation}

with c the concentration of the conductive particles in the solution. In practice it is normally more insightful to express the molar conductivity in terms of all individual specific conductivities $\lambda_{i}$ of the ions involved:

\begin{equation} \Lambda = \sum_{i} c_{i} \left | z_{i} \right | \lambda_{i}=F\sum_{i} c_{i} \left | z_{i} \right | \mu_{i} \end{equation}

The mobility $\mu _{i}$ is related to the diffusion constant $D_{i}$ by:

\begin{equation} D_{i} = RT\mu_{i} \end{equation}

with $R$ the gas constant and $T$ the temperature.

Methods for electrolyte conductivity

The measurement of the the electrolyte conductivity can be done by placing the liquid between two capacitor plates. Due to the conductive elements in the electrolyte, there is a resistive element placed in the capacitor. The electrical equivalent circuit is given in figure 8.

image41.jpg Fig. 8: Electric equivalent circuit for a conductivity cell

We are interested in the value of the resistance $R_{Cel}$ which represents the total ion concentration and is related to the conductivity $\kappa_{sol}$ by means of the cell-constant $K_{Cell}$:

\begin{equation} R_{Cell} = \frac{K_{Cell}}{\kappa_{sol}} \end{equation}

which is a geometrical constant only. For a parallel plate electrode setup the cell constant can be calculated easily:

\begin{equation} K_{Cell} = \frac{d}{A} \end{equation}

with $d$ the distance bewteen the plates and $A$ the surface area size of the plates.

The capacitors in figure 8 represent the interfering effects. The perallel capacitance $C_{Cell}$ is the result of the direct AC-coupling between the electrodes an is here equal to

\begin{equation} C_{Cell} = \frac{\epsilon A}{d} = \frac{\epsilon}{K_{Cell}} \end{equation}

with $\epsilon$ the dielectric constant of the electrolyte. The series capacitors are interface polarization effects of the electrode-electrolyte surfaces that can be simplified to

\begin{equation} C_{interface} = A \cdot C_{dl} \end{equation}

with again $A$ the surface area size of the electrode and $C_{interface}$ the double-layer capacitance as described before.

The planar interdigitated finger electrode

As an example of a practical implementationn of a conductivity cell, a planar construction will be calculated. Alternative set-ups which are commercially used are round sticks with two or four metal rings around them. What is needed for modelling such a configuration is just the cell constant and the surface area, because then all modelleing components can be calculated. The surface area is normally not so difficult to calculate, but the cell constant may involve some mathematical complexity.

image47.jpg Fig. 9: Planar interdigitated finger structure

The cellconstant can be found by means of a conformal mapping transformation. A two dimensional evaluation is in most cases sufficient. Two dimensional conformal mapping is described in books and readers about complex function theory and in specific papers about calculating cell constants7). A conformal mapping transformation is transforming a space in such a way that the Maxwell equations remain valid. As a result when we can transform the electrical fieldlines conform Maxwell, we can transform the electrode geometry accordingly. In that case, we can transform a known configuration (for example the parallele plate capacitor) to the complex geometry of interest and transform the value of the cell constant as well.

For the interdigitated finger electrode we find

\begin{equation} K_{Cell-finger} = \frac{2}{\left ( N-1 \right ) L} \cdot \frac{K \left ( k_{1} \right )}{K \left ( k_{2} \right )} \end{equation}

with

\begin{equation} K \left ( k \right ) = \int_{0}^{1} \frac{1}{ \sqrt{\left ( 1-t^{2} \right ) \left ( 1-k^{2}t^{2} \right ) } } dt\\ k_{1}=\cos \left( \frac{\pi}{2} \cdot \frac{w}{s+w}\right )\\ k_{2}=\cos \left( \frac{\pi}{2} \cdot \frac{s}{s+w}\right )\\ \end{equation}

and $S$ the spacing between the fingers, $W$ the width of the fingers, $L$ the length of the fingers and $N$ the number of fingers. To choose these geometries ($S$, $W$, $L$ and $N$) we have to optimse in such a way that the effect of the paracitic capacitances in the frequency range of interest are as small as possible.

Impedance measurements

The simulation of figure 10 is easily made from the electric equivalent circuit of figure 8. The numerical values $S = 4 \mu m$, $W = 200 \mu m$, $N = 5$ and $L = 1 mm$ were taken.

figure7_3a.jpgfigure7_3b.jpg Fig. 10: Simulation of the spectrum of a conductivity cell - Bode plot and polar plot

In the first graph we can see the modulus as a function of frequency. The working region is easily recognized: it is the ferquency range in which the modulus is only dependent on concentration. The lower boundary of the sensitive region is about $10 kHz$ and is determined by the interface capacitances and the electrolyte conductivity. The upper boundary is the result of the cell-capacitance.

In the second graph, the data is plotted as a polar plot (imaginary part and real part as a parametric plot with the frequency as the independent parameter). In such a plot, the RC-couples are easily recognized as semi-circles. Polar plots can be used to identify electrode phenomena. This method is referred to as impedance spectrostopy. A real measurement picture conform figure 10 can be made with a gain-phase analyser, for example the HP4184.

When the user is only interested in a single frequency, a Phase-Locked-Loop (PLL) can be used. In its simples form, the PLL consists of an oscillator with a DC voltage dependent frequency, the Voltage-Controlled-Oscillator (VCO), and a phase detector, which can be a simple XOR port. The XOR port compares the phase of the signal of interest with the PLL internal phase, and adjust until these are equal.

image67.jpg Fig. 11: Principle of a Phase-Locked-Loop

The signals do not have to be sinewave-shaped, there are complete integrated circuits that have VCO, phase detector and filters in a single TTL component (LM565).

Applications and examples

Construction of the Pourbaix diagram for Hydrogen Peroxide H2O2

A convenient representation of acid-base and redox equilibria in a single plot is the Pourbaix diagram8). In such a plot, equilibria are plotted in a potiential versus pH graph. The construction of such a plot is included here because it explains the value of such diagrams and it explains the relation between acid-base chemistry and electrochemistry.

The system used in this example Pourbaix diagram is the one of hydrogen peroxide ($H_{2}O_{2}$) with all numerical values taken from the CRC Handbook of Chemistry and Physics9). It was used in my thesis about detergent activity monitoring10).

Acid-base equilibrium

Decomposition of hydrogen peroxide to the peroxide anion is given by:

\begin{equation} H_{2}O_{2} \Leftrightarrow HO_{2}^{-} + H^{+} \end{equation}

with an acid constant equal to:

\begin{equation} k_{a}= \frac{\left [ HO_{2}^{-} \right ] \cdot \left [ H^{+} \right ]}{\left [ H_{2}O_{2} \right ]} = 2.4 \cdot 10^{-12}. \end{equation}

With this, we can calculate that hydrogen peroxide is only dissociated for $2.3\%$ at $pH = 10$. The logarithm of this equation gives the first condition for the Pourbaix diagram:

\begin{equation} \log \frac{\left [ HO_{2}^{-} \right ]}{\left [ H_{2}O_{2} \right ]} = -11.63+pH. \label{eq:DecompositionHydrogenPeroxide} \end{equation}

Apparently, for $pH = 11.63$ the ratio $[HO_{2}^{-}]/[H_{2}O_{2}]$ equals exactly one.

Electrochemical equilibrium

For both $HO_{2}^{-}$ and $H_{2}O_{2}$ the electrochemical interaction with water and $H^{+}$ can be described:

\begin{equation} H_{2}O_{2} + 2H^{+} + 2e^{-} \Leftrightarrow 2H_{2}O \oplus E^{0}=1.776\\ HO_{2}^{-} + 3H^{+} + 2e^{-} \Leftrightarrow 2H_{2}O \oplus E^{0}=2.119\\ \end{equation}

which results into the following implementations of the Nernst equation:

\begin{equation} E^{0'}=1.776 + \frac{RT}{2F \log \left ( e \right )} \log \left [ H_{2}O_{2} \right ] - \frac{RT}{F \log \left ( e \right )} pH \\ E^{0'}=2.119 + \frac{RT}{2F \log \left ( e \right )} \log \left [ HO_{2}^{-} \right ] - \frac{3RT}{2F \log \left ( e \right )} pH. \\ \label{eq:H2O2_HO2_WithWater} \end{equation}

The electrochemical reactions of $HO_{2}^{-}$ and $H_{2}O_{2}$ with dissolved oxygen are

\begin{equation} O_{2} + 2H^{+} + 2e^{-} \Leftrightarrow H_{2}O_{2} \oplus E^{0}=0.695 \\ H^{+} + O_{2} + 2e^{-} \Leftrightarrow HO_{2}^{-} \oplus E^{0}=0.338 \\ \end{equation}

with the corresponding potential/concentration equations

\begin{equation} E^{0'}=0.695 + \frac{RT}{2F \log \left ( e \right )} \log \frac{pO_{2}}{\left [ H_{2}O_{2} \right ]} - \frac{RT}{F \log \left ( e \right )} pH \\ E^{0'}=0.338 + \frac{RT}{2F \log \left ( e \right )} \log \frac{pO_{2}}{\left [ HO_{2}^{-} \right ]} - \frac{3RT}{2F \log \left ( e \right )} pH. \\ \label{eq:H2O2_HO2_WithOxygen} \end{equation}

The Pourbaix diagram

When equations \eqref{eq:DecompositionHydrogenPeroxide}, \eqref{eq:H2O2_HO2_WithWater} and \eqref{eq:H2O2_HO2_WithOxygen} are drawn in a single potential against pH plot, with some example values for the $O_{2}$, $H_{2}O_{2}$ and $HO_{2}^{-}$ concentrations, we get the Pourbaix diagram11).

image62.jpg Fig. 12: Pourbaix diagram for H2O2 in water of 25 °C

From the diagram we can conclude that below the lines of equation \eqref{eq:H2O2_HO2_WithWater} ((2.4) in the graph), hydrogen peroxide acts as an oxidator which oxidizes to water. Above the lines of equation \eqref{eq:H2O2_HO2_WithOxygen} ((2.6) in the graph), hydrogen peroxide is a reductor which reduces to dissolved oxygen. These two identities have a common area in which hydrogen peroxide is said to be double instable.

\begin{equation} H_{2}O_{2} + 2H^{+} + 2e^{-} \Leftrightarrow H_{2}O \\ H_{2}O_{2} \Leftrightarrow O_{2} + 2H^{+} + 2e^{-} \\ \end{equation} so: \begin{equation} 2H_{2}O_{2} \Leftrightarrow 2H_{2}O + O_{2} . \end{equation}

It appears that at a metallic surface, with an electrode potential in the area of double instability, the decomposition of hydrogen peroxide into water and oxygen is being catalyzed.

Note that the Pourbaix diagram as based on the Nernst equations shows which reactions can take plate at electrode surfaces: whether this actually happens depends on other factors.

Coulometric acid-base titration

Amperometry without a standard reference electrode

Constants

$R$ Molar gas constant $8.31441$ $J \cdot mol^{-1} K^{-1}$
$F$ Faraday's constant $9.64846 \cdot 10^{4}$ $C \cdot mol^{-1}$
$k$ Boltzmann's constant $1.38066 \cdot 10^{-23}$ $J \cdot K^{-1}$
$q$ Single electron charge $1.60218 \cdot 10^{-19}$ $C$
Tab. 3: Constants in electrochemical methods
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