# Homogeneous redox reactions separated

### An electrochemical cell consists of two half-cells.

In the case of homogeneous redox reactions separated in space, such as different electrodes immersed in a solution, the electrodes are linked by a conductive material to allow charge to flow, as illustrated in \fref{fig:klata}.

The reactions in an electrochemical cells takes place in the interfacial region between the electrode and the solution, and are heterogeneous.

\begin{figure}

\centering

\includegraphics[width=0.30\linewidth]{illustrationer/Tue/electrocell.pdf}

\caption{Schematic drawing of an electrochemical cell, where the electrodes acts as a source (reduction) and a sink (oxidation) respectively. \cite{bib:atkins}}

\label{fig:klata}

\end{figure}

The cell potential is calculated from the electrode potentials:

\begin{align}

E_{cell} &= E_{red} - E_{ox}.

\label{eq:cellpot}

\end{align}

When the sum of free energy at the electrodes is negative, the system works as a battery and energy can be harvested.

In the other case external electrical energy can be applied, enabling electrode reactions (electrolysis).

For electron transfer reactions to take place, there must be a correspondence between the donor and acceptor electron orbitals.

In the electrode this is the Fermi energy level, and in the solution the orbital of the valence electron to be given or received.

For half-reactions, such as electrochemical cell reactions, the potentials are at equilibrium at the electrode surfaces described by the Nernst equation:

\begin{align}

E_{red} &= E^\Theta _{red} - \frac{RT}{nF} \sum v_i \ln a_i \label{eq:nernsta} \\

E_{red} &= E^\Theta _{red} + \frac{RT}{nF} \ln \frac{\prod a^{v_i} _{ox_i}}{\prod a^{v_i} _{red_i}} , \label{eq:nernsta2}

\end{align}

where $E _{red}$ is the potential of the half-reaction for the reduced spices, $E^\Theta _{red}$ is the standard half-reaction potential relative to the normal hydrogen electrode, $R$ is the universal gas constant, $T$ is the absolute temperature, $n$ denotes the number of electrons transferred in the cell reaction or half-reaction, $F$ is the Faraday constant, $v_i$ is the stoichiometric number of species $i$, which is positive for reduced species (products) and negative for oxidised (reagents), $a_i$ is the activity- $\gamma _i$ activity coefficient- and $c_i$ the concentration of species $i$. \\

Using the relation $a_i = \gamma _i c_i$ the Nernst equation can be written:

\begin{align}

E_{red} &= E^{\Theta } _{red} - \frac{RT}{nF} \sum v_i \ln c_i , \label{eq:nernstc}

\end{align}

where $E^{\Theta } _{red}$ is the formal potential for the reduced spices half-reaction, which is the potential that is measured in an electrochemical cell relative to a reference electrode.

Through the cell potential the maximum work the cell can supply can be calculated as the change in free Gibbs energy:

\begin{align}

\Delta G &= - n F E_{cell}.

\label{eq:gibbs}

\end{align}

There are two main scenarios to be considered in electrochemical cell reactions, galvanic- and electrolytic cell reactions.

In a galvanic cell the chemical energy is transformed into electron movement.

As the reaction is spontaneous there is an excess of electrons at the anode, giving it an overall negative charge and the cathode a positive.

In an electrolytic cell, electron movement is transformed into chemical energy.

The reaction is forced to occur, so there is a shortage of electrons at the anode, giving it an overall positive charge and a negative at the cathode.\\

\cite{bib:electrochemistry}

\subsection{The Electrode}\label{subsec:electrodeclassification}

The electrodes can be made of a variety of different materials, dependent on the type of reaction and solution being used.\\

\hspace*{0.5cm}\textbf{Type 1} are electrodes with a solution contain ions of the same material, e.g.\\

Cu | Cu$^{+2}$ or the hydrogen electrode H$_2$ | H$^+$.\\

\hspace*{0.5cm}\textbf{Type 2} are metallic electrodes in contact with a solution containing anions which can form soluble salt with the metal ions from the electrode. One example being the calomel electrode Hg | Hg$_2$Cl$_2$ | Cl$^-$.

Other examples are Ag | AgCl | Cl$^-$ and Hg | HgO | OH$^-$.\\

\hspace*{0.5cm}\textbf{Type 3} are redox electrodes which acts as a source or sink for electrons, and do not participate in the reaction.

The electrode materials are usually noble such as gold, platinum and mercury, but also glassy carbon, graphite and semiconductor oxides.

This is to avoid to much interference from bond formation between the electrode surface and species in solution, giving rise to non-faradaic currents.

Examples are Pt electrodes in a solution containing Fe(CN)$_6 ^{-3/-4}$.\\

\hspace*{0.5cm}Finally there are type 4, the electrodes that cannot be grouped with the already mentioned.\\

\cite{bib:electrochemistry}

\begin{figure}[ht]

%\strictpagechecktrue

\begin{minipage}[t]{0.5\linewidth}

\captionwidth{1\linewidth}

\centering

\includegraphics[width=0.70\linewidth]{llustrationer/Tue/hydrogen.jpg}

\caption{Hydrogen electrode. \cite{bib:electrochemistry}}

\label{fig:hydrogenelectrode}

\end{minipage}

%

\hspace{0.5cm}

\begin{minipage}[t]{0.5\linewidth}

\captionwidth{1\linewidth}

\centering

\includegraphics[width=0.70\linewidth]{llustrationer/Tue/calomel.jpg}

\caption{Calomel electrode. \cite{bib:electrochemistry}}

\label{fig:calomel}

\end{minipage}

\end{figure}

\subsection{The Reference Electrode}\label{subsec:refelectrode}

As potentials only can be defined as a difference to a chosen reference, reference electrodes are needed in electrochemical experiments.

A reference electrode should have a potential stable over time, at different temperatures and not be disturbed by the passage of a small current through the system.

The three main types of electrodes that can be used as reference electrodes:\\

\hspace*{0.5cm}\textbf{Type 1} e.g. the hydrogen electrode.

The reaction is H$^+$ + e$^-$ $\rightarrow$ $1/2$Hswe$_2$ and is normally catalysed by platinum.

The results are highly reproducible and the electrode used to define the standard electrode potential scale.\\

\hspace*{0.5cm}\textbf{Type 2} e.g. the calomel electrode with the reaction Hg | Hg$_2$Cl$_2$ | Cl$^-$ are good reference electrodes.\\

\hspace*{0.5cm}\textbf{Others} e.g. glass electrodes, Type 3 electrodes, etc.\\

\hspace*{0.5cm}A precaution which should be taken when reference electrodes are used together with complexes where there is a chance of the formation of poorly soluble salts.

The problem arises in many metallic hydroxides at higher concentrations which limits the usefulness of these.

The problem can sometimes be countered by using an alkaline solution, in other cases e.g. mercury oxide can be used.

Quasi-reference electrodes such as platinum or silver wires or pools of mercury can also be used. The advantage of these reference electrodes are low resistance, but the potential may vary.\\

\cite{bib:electrochemistry}

\subsection{Movement of Ions and Conductivity}

The two major causes for ion movement in the electrolyte solution arises due to diffusion and migration.

Diffusion is due to a concentration gradient, and is described by Fick's first law:

\begin{align}

J_i &= -D_i \frac{\delta c_i}{\delta x},

\label{eq:ficks1}

\end{align}

where $J _i$ is flux of spices $i$ with the concentration $c_i$ in direction $x$.

$\delta c_i / \delta x$ is the concentration gradient and $D_i$ is the diffusion coefficient i.e. the proportionality factor between the flux and the concentration gradient.

If an electric field of field strength $\textbf{E} = \frac{\delta \phi}{\delta x}$, where $\delta \phi / \delta x$ is the electric field gradient, the flux becomes:

\begin{align}

J_i &= -D_i \frac{\delta c_i}{\delta x} - z_i c_i \frac{F}{RT} \textbf{E},

\label{eq:ficks1ext}

\end{align}

where $z_i$ is the charge of spices $i$.

Three forces counteract the electric force:\\

\hspace*{0.5cm}Firstly, there is a friction force which is dependent on the size of the solvated ion.\\

\hspace*{0.5cm}Secondly, there is an asymmetric effect.

As ions move in the electrolytic solution with an applied electrical potential, the atmosphere around the ions are disorted, making it asymmetric.

As the ions move toward the oppositely charged electrodes they will drag ions with the opposite charge to themselves along.

In result the charge density behind the ions are greater than in the front, slowing the movement.

The higher the concentration of the solution, the greater the resistance is to the moving ions.\\

\hspace*{0.5cm}Thirdly, there is an electrophoretic effect.

As the ionic atmosphere is disorted due to the applied potential, solvent molecules will be dragged along because of the attractive forces between ions and solvent molecules.

As a consequence the movement of the ion toward the electrode with opposite charge is slowed, and there is a net flow of solvent molecules in the opposite direction.\\

These three forces leads to a maximum velocity for each ion in the solution.

By measuring these velocities, information about the structure of the solution can be obtained.

If the cation and anion velocity is different, it gives rise to a potential, known as the liquid junction potential.\\

The movement of the ions in a solution under an applied electrical field, results in conductivity.

The force $\textbf{F}$ applied to an ion under an electric field is:

\begin{align}

\textbf{F} &= z e \textbf{E},

\label{eq:forceE}

\end{align}

where $e$ is the charge of an electron.

In solution this force is counterbalanced by viscous forces, these can be described by Stoke's law:

\begin{align}

\textbf{F} &= 6 \pi \eta r \textbf{v},

\label{eq:stokes}

\end{align}

where $\eta$ is the viscosity of the solution, $r$ is the radius of the ion when solvated and $\textbf{v}$ is the velocity.

When neglecting all other slowing effects, combining \eref{eq:forceE} and \eref{eq:stokes} the maximum velocity becomes:

\begin{align}

\textbf{v} &= \frac{z e \textbf{E}}{6 \pi \eta r} \\

&= u E,

\label{eq:vmax}

\end{align}

where $u$ is the ion mobility and is the proportionality coefficient between velocity and the electric field strength.

By looking at the flux of charge, a relationship between mobility and charge can be obtained:

\begin{align}

j &= z e \textbf{v} c N_A,

\label{eq:flux1}

\end{align}

where $N_A$ is the Avogadro constant.

By defining $F = e N_A$ as the charge of one mole of electrons and use \eref{eq:vmax}:

\begin{align}

j &= z \textbf{v} c F \\

&= z c u F \textbf{E}.

\label{eq:flux2}

\end{align}

For two parallel electrodes the current $I$ can be related to the flux as:

\begin{align}

\textbf{I} &= jA \\

&= \kappa \frac{\Delta \phi A}{l} \\

&= \kappa \textbf{E} A,

\label{eq:current}

\end{align}

where $\kappa$ is the conductivity, $A$ the electrode area with a potential difference of $\Delta \phi$, separated by a distance $l$.

The strength of the applied electric field is $\textbf{E} = \Delta \phi / l$.

By combining \eref{eq:flux2} and \eref{eq:current} the conductivity for each ion can be calculated:

\begin{align}

\kappa _i &= z_i c_i u_i F,

\label{eq:eachion}

\end{align}

which for the whole solution gives:

\begin{align}

\kappa &= F \sum _i |z_i| c_i u_i.

\end{align}

The molar conductivity of a given ion, $\lambda _i$, is:

\begin{align}

\lambda _i &= \frac{\kappa _i}{c _i} \\

&= z_i u_i F

\label{eq:molarconductivityion}

\end{align}

and the molar conductivity of the electrolyte becomes:

\begin{align}

\Lambda &= \sum _i \lambda _i

&= \sum _i \frac{\kappa _i}{c _i}

\label{eq:molarconductivityelyte}

\end{align}

\cite{bib:electrochemistry}

\subsection{The Interfacial Region}

Se bog for reactions at the interfaces, den er bedre til dette emne.

\subsection{Mechanism of Electron Transfer}

In a system with products and reactants at equilibrium the reaction rates for formation and degradation of product are equal.

Such systems can be seen as limiting cases, and for half-reactions at an electrode the expression for the equilibrium is the Nernst equation.

A half-reaction with electron transfer without chemical transformation is:\\

Fe$^{3+}$(aq) + e$^-$(electrode) $\rightleftharpoons$ Fe$^{2+}$(aq)\\

\begin{figure}

\centering

\includegraphics[width=0.50\linewidth]{illustrationer/Tue/etransfer.jpg}

\caption{Schematic drawing of a electron transfer without chemical transformation at the electrode solution interface. \cite{bib:electrochemistry}}

\label{fig:etransferfe}

\end{figure}

The equilibirums are shown in \

Mechanism of electron transfer at the electrode (72) og lidt fra 75.

Relation between current and reation rate (exchange current).

\subsection{Mass Transport}

Mere om dette, skal evt. bare sættes ind ovenover.

\subsection{Planar Electrode Constant Current}

\subsection{Convection and Diffusion}

Mere om dette, skal evt. bare sættes ind ovenover.

Udvid ligning for I med formel side .

\subsection{Reversible Reaction}

Side 106 har en del om netop vores system.

\subsection{The Tafel Law}

Beskrivelse af hvorfor vi får en eksponentiel region.

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