\Section[
    \eng{Electrochemistry}
    \ger{Elektrochemie}
]{el}
\begin{formula}{chemical_potential}
    \desc{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{gibbs_free_energy}, \QtyRef{amount}}
    \desc[german]{Chemisches Potential}{der Spezies $i$\\Involvierte Energie, wenn sich die Teilchenzahl ändert}{}
    \quantity{\mu}{\joule\per\mol;\joule}{is}
    \eq{
        \mu_i \equiv \pdv{G}{n_i}_{n_j\neq n_i,p,T}
    }
\end{formula}

\begin{formula}{standard_chemical_potential}
    \desc{Standard chemical potential}{In equilibrium}{\QtyRef{chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, \QtyRef{activity}}
    \desc[german]{Standard chemisches Potential}{}{}
    \eq{\mu_i = \mu_i^\theta + RT \Ln{a_i}}
\end{formula}

\begin{formula}{chemical_equilibrium}
    \desc{Chemical equilibrium}{}{\QtyRef{chemical_potential}, \QtyRef{stoichiometric_coefficient}}
    \desc[german]{Chemisches Gleichgewicht}{}{}
    \eq{\sum_\text{\GT{products}} \nu_i \mu_i = \sum_\text{\GT{educts}} \nu_i \mu_i}
\end{formula}

\begin{formula}{activity}
    \desc{Activity}{relative activity}{\QtyRef{chemical_potential}, \QtyRef{standard_chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}}
    \desc[german]{Aktivität}{Relative Aktivität}{}
    \quantity{a}{}{s}
    \eq{a_i = \Exp{\frac{\mu_i-\mu_i^\theta}{RT}}}
\end{formula}

\begin{formula}{electrochemical_potential}
    \desc{Electrochemical potential}{Chemical potential with electrostatic contributions}{\QtyRef{chemical_potential}, $z$ valency (charge), \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvani Potential)}
    \desc[german]{Elektrochemisches Potential}{Chemisches Potential mit elektrostatischen Enegiebeiträgen}{\QtyRef{chemical_potential}, $z$ Ladungszahl, \ConstRef{faraday}, \QtyRef{electric_scalar_potential} (Galvanisches Potential)}
    \quantity{\muecp}{\joule\per\mol;\joule}{is}
    \eq{\muecp_i \equiv \mu_i + z_i F \phi}
\end{formula}


\Subsection[
    \eng{Electrochemical cell}
    \ger{Elektrochemische Zelle}
]{cell}
    \eng[galvanic]{galvanic}
    \ger[galvanic]{galvanisch}
    \eng[electrolytic]{electrolytic}
    \ger[electrolytic]{electrolytisch}

    \Eng[working_electrode]{Working electrode}
    \Eng[counter_electrode]{Counter electrode}
    \Eng[reference_electrode]{Reference electrode}
    \Ger[working_electrode]{Working electrode}
    \Ger[counter_electrode]{Gegenelektrode}
    \Ger[reference_electrode]{Referenzelektrode}
    \Eng[potentiostat]{Potentiostat}
    \Ger[potentiostat]{Potentiostat}

    \begin{formula}{schematic}
        \desc{Schematic}{}{}
        \desc[german]{Aufbau}{}{}
        \begin{tikzpicture}[scale=1.0,transform shape]
            \pgfmathsetmacro{\W}{6}
            \pgfmathsetmacro{\H}{3}
            \pgfmathsetmacro{\elW}{\W/20}

            \pgfmathsetmacro{\REx}{1/6*\W}
            \pgfmathsetmacro{\WEx}{3/6*\W}
            \pgfmathsetmacro{\CEx}{5/6*\W}

            \fill[bg-blue] (0,0) rectangle (\W, \H/2);
            \draw[ultra thick] (0,0) rectangle (\W,\H);
            % Electrodes
            \draw[thick, fill=bg-gray]   (\REx-\elW,\H/5) rectangle (\REx+\elW,\H);
            \draw[thick, fill=bg-purple] (\WEx-\elW,\H/5) rectangle (\WEx+\elW,\H);
            \draw[thick, fill=bg-yellow] (\CEx-\elW,\H/5) rectangle (\CEx+\elW,\H);
            \node at (\REx,3*\H/5) {R};
            \node at (\WEx,3*\H/5) {W};
            \node at (\CEx,3*\H/5) {C};

            % potentiostat
            \pgfmathsetmacro{\potH}{\H+0.5+2}
            \pgfmathsetmacro{\potM}{\H+0.5+1}
            \draw[thick] (0,\H+0.5) rectangle (\W,\potH);
            % Wires
            \draw (\REx,\H) -- (\REx,\potM) to[voltmeter,-o] (\WEx,\potM) to[european voltage source] (\WEx+1/6*\W,\potM) to[ammeter] (\CEx,\potM);
            \draw (\WEx,\H) -- (\WEx,\H+1.5);
            \draw (\CEx,\H) -- (\CEx,\H+1.5);

            % labels
            \node[anchor=west, align=left] at (\W+0.2, 1*\H/4) {{\color{bg-gray}  \blacksquare} \GT{reference_electrode}};
            \node[anchor=west, align=left] at (\W+0.2, 2*\H/4) {{\color{bg-purple}\blacksquare} \GT{working_electrode}};
            \node[anchor=west, align=left] at (\W+0.2, 3*\H/4) {{\color{bg-yellow}\blacksquare} \GT{counter_electrode}};
            \node[anchor=west, align=left] at (\W+0.2, \potM) {\GT{potentiostat}};
    \end{tikzpicture}
    \end{formula}

    \begin{formula}{cell}
        \desc{Electrochemical cell types}{}{}
        \desc[german]{Arten der Elektrochemische Zelle}{}{}
        \ttxt{
            \eng{
                \begin{itemize}
                    \item Electrolytic cell: Uses electrical energy to force a chemical reaction
                    \item Galvanic cell: Produces electrical energy through a chemical reaction
                \end{itemize}
            }
            \ger{
                \begin{itemize}
                    \item Elektrolytische Zelle: Nutzt elektrische Energie um eine Reaktion zu erzwingen
                    \item Galvanische Zelle: Produziert elektrische Energie durch eine chemische Reaktion 
                \end{itemize}
            }
        }
    \end{formula}

    % todo group together
    \begin{formula}{faradaic}
        \desc{Faradaic process}{}{}
        \desc[german]{Faradäischer Prozess}{}{}
        \ttxt{
            \eng{Charge transfers between the electrode bulk and the electrolyte.}
            \ger{Ladung wird zwischen Elektrode und dem Elektrolyten transferiert.}
        }
    \end{formula}
    \begin{formula}{non-faradaic}
        \desc{Non-Faradaic (capacitive) process}{}{}
        \desc[german]{Nicht-Faradäischer (kapazitiver) Prozess}{}{}
        \ttxt{
            \eng{Charge is stored at the electrode-electrolyte interface.}
            \ger{Ladung lagert sich am Elektrode-Elektrolyt Interface an.}
        }
    \end{formula}


    \begin{formula}{electrode_potential}
        \desc{Electrode potential}{}{}
        \desc[german]{Elektrodenpotential}{}{}
        \quantity{E}{\volt}{s}
    \end{formula}

    \begin{formula}{standard_cell_potential}
        \desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{gibbs_free_energy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
        \desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{gibbs_free_energy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
        \eq{E^\theta_\text{rev} = \frac{-\Delta_\txR G^\theta}{nF}}
    \end{formula}


    \begin{formula}{nernst_equation}
        \desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \secEqRef{standard_cell_potential}, \ConstRef{universal_gas}, \ConstRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
        \desc[german]{Nernst-Gleichung}{Elektrodenpotential für eine Halbzellenreaktion}{}
        \eq{E = E^\theta + \frac{RT}{zF} \Ln{\frac{ \left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{oxidized}}{\left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{reduced}}}}
    \end{formula}

    \begin{formula}{cell_efficiency}
        \desc{Thermodynamic cell efficiency}{}{$P$ \fqEqRef{ed:el:power}}
        \desc[german]{Thermodynamische Zelleffizienz}{}{}
        \eq{
            \eta_\text{cell} &= \frac{P_\text{obtained}}{P_\text{maximum}} = \frac{E_\text{cell}}{E_\text{cell,rev}} & & \text{\gt{galvanic}} \\
            \eta_\text{cell} &= \frac{P_\text{minimum}}{P_\text{applied}} = \frac{E_\text{cell,rev}}{E_\text{cell}} & & \text{\gt{electrolytic}}
        }
    \end{formula}


\Subsection[
    \eng{Ionic conduction in electrolytes}
    \ger{Ionische Leitung in Elektrolyten}
]{ion_cond}
    \eng[z]{charge number}
    \ger[z]{Ladungszahl}
    \eng[of_i]{of ion $i$}
    \ger[of_i]{des Ions $i$}
     
    \begin{formula}{diffusion}
        \desc{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_constant} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
        \desc[german]{Diffusion}{durch Konzentrationsgradienten}{}
        \eq{ i_\text{diff} = \sum_i -z_i F D_i \left(\odv{c_i}{x}\right) }
    \end{formula}

    \begin{formula}{migration}
        \desc{Migration}{caused by potential gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ potential gradient in the solution}
        \desc[german]{Migration}{durch Potentialgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, \QtyRef{mobility} \gt{of_i}, $\nabla\phi_\txs$ Potentialgradient in der Lösung}
        \eq{ i_\text{mig} = \sum_i -z_i^2 F^2 \, c_i \, \mu_i \, \nabla\Phi_\txs }
    \end{formula}

    \begin{formula}{convection}
        \desc{Convection}{caused by pressure gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} in flowing electrolyte}
        \desc[german]{Convection}{durch Druckgradienten}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{concentration} \gt{of_i}, $v_i^\text{flow}$ \qtyRef{velocity} \gt{of_i} im fliessenden Elektrolyt}
        \eq{ i_\text{conv} = \sum_i -z_i F \, c_i \, v_i^\text{flow} }
    \end{formula}

    \begin{formula}{ionic_mobility}
        \desc{Ionic  mobility}{}{$v_\pm$ steady state drift \qtyRef{velocity}, $\phi$ \qtyRef{electric_scalar_potential}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{charge}, \QtyRef{viscosity}, $r_\pm$ ion radius}
        \desc[german]{Ionische Moblilität}{}{}
        \quantity{u_\pm}{\cm^2\mol\per\joule\s}{}
        \eq{u_\pm = - \frac{v_\pm}{\nabla \phi \,z_\pm F} = \frac{e}{6\pi F \eta_\text{dyn} r_\pm}}
    \end{formula}

    \begin{formula}{stokes_friction}
        \desc{Stokes's law}{Frictional force exerted on spherical objects moving in a viscous fluid at low Reynolds numbers}{$r$ particle radius, \QtyRef{viscosity}, $v$ particle \qtyRef{velocity}}
        \desc[german]{Gesetz von Stokes}{Reibungskraft auf ein sphärisches Objekt in einer Flüssigkeit bei niedriger Reynolds-Zahl}{$r$ Teilchenradius, \QtyRef{viscosity}, $v$ Teilchengeschwindigkeit}
        \eq{F_\txR = 6\pi\,r \eta v}
    \end{formula}


    \begin{formula}{ionic_conductivity}
        \desc{Ionic conductivity}{}{\ConstRef{faraday}, $z_i$, $c_i$, $u_i$ charge number, \qtyRef{concentration} and \qtyRef{ionic_mobility} of the positive (+) and negative (-) ions}
        \desc[german]{Ionische Leitfähigkeit}{}{\ConstRef{faraday}, $z_i$, $c_i$, $u_i$ Ladungszahl, \qtyRef{concentration} und \qtyRef{ionic_mobility} der positiv (+) und negativ geladenen Ionen}
        \quantity{\kappa}{\per\ohm\cm=\siemens\per\cm}{}
        \eq{\kappa = F^2 \left(z_+^2 \, c_+  \, u_+ + z_-^2 \, c_- \, u_-\right)}
    \end{formula}

    \begin{formula}{ionic_resistance}
        \desc{Ohmic resistance of ionic current flow}{}{$L$ \qtyRef{length}, $A$ \qtyRef{area}, \QtyRef{ionic_conductivity}}
        \desc[german]{Ohmscher Widerstand für Ionen-Strom}{}{}
        \eq{R_\Omega = \frac{L}{A\,\kappa}}
    \end{formula}

    \begin{formula}{transference}
        \desc{Transference number}{Ion transport number \\Fraction of the current carried by positive / negative ions}{$i_{+/-}$ current through positive/negative charges}
        \desc[german]{Überführungszahl}{Anteil der positiv / negativ geladenen Ionen am Gesamtstrom}{$i_{+/-}$ Strom durch positive / negative Ladungn}
        \eq{t_{+/-} = \frac{i_{+/-}}{i_+ + i_-}}
    \end{formula}

    \eng[csalt]{electrolyte \qtyRef{concentration}}
    \eng[csalt]{\qtyRef{concentration} des Elektrolyts}
    \begin{formula}{molar_conductivity}
        \desc{Molar conductivity}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{csalt}}
        \desc[german]{Molare Leitfähigkeit}{}{\QtyRef{ionic_conductivity}, $c_\text{salt}$ \gt{salt}}
        \quantity{\Lambda_\txM}{\siemens\cm^2\per\mol=\ampere\cm^2\per\volt\mol}{ievs}
        \eq{\Lambda_\txM = \frac{\kappa}{c_\text{salt}}}
    \end{formula}

    \begin{formula}{kohlrausch_law}
        \desc{Kohlrausch's law}{For strong electrolytes}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} at infinite dilution, $c_\text{salt}$ \gt{csalt}, $K$ \GT{constant}}
        \desc[german]{}{}{$\Lambda_\txM^0$ \qtyRef{molar_conductivity} bei unendlicher Verdünnung, $\text{salt}$ \gt{csalt},$K$ \GT{constant}}
        \eq{\Lambda_\txM = \Lambda_\txM^0 - K \sqrt{c_\text{salt}}}
        \fig{img/ch_kohlrausch.pdf}
    \end{formula}
    
% Electrolyte conductivity
    \begin{formula}{molality}
        \desc{Molality}{Amount per mass}{\QtyRef{amount} of the solute, \QtyRef{mass} of the solvent}
        \desc[german]{Molalität}{Stoffmenge pro Masse}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{mass} des Lösungsmittels}
        \quantity{b}{\mol\per\kg}{}
        \eq{b = \frac{n}{m}}
    \end{formula}

    \begin{formula}{molarity}
        \desc{Molarity}{Amount per volume\\\qtyRef{concentration}}{\QtyRef{amount} of the solute, \QtyRef{volume} of the solvent}
        \desc[german]{Molarität}{Stoffmenge pro Volumen\\\qtyRef{concentration}}{\QtyRef{amount} des gelösten Stoffs, \QtyRef{volume} des Lösungsmittels}
        \quantity{c}{\mol\per\litre}{}
        \eq{c = \frac{n}{V}}
    \end{formula}

    \begin{formula}{ionic_strength}
        \desc{Ionic strength}{Measure of the electric field in a solution through solved ions}{\QtyRef{molality}, \QtyRef{molarity}, $z$ \qtyRef{charge_number}}
        \desc[german]{Ionenstärke}{Maß einer Lösung für die elektrische Feldstärke durch gelöste Ionen}{}
        \quantity{I}{\mol\per\kg;\mol\per\litre}{}
        \eq{I_b &= \frac{1}{2} \sum_i b_i z_i^2 \\ I_c &= \frac{1}{2} \sum_i c_i z_i^2}
    \end{formula}

    \begin{formula}{debye_screening_length}
        \desc{Debye screening length}{}{\ConstRef{avogadro}, \ConstRef{charge}, \QtyRef{ionic_strength}, \QtyRef{permittivity}, \ConstRef{boltzmann}, \QtyRef{temperature}}
        \desc[german]{Debye-Länge / Abschirmlänge}{}{}
        \eq{\lambda_\txD = \sqrt{\frac{\epsilon \kB T}{2\NA e^2 I_C}}}
    \end{formula}

    \begin{formula}{mean_ionic_activity}
        \desc{Mean ionic activity coefficient}{Accounts for decreased reactivity because ions must divest themselves of their ion cloud before reacting}{\QtyRef{activity}, $m_i$ \qtyRef{molality}, $m_0 = \SI{1}{\mol\per\kg}$}
        \desc[german]{Mittlerer ionischer Aktivitätskoeffizient}{Berücksichtigt dass Ionen sich erst von ihrer Ionenwolke lösen müssen, bevor sie reagieren können}{}
        \quantity{\gamma}{}{s}
        \eq{\gamma_\pm = \left(\gamma_+^{\nu_+} \, \gamma_-^{\nu_-}\right)^{\frac{1}{\nu_+ + \nu_-}}}
        \eq{a_i \equiv \gamma_i \frac{m_i}{m^0}}
    \end{formula}

    \begin{formula}{debye_hueckel_law}
        \desc{Debye-Hückel limiting law}{For an infinitely dilute solution}{\QtyRef{mean_ionic_activity}, $A$ solvent dependant constant, $z$ \qtyRef{charge_number}, \QtyRef{ionic_strength} in [\si{\mol\per\kg}]}
        \desc[german]{Debye-Hückel Gesetz}{Für eine unendlich verdünnte Lösung}{}
        \eq{\Ln{\gamma_{\pm}} = -A \abs{z_+ \, z_-} \sqrt{I_b}}
    \end{formula} 

\Subsection[
    \eng{Kinetics}
    \ger{Kinetik}
]{kin}
    \begin{formula}{transfer_coefficient}
        \desc{Transfer coefficient}{}{}
        \desc[german]{Durchtrittsfaktor}{Transferkoeffizient\\Anteil des Potentials der sich auf die freie Reaktionsenthalpie des anodischen Prozesses auswirkt}{}
        \eq{
            \alpha_\txA &= \alpha \\
            \alpha_\txC &= 1-\alpha
        }
    \end{formula}

    \begin{formula}{overpotential}
        \desc{Overpotential}{}{}
        \desc[german]{Überspannung}{}{}
        \ttxt{
            \eng{Potential deviation from the equilibrium cell potential}
            \ger{Abweichung der Spannung von der Zellspannung im Gleichgewicht}
        }
    \end{formula}
    \begin{formula}{activation_overpotential}
        \desc{Activation verpotential}{}{$E_\text{electrode}$ potential at which the reaction starts $E_\text{ref}$ thermodynamic potential of the reaction}
        \desc[german]{Aktivierungsüberspannung}{}{$E_\text{electrode}$ Potential bei der die Reaktion beginnt, $E_\text{ref}$ thermodynamisches Potential der Reaktion}
        \eq{\eta_\text{act} = E_\text{electrode} - E_\text{ref}}
    \end{formula}

    \Subsubsection[
        \eng{Mass transport}
        \ger{Massentransport}
    ]{mass}
        \begin{formula}{concentration_overpotential}
            \desc{Concentration overpotential}{Due to concentration gradient near the electrode, the ions need to \hyperref[f:ch:el:ion_cond:diffusion]{diffuse} to the electrode before reacting}{\ConstRef{universal_gas}, \QtyRef{temperature}, $\c_{0/\txS}$ ion concentration in the electrolyte / at the double layer, $z$ \qtyRef{charge_number}, \ConstRef{faraday}}
            \desc[german]{Konzentrationsüberspannung}{Durch einen Konzentrationsgradienten an der Elektrode müssen Ionen erst zur Elektrode \hyperref[f:ch:el:ion_cond:diffusion]{diffundieren}, bevor sie reagieren können}{}
            \eq{
                \eta_\text{conc,anodic} &= -\frac{RT}{\alpha \,zF} \ln \left(\frac{c_\text{red}^0}{c_\text{red}^\txS}\right) \\
                \eta_\text{conc,cathodic} &= -\frac{RT}{(1-\alpha) zF} \ln \left(\frac{c_\text{ox}^0}{c_\text{ox}^\txS}\right)
            }
        \end{formula}

        \begin{formula}{diffusion_overpotential}
            \desc{Diffusion overpotential}{Due to mass transport limitations}{$j_\infty$ \secEqRef{limiting_current}, $j_\text{meas}$ measured \qtyRef{current_density}, \ConstRef{universal_gas}, \QtyRef{temperature}, $n$ \qtyRef{charge_number}, \ConstRef{faraday}}
            \desc[german]{Diffusionsüberspannung}{Durch Limit des Massentransports}{}
            % \eq{\eta_\text{diff} = \frac{RT}{nF} \ln \left( \frac{\cfrac{c^\txs_\text{ox}}{c^0_\text{ox}}}{\cfrac{c^\txs_\text{red}}{c^0_\text{red}}} \right)}
            \eq{\eta_\text{diff} = \frac{RT}{nF} \Ln{\frac{j_\infty}{j_\infty - j_\text{meas}}}}
        \end{formula}

        % 1: ion radius
		% 2: ion color
		% 3: ion label
		% 4: N solvents, leave empty for none
		% 5: solvent radius 6: solvent color
		% 7:position
        \newcommand{\drawIon}[7]{%
            \fill[#2] (#7) circle[radius=#1] node[fg0] {#3};
            \ifstrempty{#4}{}{
                \foreach \j in {1,...,#4} {
                    \pgfmathsetmacro{\angle}{\j * 360/#4}
                    \fill[#6] (#7) ++(\angle:#1 + #5) circle[radius=#5];
                }
            }
        }
        \newcommand{\drawAnion}[1]{\drawIon{\Ranion}{bg-blue}{-}{}{}{}{#1}}
        \newcommand{\drawCation}[1]{\drawIon{\Rcation}{bg-red}{+}{}{}{}{#1}}
        \newcommand{\drawAnionSolved}[1]{\drawIon{\Ranion}{bg-blue}{-}{6}{\Rsolvent}{fg-blue!50!bg2}{#1}}

        \Eng[electrode]{Electrode}
        \Ger[electrode]{Elektrode}
        \Eng[nernst_layer]{Nernst layer}
        \Ger[nernst_layer]{Nernst-Schicht}
        \Eng[electrolyte]{Electrolyte}
        \Ger[electrolyte]{Elektrolyt}
        \Eng[c_surface]{surface \qtyRef{concentration}}
        \Eng[c_bulk]{bulk \qtyRef{concentration}}
        \Ger[c_surface]{Oberflächen-\qtyRef{concentration}}
        \Ger[c_bulk]{Bulk-\qtyRef{concentration}}

        \begin{formula}{diffusion_layer}
            \desc{Cell layers}{}{IHP/OHP inner/outer Helmholtz-plane, $c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}}
            \desc[german]{Zellschichten}{}{IHP/OHP innere/äußere Helmholtzschicht, $c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}}
            \begin{tikzpicture}
                \tikzset{
                    label/.style={color=fg1,anchor=center,rotate=90},
                }
                \pgfmathsetmacro{\Ranion}{0.15}
                \pgfmathsetmacro{\Rcation}{0.2}
                \pgfmathsetmacro{\Rsolvent}{0.06}

                \pgfmathsetmacro{\tkW}{8}               % Total width
                \pgfmathsetmacro{\tkH}{4}               % Total height
                \pgfmathsetmacro{\edW}{1}               % electrode width              
                \pgfmathsetmacro{\hhW}{4*\Rsolvent+2*\Ranion} % helmholtz width
                \pgfmathsetmacro{\ndW}{3}               % nernst diffusion with
                \pgfmathsetmacro{\eyW}{\tkW-\edW-\hhW-\ndW}     % electrolyte width
                \pgfmathsetmacro{\edX}{0}               % electrode width              
                \pgfmathsetmacro{\hhX}{\edW}            % helmholtz width
                \pgfmathsetmacro{\ndX}{\edW+\hhW}       % nernst diffusion with
                \pgfmathsetmacro{\eyX}{\tkW-\eyW}       % electrolyte width

                \path[fill=bg-orange]        (\edX,0) rectangle (\edX+\edW,\tkH);
                \path[fill=bg-green!90!bg0]  (\hhX,0) rectangle (\hhX+\hhW,\tkH);
                \path[fill=bg-green!60!bg0]  (\ndX,0) rectangle (\ndX+\ndW,\tkH);
                \path[fill=bg-green!20!bg0]  (\eyX,0) rectangle (\eyX+\eyW,\tkH);
                \draw (\ndX,2) -- (\eyX,3) -- (\tkW,3);
                % axes
                \draw[->] (0,0) -- (\tkW+0.2,0) node[anchor=north] {$x$};
                \draw[->] (0,0) -- (0,\tkH+0.2) node[anchor=east]  {$c$};
                \tkYTick{2}{$c^\txS$};
                \tkYTick{3}{$c^0$};
                \foreach \i in {1,...,5} {
                    \drawCation{\edW-\Ranion, \tkH * \i /6}
                    \drawAnionSolved{\edW+\Rcation+2*\Rsolvent, \tkH * \i /6}
                }
                \drawCation{\ndX+\ndW * 0.1, \tkH * 2/10}
                \drawCation{\ndX+\ndW * 0.15, \tkH * 4/10}
                \drawCation{\ndX+\ndW * 0.1, \tkH * 6/10}
                \drawCation{\ndX+\ndW * 0.1, \tkH * 9/10}
                \drawAnion{ \ndX+\ndW * 0.2, \tkH * 7/10}
                \drawAnion{ \ndX+\ndW * 0.4, \tkH * 4/10}
                \drawAnion{ \ndX+\ndW * 0.3, \tkH * 3/10}
                \drawAnion{ \ndX+\ndW * 0.5, \tkH * 6/10}
                \drawAnion{ \ndX+\ndW * 0.8, \tkH * 3/10}
                \drawAnion{ \ndX+\ndW * 0.3, \tkH * 1/10}
                \drawAnion{ \ndX+\ndW * 0.4, \tkH * 9/10}
                \drawAnion{ \ndX+\ndW * 0.6, \tkH * 7/10}
                \drawCation{\ndX+\ndW * 0.3, \tkH * 3/10}
                \drawCation{\ndX+\ndW * 0.6, \tkH * 8/10}
                \draw (\edX+\Rcation, 0) -- ++(0, -0.5)         node[anchor=west,rotate=-45] {\GT{electrode}};
                \draw (\edX+\edW-\Rcation, 0) -- ++(0, -0.5)    node[anchor=west,rotate=-45] {{IHP}};
                \draw (\hhX+\hhW/2, 0) -- ++(0, -0.5)           node[anchor=west,rotate=-45] {{OHP}};
                \draw (\ndX+\ndW/2, 0) -- ++(0, -0.5)           node[anchor=west,rotate=-45] {\GT{nernst_layer}};
                \draw (\eyX+\eyW/2, 0) -- ++(0, -0.5)           node[anchor=west,rotate=-45] {\GT{electrolyte}};
                % TODO
            \end{tikzpicture}
        \end{formula}


        \begin{formula}{diffusion_layer_thickness}
            \desc{Nerst Diffusion layer thickness}{}{$c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}}
            \desc[german]{Dicke der Nernstschen Diffusionsschicht}{}{}
            \eq{\delta_\txN = \frac{c^0 - c^\txS}{\odv{c}{x}_{x=0}}}
        \end{formula}
        
        \begin{formula}{limiting_current}
            \desc{(Limiting) current density}{}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \secEqRef{diffusion_layer_thickness}}
            % \desc[german]{Limitierender Strom}{}{}
            \eq{
                \abs{j} &= nFD \frac{c^0-c^\txS}{\delta_\text{diff}}
                \shortintertext{\GT{for} $c^\txS \to 0$}
                \abs{j_\infty} &= nFD \frac{c^0}{\delta_\text{diff}}
            }
        \end{formula}

        \begin{formula}{relation?}
            \desc{Current - concentration relation}{}{$c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}, $j$ \secEqRef{limiting_current}}
            \desc[german]{Strom - Konzentrationsbeziehung}{}{}
            \eq{\frac{j}{j_\infty} = 1 - \frac{c^\txS}{c^0}}
        \end{formula}

        \begin{formula}{kinetic_current}
            \desc{Kinetic current density}{}{$j_\text{meas}$ measured \qtyRef{current_density}, $j_\infty$ \secEqRef{limiting_current}}
            \desc[german]{Kinetische Stromdichte}{}{$j_\text{meas}$ gemessene \qtyRef{current_density}, $j_\infty$ \secEqRef{limiting_current}}
            \eq{j_\text{kin} = \frac{j_\text{meas} j_\infty}{j_\infty - j_\text{meas}}}
        \end{formula}

    \begin{formula}{roughness_factor}
        \desc{Roughness factor}{Surface area related to electrode geometry}{}
        \eq{\rfactor}
    \end{formula}   

    \begin{formula}{butler_volmer}
        \desc{Butler-Volmer equation}{Reaction kinetics near the equilibrium potentential}
        {$j$ \qtyRef{current_density}, $j_0$ exchange current density, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ cathodic/anodic charge transfer coefficient, $\text{rf}$ \secEqRef{roughness_factor}}
        %Current through an electrode iof a unimolecular redox reaction with both anodic and cathodic reaction occuring on the same electrode
        \desc[german]{Butler-Volmer-Gleichung}{Reaktionskinetik in der Nähe des Gleichgewichtspotentials}
        {$j$ \qtyRef{current_density}, $j_0$ Austauschstromdichte, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ Ladungstransferkoeffizient an der Kathode/Anode, $\text{rf}$ \secEqRef{roughness_factor}}
        \begin{gather}
            j = j_0 \,\rfactor\, \left[ \Exp{\frac{(1-a_\txC) z F \eta}{RT}} - \Exp{-\frac{\alpha_\txC z F \eta}{RT}}\right]
            \intertext{\GT{with}}
            \alpha_\txA = 1 - \alpha_\txC
        \end{gather}
        \separateEntries
        \fig{img/ch_butler_volmer.pdf}
    \end{formula}

    % \Subsubsection[
    %     \eng{Tafel approximation}
    %     \ger{Tafel Näherung}
    % ]{tafel}

    % \begin{formula}{slope}
    %     \desc{Tafel slope}{}{}
    %     \desc[german]{Tafel Steigung}{}{}
    %     \eq{}
    % \end{formula}

    \begin{formula}{equation}
        \desc{Tafel approximation}{For slow kinetics: $\abs{\eta} > \SI{0.1}{\volt}$}{}
        \desc[german]{Tafel Näherung}{Für langsame Kinetik: $\abs{\eta} > \SI{0.1}{\volt}$}{}
        \eq{
            \Log{j} &\approx \Log{j_0} + \frac{\alpha_\txC zF \eta}{RT\ln(10)} && \eta \gg \SI{0.1}{\volt}\\
            \Log{\abs{j}} &\approx \Log{j_0} - \frac{(1-\alpha_\txC) zF \eta}{RT\ln(10)} && \eta \ll -\SI{0.1}{\volt}
        }
        \fig{img/ch_tafel.pdf}
    \end{formula}



\Subsection[
    \eng{Techniques}
    \ger{Techniken}
]{tech}

    \Subsubsection[
        \eng{Reference electrodes}
        \ger{Referenzelektroden}
    ]{ref}
        \begin{ttext}
            \eng{Defined as reference for measuring half-cell potententials}
            \ger{Definiert als Referenz für Messungen von Potentialen von Halbzellen}
        \end{ttext}

        \begin{formula}{she}
            \desc{Standard hydrogen elektrode (SHE)}{}{$p=\SI{e5}{\pascal}$, $a_{\ce{H+}}=\SI{1}{\mol\per\litre}$ (\Rightarrow $\pH=0$)}
            \desc[german]{Standardwasserstoffelektrode (SHE)}{}{}
            \ttxt{
                \eng{Potential of the reaction: \ce{2H^+ +2e^- <--> H2}}
                \ger{Potential der Reaktion: \ce{2H^+ +2e^- <--> H2}}
            }
        \end{formula}

        \begin{formula}{rhe}
            \desc{Reversible hydrogen electrode (RHE)}{RHE Potential does not change with the pH value}{$E^0\equiv \SI{0}{\volt}$, \QtyRef{activity}, \QtyRef{pressure}, \GT{see} \fqEqRef{ch:el:cell:nernst_equation}}
            \desc[german]{Reversible Wasserstoffelektrode (RHE)}{Potential ändert sich nicht mit dem pH-Wert}{}
            \eq{
                E_\text{RHE} &= E^0 + \frac{RT}{F} \Ln{\frac{a_{\ce{H^+}}}{p_{\ce{H2}}}}
                          % \\ &= \SI{0}{\volt} - \SI{0.059}{\volt}
            }
        \end{formula}
    
    

    \Subsubsection[
        \eng{Cyclic voltammetry}
        \ger{Zyklische Voltammetrie}
    ]{cv}
        \begin{bigformula}{duck}
            \desc{Cyclic voltammogram}{}{}
            % \desc[german]{}{}{}

            \begin{minipage}{0.44\textwidth}
                
                \begin{tikzpicture}
                    \pgfmathsetmacro{\Ax}{-2.3}
                    \pgfmathsetmacro{\Ay}{ 0.0}
                    \pgfmathsetmacro{\Bx}{ 0.0}
                    \pgfmathsetmacro{\By}{ 1.0}
                    \pgfmathsetmacro{\Cx}{ 0.4}
                    \pgfmathsetmacro{\Cy}{ 1.5}
                    \pgfmathsetmacro{\Dx}{ 2.0}
                    \pgfmathsetmacro{\Dy}{ 0.5}
                    \pgfmathsetmacro{\Ex}{ 0.0}
                    \pgfmathsetmacro{\Ey}{-1.5}
                    \pgfmathsetmacro{\Fx}{-0.4}
                    \pgfmathsetmacro{\Fy}{-2.0}
                    \pgfmathsetmacro{\Gx}{-2.3}
                    \pgfmathsetmacro{\Gy}{-0.3}
                    \pgfmathsetmacro{\x}{3}
                    \pgfmathsetmacro{\y}{3}
                    \begin{axis}[ymin=-\y,ymax=\y,xmax=\x,xmin=-\x,
                        % equal axis,
                        minor tick num=1,
                        xlabel={$E$}, xlabel style={at={(axis description cs:0.5,-0.06)}},
                        ylabel={$j$}, ylabel style={at={(axis description cs:-0.06,0.5)}},
                        anchor=center, at={(0,0)},
                        axis equal image,clip=false,
                    ]
                        % CV with beziers
                        \draw[thick, fg-blue]                                                       (axis cs:\Ax,\Ay) coordinate (A) node[left]  {A}
                            ..controls (axis cs:\Ax+1.8, \Ay+0.0) and (axis cs:\Bx-0.2, \By-0.4) .. (axis cs:\Bx,\By) coordinate (B) node[left]  {B}
                            ..controls (axis cs:\Bx+0.1, \By+0.2) and (axis cs:\Cx-0.3, \Cy+0.0) .. (axis cs:\Cx,\Cy) coordinate (C) node[above] {C}
                            ..controls (axis cs:\Cx+0.5, \Cy+0.0) and (axis cs:\Dx-1.3, \Dy+0.1) .. (axis cs:\Dx,\Dy) coordinate (D) node[right] {D}
                            ..controls (axis cs:\Dx-2.0, \Dy-0.1) and (axis cs:\Ex+0.3, \Ey+0.8) .. (axis cs:\Ex,\Ey) coordinate (E) node[right] {E}
                            ..controls (axis cs:\Ex-0.1, \Ey-0.2) and (axis cs:\Fx+0.2, \Fy+0.0) .. (axis cs:\Fx,\Fy) coordinate (F) node[below] {F}
                            ..controls (axis cs:\Fx-0.2, \Fy+0.0) and (axis cs:\Gx+1.5, \Gy-0.2) .. (axis cs:\Gx,\Gy) coordinate (G) node[left]  {G};
                        \node[above] at (A) {\rightarrow};

                        \draw[dashed, fg2] (axis cs: \Bx,\By) -- (axis cs: \Ex, \Ey);

                        \draw[->] (axis cs:-\x-0.6, 0.4) -- (axis cs:-\x-0.6, \y) node[left=0.3cm, anchor=east, rotate=90] {Cath / Red};
                        \draw[->] (axis cs:-\x-0.6,-0.4) -- (axis cs:-\x-0.6,-\y) node[left=0.3cm, anchor=west, rotate=90] {An / Ox};
                    \end{axis}
                \end{tikzpicture}
            \end{minipage}
            \begin{minipage}{0.55\textwidth}
                \begin{ttext}
                    \eng{\begin{itemize}
                        \item {\color{fg-blue}A-D}: Diffusion layer growth \rightarrow decreased current after peak
                        \item {\color{fg-blue}D}: Switching potential
                        \item {\color{fg-blue}B,E}: Equal concentrations of reactants
                        \item {\color{fg-blue}C,F}: Formal potential of redox pair: $E \approx \frac{E_\txC - E_\txF}{2}$
                        \item {\color{fg-blue}C,F}: Peak separation for reverisble processes: $\Delta E_\text{rev} = E_\txC - E_\txF = n\,\SI{59}{\milli\volt}$
                        \item Information about surface chemistry
                        \item Double-layer capacity (horizontal lines): $I = C v$
                    \end{itemize}}
                \end{ttext}
            \end{minipage}
        \end{bigformula}

        \begin{formula}{charge}
            \desc{Charge}{Area under the curve}{$v$ \qtyRef{scan_rate}}
            \desc[german]{Ladung}{Fläche unter der Kurve}{}
            \eq{q = \frac{1}{v} \int_{E_1}^{E_2}j\,\d E}
        \end{formula}

        \begin{formula}{peak_current}
            \desc{Randles-Sevcik equation}{For reversible faradaic reaction.\\Peak current depends on square root of the scan rate}{$n$ \qtyRef{charge_number}, \ConstRef{faraday}, $A$ electrode surface area, $c^0$ bulk \qtyRef{concentration}, $v$ \qtyRef{scan_rate}, $D_\text{ox}$ \qtyRef{diffusion_coefficient} of oxidized analyte, \ConstRef{universal_gas}, \QtyRef{temperature}}
            \desc[german]{Randles-Sevcik Gleichung}{Für eine reversible, faradäische Reaktion\\Spitzenstrom hängt von der Wurzel der Scanrate ab}{}
            \eq{i_\text{peak} = 0.446\,nFAc^0 \sqrt{\frac{nFvD_\text{ox}}{RT}}}
        \end{formula}

        \begin{hiddenformula}{scan_rate}
            \desc{Scan rate}{}{}
            \desc[german]{Scanrate}{}{}
            \quantity{v}{\volt\per\s}{s}
        \end{hiddenformula}
        

        \begin{formula}{upd}
            \desc{Underpotential deposition (UPD)}{}{}
            \desc[german]{}{}{}
            \ttxt{Reversible deposition of metal onto a foreign metal electrode at potentials positive of the Nernst potential \TODO{clarify}}
        \end{formula}

    \Subsubsection[
        \eng{Rotating disk electrodes}
        % \ger{}
    ]{rde} \abbrLink{rde}{RDE}
        \begin{formula}{viscosity}
            \desc{Dynamic viscosity}{}{}
            \desc[german]{Dynamisch Viskosität}{}{}
            \quantity{\eta,\mu}{\pascal\s=\newton\s\per\m^2=\kg\per\m\s}{}
        \end{formula}

        \begin{formula}{kinematic_viscosity}
            \desc{Kinematic viscosity}{\qtyRef{viscosity} related to density of a fluid}{\QtyRef{viscosity}, \QtyRef{density}}
            \desc[german]{Kinematische Viskosität}{\qtyRef{viscosity} im Verhältnis zur Dichte der Flüssigkeit}{}
            \quantity{\nu}{\cm^2\per\s}{}
            \eq{\nu = \frac{\eta}{\rho}}
        \end{formula}

        \begin{formula}{diffusion_layer_thickness}
            \desc{Diffusion layer thickness}{\TODO{Where does 1.61 come from}}{$D$ \qtyRef{diffusion_coefficient}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
            \desc[german]{Diffusionsschichtdicke}{}{}
            \eq{\delta_\text{diff}= 1.61 D{^\frac{1}{3}} \nu^{\frac{1}{6}} \omega^{-\frac{1}{2}}}
        \end{formula}

        \begin{formula}{limiting_current}
            \desc{Limiting current density}{for a \abbrRef{rde}}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \secEqRef{diffusion_layer_thickness}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
            % \desc[german]{Limitierender Strom}{}{}
            \eq{j_\infty = nFD \frac{c^0}{\delta_\text{diff}} = \frac{1}{1.61} nFD^{\frac{2}{3}}  v^{\frac{-1}{6}} c^0 \sqrt{\omega}}
        \end{formula}

    \Subsubsection[
        \eng{AC-Impedance}
        \ger{AC-Impedanz}
    ]{ac}
        \begin{formula}{nyquist}
            \desc{Nyquist diagram}{Real and imaginary parts of \qtyRef{impedance} while varying the frequency}{}
            \desc[german]{Nyquist-Diagram}{Real und Imaginaärteil der \qtyRef{impedance} während die Frequenz variiert wird}{}
            \fig{img/ch_nyquist.pdf}
        \end{formula}

        \begin{formula}{tlm}
            \desc{Transmission line model}{Model of porous electrodes as many slices}{$R_\text{ion}$ ion conduction resistance in electrode slice, $R$ / $C$ resistance / capacitance of electode slice}
            % \desc[german]{}{}{}
            \ctikzsubcircuitdef{rcpair}{in, out}{%
                coordinate(#1-in)
                (#1-in) -- ++(0, -\rcpairH) 
                -- ++(\rcpairW, 0) to[R, l=$R$] ++(0,-\rcpairL) -- ++(-\rcpairW, 0)   
                -- ++(0,-\rcpairH) coordinate (#1-out) ++(0,\rcpairH) 
                -- ++(-\rcpairW, 0) to[C, l=$C$] ++(0,\rcpairL) -- ++(\rcpairW,0)
                (#1-out)
            }

            \pgfmathsetmacro{\rcpairH}{0.5}
            \pgfmathsetmacro{\rcpairW}{0.5}
            \pgfmathsetmacro{\rcpairL}{1.8}
            \ctikzsubcircuitactivate{rcpair}
            \pgfmathsetmacro{\rcpairD}{3.0} % distance
            \centering
            \begin{circuitikz}[/tikz/circuitikz/bipoles/length=1cm,scale=0.7]
            \draw (0,0) to[R,l=$R_\text{electrolyte}$] ++(2,0) -- ++(1,0)
            \rcpair{rc1}{} (rc1-in) to[R,l=$R_\text{ion}$] ++(\rcpairD,0) \rcpair{rc2}{} (rc2-in) to[R,l=$R_\text{ion}$] ++(\rcpairD,0) ++(\rcpairD,0) \rcpair{rc3}{};
                \draw[dashed] (rc2-in) ++(\rcpairD,0) -- (rc3-in) (rc2-out) ++(\rcpairD,0) -- (rc3-out);
                \draw (rc1-out) -- (rc2-out) -- ++(\rcpairD,0) (rc3-out) -- ++(\rcpairD/2,0);
            \end{circuitikz}
            \fig{img/ch_nyquist_tlm.pdf}
        \end{formula}

