Electrochemistry with Periodic Signals

AC Voltage and current

  • AC:

    • U = U\(_0\)sin\(\omega\)t

  • Amplitude:

    • U\(_0\)
    • peak to peak voltage U\(_{p-p}\) = 2U\(_0\)
    • RMS: U\(_{rms}\) = U\(_0\)/\(\sqrt{2}\)

  • Resulting AC current can have different amplitude because of impedance:

    • \(I=I_{0} \sin (\omega t+\theta)\)

  • Preiodicity: \(E(t+P)=E(t)\)

  • Sinusiodal case:

    • \(\Delta E(t)=|E| \sin \left\{\omega t+\varphi_{E}\right\}\)
    • \(I(t)=|I| \sin \left\{\omega t+\varphi_{I}\right\}\)

  • Angular frequency is \(\omega=2 \pi f\)

  • Phase angle is \(\varphi\)

  • What is the periodicity P? Maybe peak to peak, known as \(2\pi\)

Conductivity cell

  • Calibrate geometrical factors by using 1M KCl sol: \(\kappa=\frac{L}{A R}=\frac{L|I|}{A|E|}\)
  • AC measurement: Current \(I(t)=|I| \sin \left\{\omega t+\varphi_{I}\right\}\)
  • Will generate voltages over an electrochemical cell
  • Over R: \(E_{R}(t)=R_{\mathrm{u}} I(t)=R_{\mathrm{u}}|I| \sin \left\{\omega t+\varphi_{I}\right\}\)
  • Over C: \(E_{C}(t)=\frac{Q(t)}{C}=\frac{1}{C} \int|I| \sin \left\{\omega t+\varphi_{I}\right\} \mathrm{d} t=\frac{|I|}{\omega C} \cos \left\{\omega t+\varphi_{I}\right\}=\frac{|I|}{\omega C} \sin \left\{\omega t+\varphi_{I}+\frac{\pi}{2}\right\}\)

Faradaic effects of AC, Impedance

  • The aim of this subsection is to show that there is two contributions to electrode impedance: Kinetics and Transport
  • Apply AC current, measure AC voltage, or vice versa
  • Impedance Z (= R in DC) > = |E|/|I|
  • Faradaic effects:
    • Apply (small) AC voltage \(\Delta E(t)=|E| \sin \left\{\omega t+\varphi_{E}\right\}\)
    • Reaction \(\mathrm{R}(\) soln \() \rightleftarrows \mathrm{e}^{-}+\mathrm{O}(\) soln \()\)
    • Response on concentrations \(c_{\mathrm{i}}^{\mathrm{s}}(t)=c_{\mathrm{i}}^{\mathrm{b}}+\left|c_{\mathrm{i}}^{\mathrm{s}}\right| \sin \left\{\omega t+\varphi_{\mathrm{i}}^{\mathrm{s}}\right\} \quad \mathrm{i}=\mathrm{R}, \mathrm{O}\)
  • Oscillating current changes the concentrations of reactants and products:
    • \(c_{\mathrm{R}}^{\mathrm{s}}(t)=c_{\mathrm{R}}^{\mathrm{b}}-\frac{|I| \sin \left\{\omega t+\varphi_{I}-\pi / 4\right\}}{F A \sqrt{D_{\mathrm{R}} \omega}} ; \quad c_{\mathrm{O}}^{\mathrm{s}}(t)=c_{\mathrm{O}}^{\mathrm{b}}+\frac{|I| \sin \left\{\omega t+\varphi_{I}-\pi / 4\right\}}{F A \sqrt{D_{\mathrm{O}} \omega}}\)
    • \(\frac{c_{\mathrm{R}}^{\mathrm{s}}(t)}{c_{\mathrm{R}}^{\mathrm{b}}}=1-\frac{\left|c_{\mathrm{R}}^{\mathrm{s}}\right|}{c_{\mathrm{R}}^{\mathrm{b}}} \sin \left\{\omega t+\varphi_{I}-\pi / 4\right\} ; \quad \frac{c_{\mathrm{O}}^{\mathrm{s}}(t)}{c_{\mathrm{O}}^{\mathrm{b}}}=1+\frac{\left|c_{\mathrm{O}}^{\mathrm{s}}\right|}{c_{\mathrm{O}}^{\mathrm{b}}} \sin \left\{\omega t+\varphi_{I}-\pi / 4\right\}\)
    • Product and reactant concentrations are shifted -45 deg and +135 deg respectively, 180 deg apart.
  • \(\frac{R T}{F A i_{\mathrm{n}}}=\frac{R T}{F^{2} A k^{0 \prime}\left(c_{\mathrm{R}}^{\mathrm{b}}\right)^{\alpha}\left(c_{\mathrm{O}}^{\mathrm{b}}\right)^{1-\alpha}}=R_{\mathrm{ct}}\) charge-transfer resistor
  • \(\frac{R T \sqrt{\omega}}{F|I|}\left(\frac{\left|c_{\mathrm{R}}^{\mathrm{s}}\right|}{c_{\mathrm{R}}^{\mathrm{b}}}+\frac{\left|c_{\mathrm{O}}^{\mathrm{s}}\right|}{c_{\mathrm{O}}^{\mathrm{b}}}\right)=\frac{R T}{F^{2} A}\left(\frac{1}{c_{\mathrm{R}}^{\mathrm{b}} \sqrt{D_{\mathrm{R}}}}+\frac{1}{c_{\mathrm{O}}^{\mathrm{b}} \sqrt{D_{\mathrm{O}}}}\right)= W\) Warburg element
  • W has units of Ohms s\(^{-1/2}\)
    • Reflects a combination of Diffusion and Chemical capacitance
    • Appears in AC circuits
    • Imposes complex impedance Z_{\mathrm{W}}(\mathrm{ohm})=W / \mathrm{V}(2 \omega)-\mathrm{j} W / \mathrm{V}(2 \omega)
    • Has a constant, frequency independed phase shift of \(-\pi / 4\left(-45^{\circ}\right)\)
  • Revesibility index \(\lambda\)
    • \(\lambda=\frac{W}{R_{\mathrm{ct}} \sqrt{\omega}}=\frac{k^{\circ}}{\sqrt{\omega}}\left[\frac{1}{\sqrt{D_{\mathrm{R}}}}\left(\frac{c_{\mathrm{O}}^{\mathrm{b}}}{c_{\mathrm{R}}^{\mathrm{b}}}\right)^{1-\alpha}+\frac{1}{\sqrt{D_{\mathrm{O}}}}\left(\frac{c_{\mathrm{R}}^{\mathrm{b}}}{c_{\mathrm{O}}^{\mathrm{b}}}\right)^{\alpha}\right]\)
    • Reversible: R\(_{ct}\) very small, \(\lambda >> 1\)
    • Quasireversible: \(\lambda \approx 1\)
    • Irreversible: R\(_{ct}\) large, \(\lambda << 1\), No Faradaic periodicity

Impedance spectrometer

  • 0-3V rms AC 0.01mHz-32MHz, -42 - +42 V DC
  • Electrode electrical elements:
    • R\(_u\) uncompensated electrolyte resistance (ohmic)
    • R\(_{ct}\) Charge transfer resistance (kinetics)
    • W Warburg impedance (diffusion)
    • C Capacitance (double layers)

The Z semicircle (RC)

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