Correction to: Electrochemical Impedance Spectroscopy and its Applications

Correction to: A. Lasia, Electrochemical Impedance Spectroscopy and its Applicationshttps://doi.org/10.1007/978-1-4614-8933-7

Unfortunately the original version of the book had been published with several incorrect equations. The corrected equations have been listed below.

p. 39, Eq. (2.113)

$$ {\displaystyle \begin{array}{c}i(t)=\frac{E_0}{R}\frac{1}{1+\frac{1}{{\left(\omega RC\right)}^2}}\left[\tan \left(\varphi \right)\cos \left(\omega t\right)+\sin \left(\omega t\right)\right]=\\ {}=\frac{E_0}{R}\cos \left(\varphi \right)\left[\frac{\cos \left(\omega t\right)\sin \left(\varphi \right)}{\cos \left(\varphi \right)}+\sin \left(\omega t\right)\right]\\ {}=\frac{E_0}{R}\left[\cos \left(\omega t\right)\sin \left(\varphi \right)+\sin \left(\omega t\right)\cos \left(\varphi \right)\right]\end{array}} $$

(2.113)

p. 40. Eq. (2.118)

$$ {\displaystyle \begin{array}{c}i(t)=\frac{E_0}{R}\frac{1}{1+\frac{1}{{\left(\omega RC\right)}^2}}\left[\cos \left(\omega t\right)-\tan \left(\varphi \right)\sin \left(\omega t\right)\right]=\\ {}=\frac{E_0}{R}\frac{1}{1+\frac{1}{{\left(\omega RC\right)}^2}}\left[\frac{\cos \left(\omega t\right)\cos \left(\omega t\right)-\sin \left(\varphi \right)\sin \left(\omega t\right)}{\cos \left(\varphi \right)}\right]\end{array}} $$

(2.118)

p. 46, Eq. (2.135)–(2.136)

$$ {\displaystyle \begin{array}{c}\hat{Z}\left( j\omega \right)={R}_0+\frac{R_1}{1+ j\omega {R}_1{C}_1}={R}_0+\frac{R_1\left(1- j\omega {R}_1{C}_1\right)}{1+{\left(\omega {R}_1{C}_1\right)}^2}=\\ {}={R}_0+\frac{R_1}{1+{\left(\omega {R}_1{C}_1\right)}^2}-j\frac{\omega {R}_1^2{C}_1}{1+{\left(\omega {R}_1{C}_1\right)}^2}\end{array}} $$

(2.135)

$$ {\displaystyle \begin{array}{c}\operatorname{Re}\left[\hat{Z}\left( j\omega \right)\right]={R}_0+\frac{R_1}{1+{\left(\omega {R}_1{C}_1\right)}^2}\\ {}\operatorname{Im}\left[\hat{Z}\left( j\omega \right)\right]=-\frac{\omega {R}_1^2{C}_1}{1+{\left(\omega {R}_1{C}_1\right)}^2}\end{array}} $$

(2.136)

p. 50, Eq. (2.140)

$$ {\displaystyle \begin{array}{c}\left|Z\right|=\sqrt{R^2+\frac{1}{{\left(\omega C\right)}^2}}\\ {}\varphi =\mathrm{atan}\left(\frac{Z^{{\prime\prime} }}{Z^{\prime }}\right)=\mathrm{atan}\left(-\frac{1}{\omega RC}\right)=-\mathrm{atan}\left(\frac{1}{\omega RC}\right)\end{array}} $$

(2.140)

p. 92, Eq. (4.39).

$$ {C}_{\mathrm{O}}(0)=1-\frac{i}{i_{\mathrm{lim}}}\mathrm{and}\;{C}_{\mathrm{R}}(0)=\frac{i}{i_{\mathrm{lim}}} $$

(4.39)

p. 92, Eq. (4.41)

$$ {\displaystyle \begin{array}{c}{\hat{Z}}_{\mathrm{W}}={\hat{Z}}_{\mathrm{W},\mathrm{O}}+{\hat{Z}}_{\mathrm{W},\mathrm{R}}=\frac{RT}{n^2{F}^2\sqrt{j\omega}}\left(\frac{1}{\sqrt{D_{\mathrm{O}}}{C}_{\mathrm{O}}(0)}+\frac{1}{\sqrt{D_{\mathrm{R}}}{C}_{\mathrm{R}}(0)}\right)\\ {}=\frac{\sigma^{\prime }}{\sqrt{j\omega}}=\frac{\sqrt{2}}{\sqrt{j}}\frac{\sigma }{\sqrt{\omega }}=\sigma \left(1-j\right)\end{array}} $$

(4.41)

p. 105, Eq. (4.76)

$$ {\lim}_{\omega \to 0}\left({\hat{Z}}_{\mathrm{t}}\right)={R}_{\mathrm{S}}+{R}_{\mathrm{ct}}+\frac{\sigma^{\prime }l}{\sqrt{D_{\mathrm{O}}}} $$

(4.76)

p. 113, Eq. (4.106)

$$ {\displaystyle \begin{array}{c}{\hat{Z}}_{\mathrm{W}}=\frac{RT}{n^2{F}^2\sqrt{D_{\mathrm{R}}}{C}_{\mathrm{R}}(0)}\frac{1}{\sqrt{j\omega}\coth \left(\sqrt{\frac{j\omega}{D_{\mathrm{R}}}{r}_0}\right)-\frac{\sqrt{D_{\mathrm{R}}}}{r_0}}=\\ {}=\frac{\sigma^{\prime }}{\sqrt{j\omega}}\frac{1}{\coth \left(\sqrt{\frac{j\omega}{D_{\mathrm{R}}}{r}_0}\right)-\frac{\sqrt{D_{\mathrm{R}}}}{r_0\sqrt{j\omega}}}=\\ {}=\frac{\sigma^{\prime }{r}_0}{\sqrt{D_{\mathrm{R}}}}\frac{1}{\left(\sqrt{\frac{j\omega}{D_{\mathrm{R}}}{r}_0}\right)\coth \left(\sqrt{\frac{j\omega}{D_{\mathrm{R}}}{r}_0}\right)-1}\end{array}} $$

(4.106)

p. 141, Eq. (5.71)

$$ {\mathrm{B}}_{\mathrm{ads}}+\mathrm{e}\underset{{\overleftarrow{k}}_{-2}}{\overset{{\overrightarrow{k}}_2}{\rightleftarrows }}{\mathrm{C}}_{\mathrm{ads}} $$

(5.71)

p. 142, Eq. (5.80)

$$ \tilde{i}=-F\left[{\left(\frac{\partial {r}_0}{\partial \eta}\right)}_{\theta_{\mathrm{B}},{\theta}_{\mathrm{C}}}\tilde{\eta}+{\left(\frac{\partial {r}_0}{\partial {\theta}_{\mathrm{B}}}\right)}_{\eta, {\theta}_{\mathrm{C}}}{\tilde{\theta}}_{\mathrm{B}}+{\left(\frac{\partial {r}_0}{\partial {\theta}_{\mathrm{C}}}\right)}_{\eta, {\theta}_{\mathrm{B}}}{\tilde{\theta}}_{\mathrm{C}}\right] $$

(5.80)

p. 150, Eq. (6.19), (6.20), (6.21) (errors in signs)

$$ {\displaystyle \begin{array}{c}-\left(\frac{\partial {r}_0}{\partial \eta}\right)=\frac{1}{F}\frac{\tilde{i}}{\tilde{\eta}}+\left(\frac{\partial {r}_0}{\partial {\theta}_{\mathrm{B}}}\right)\frac{{\tilde{\theta}}_{\mathrm{B}}}{\tilde{\eta}}+\left(\frac{\partial {r}_0}{\partial {C}_{\mathrm{A}}}\right)\frac{{\tilde{C}}_{\mathrm{A}}(0)}{\tilde{\eta}}+\left(\frac{\partial {r}_0}{\partial {C}_{\mathrm{C}}}\right)\frac{{\tilde{C}}_{\mathrm{C}}(0)}{\tilde{\eta}}\\ {}0=-\frac{1}{2F}\frac{\tilde{i}}{\tilde{\eta}}+\sqrt{j\omega {D}_{\mathrm{A}}}\frac{{\tilde{C}}_{\mathrm{A}}(0)}{\tilde{\eta}}\\ {}0=\frac{1}{2F}\frac{\tilde{i}}{\tilde{\eta}}+\sqrt{j\omega {D}_{\mathrm{C}}}\frac{{\tilde{C}}_{\mathrm{C}}(0)}{\tilde{\eta}}\\ {}-\left(\frac{\partial {r}_1}{\partial \eta}\right)=-{\Gamma}_{\infty } j\omega \frac{{\tilde{\theta}}_{\mathrm{B}}}{\tilde{\eta}}+\left(\frac{\partial {r}_1}{\partial {\theta}_{\mathrm{B}}}\right)\frac{{\tilde{\theta}}_{\mathrm{B}}}{\tilde{\eta}}+\left(\frac{\partial {r}_1}{\partial {C}_{\mathrm{A}}}\right)\frac{{\tilde{C}}_{\mathrm{A}}(0)}{\tilde{\eta}}+\left(\frac{\partial {r}_1}{\partial {C}_{\mathrm{C}}}\right)\frac{{\tilde{C}}_{\mathrm{C}}(0)}{\tilde{\eta}}\end{array}} $$

(6.19)

$$ \left[\begin{array}{c}-\frac{\partial {r}_0}{\partial \eta}\\ {}0\\ {}0\\ {}-\frac{\partial {r}_1}{\partial \eta}\end{array}\right]=\left[\begin{array}{cccc}\frac{1}{F}& \frac{\partial {r}_0}{\mathrm{\partial \Theta }}& \frac{\partial {r}_0}{\partial {C}_A}& \frac{\partial {r}_0}{\partial {C}_C}\\ {}\frac{1}{2F}& 0& \sqrt{j\omega {D}_A}& 0\\ {}\frac{1}{2F}& 0& 0& \sqrt{j\omega {D}_C}\\ {}0& \frac{\partial {r}_1}{\mathrm{\partial \Theta }}-{\Gamma}_{\infty } j\omega & \frac{\partial {r}_1}{\partial {C}_A}& \frac{\partial {r}_1}{\partial {C}_C}\end{array}\right]\left[\begin{array}{c}\frac{\tilde{i}}{\tilde{\eta}}\\ {}\frac{\tilde{\Theta}}{\tilde{\eta}}\\ {}\frac{{\tilde{C}}_C(0)}{\tilde{\eta}}\\ {}\frac{{\tilde{C}}_A(0)}{\tilde{\eta}}\end{array}\right] $$

(6.20)

$$ A=\left|\begin{array}{cccc}\frac{1}{F}& \frac{\partial {r}_0}{\partial {\theta}_{\mathrm{B}}}& \frac{\partial {r}_0}{\partial {C}_{\mathrm{A}}}& \frac{\partial {r}_0}{\partial {C}_{\mathrm{C}}}\\ {}\frac{1}{-2F}& 0& \sqrt{j\omega {D}_{\mathrm{A}}}& 0\\ {}\frac{1}{2F}& 0& 0& \sqrt{j\omega {D}_C}\\ {}0& \frac{\partial {r}_1}{\partial {\theta}_{\mathrm{B}}}-{\Gamma}_{\infty } j\omega & \frac{\partial {r}_1}{\partial {C}_{\mathrm{A}}}& \frac{\partial {r}_1}{\partial {C}_{\mathrm{C}}}\end{array}\right| $$

(6.21)

p. 153, Eq. (6.22), (6.23), (6.24)

$$ T=\left|\begin{array}{cccc}-\frac{\partial {r}_0}{\partial \eta }& \frac{\partial {r}_0}{\partial {\theta}_{\mathrm{B}}}& \frac{\partial {r}_0}{\partial {C}_{\mathrm{A}}}& \frac{\partial {r}_0}{\partial {C}_{\mathrm{C}}}\\ {}0& 0& \sqrt{j\omega {D}_{\mathrm{A}}}& 0\\ {}0& 0& 0& \sqrt{j\omega {D}_{\mathrm{C}}}\\ {}-\frac{\partial {r}_1}{\partial \eta }& \frac{\partial {r}_1}{\partial {\theta}_{\mathrm{B}}}-{\Gamma}_{\infty } j\omega & \frac{\partial {r}_1}{\partial {C}_{\mathrm{A}}}& \frac{\partial {r}_1}{\partial {C}_{\mathrm{C}}}\end{array}\right| $$

(6.22)

$$ {\displaystyle \begin{array}{c}T=\sqrt{D_{\mathrm{A}}{D}_{\mathrm{C}}}\left[{\Gamma}_{\infty}\frac{\partial {r}_0}{\partial \eta }{\left( j\omega \right)}^2+\left(-\frac{\partial {r}_0}{\partial \eta}\frac{\partial {r}_1}{\partial \theta }+\frac{\partial {r}_0}{\partial \theta}\frac{\partial {r}_1}{\partial \eta}\right)\left( j\omega \right)\right]\\ {}={a}_4{\left( j\omega \right)}^2+{a}_2\left( j\omega \right)\end{array}} $$

(6.23)

$$ {\displaystyle \begin{array}{c}A=-\frac{1}{2F}\left[\begin{array}{l}2\sqrt{D_{\mathrm{A}}{D}_{\mathrm{C}}}{\Gamma}_{\infty }{\left( j\omega \right)}^2\\ {}+{\Gamma}_{\infty}\left(\sqrt{D_{\mathrm{C}}}\frac{\partial {r}_0}{\partial {C}_{\mathrm{A}}}-\sqrt{D_{\mathrm{A}}}\frac{\partial {r}_0}{\partial {C}_{\mathrm{C}}}\right){\left( j\omega \right)}^{3/2}\\ {}-2\sqrt{D_A{D}_C}\frac{\partial {r}_1}{\partial \theta}\left( j\omega \right)\\ {}+\left(\begin{array}{l}-\sqrt{D_C}\frac{\partial {r}_0}{\partial {C}_{\mathrm{A}}}\frac{\partial {r}_1}{\partial \theta }+\sqrt{D_A}\frac{\partial {r}_0}{\partial {C}_{\mathrm{C}}}\frac{\partial {r}_1}{\partial \theta}\\ {}+\sqrt{D_C}\frac{\partial {r}_0}{\partial \theta}\frac{\partial {r}_1}{\partial {C}_{\mathrm{A}}}-\sqrt{D_A}\frac{\partial {r}_0}{\partial \theta}\frac{\partial {r}_1}{\partial {C}_{\mathrm{C}}}\end{array}\right){\left( j\omega \right)}^{1/2}\end{array}\right]\\ {}=\frac{1}{2F}\left[{b}_4{\left( j\omega \right)}^2+{b}_3{\left( j\omega \right)}^{3/2}+{b}_2\left( j\omega \right)+{b}_1{\left( j\omega \right)}^{1/2}\right]\end{array}} $$

(6.24)

p. 152, Eq. (6.27)

$$ {\displaystyle \begin{array}{c}{\hat{Z}}_{\mathrm{f}}=\frac{1}{2F}\frac{b_4}{a_4}+\frac{1}{2F}\frac{b_4}{a_4}\left[\frac{\frac{b_3}{b_4}\left( j\omega \right)+\left(\frac{b_2}{b_4}-\frac{a_2}{a_4}\right){\left( j\omega \right)}^{1/2}+\frac{b_1}{b_4}}{{\left( j\omega \right)}^{3/2}+\frac{a_2}{a_4}{\left( j\omega \right)}^{1/2}}\right]\\ {}={R}_{\mathrm{ct}}+\frac{1}{2F}\left[\frac{\frac{b_3}{a_4}\left( j\omega \right)+\left(\frac{b_2}{a_4}-\frac{a_2{b}_4}{a_4^2}\right){\left( j\omega \right)}^{1/2}+\frac{b_1}{a_4}}{{\left( j\omega \right)}^{1/2}\left[\left( j\omega \right)+\frac{a_2}{a_4}\right]}\right]\\ {}={R}_{\mathrm{ct}}+\frac{1}{2F}\left[\frac{\frac{b_3}{a_2}\left( j\omega \right)+\left(\frac{b_2}{a_2}-\frac{b_4}{a_4}\right){\left( j\omega \right)}^{1/2}+\frac{b_1}{a_2}}{{\left( j\omega \right)}^{1/2}\left[\left( j\omega \right)\frac{a_4}{a_2}+1\right]}\right]\end{array}} $$

(6.27)

p. 152, Eq. (6.28)

$$ {\displaystyle \begin{array}{c}{\hat{Z}}_{\mathrm{f}}={R}_{\mathrm{ct}}+\frac{1}{j\omega {C}_{\mathrm{p}}+\frac{1}{R_{\mathrm{p}}}}+{Z}_{\mathrm{W}}={R}_{\mathrm{ct}}+\frac{1}{j\omega {C}_{\mathrm{p}}+\frac{1}{R_{\mathrm{p}}}}+\frac{\sigma }{\sqrt{j\omega}}\\ {}={R}_{\mathrm{ct}}+\frac{\left( j\omega \right){R}_p{C}_p\omega +{R}_p{\left( j\omega \right)}^{1/2}+\sigma }{{\left( j\omega \right)}^{1/2}\left[\left( j\omega \right){R}_p{C}_p+1\right]}\end{array}} $$

(6.28)

p. 153, Eq. (6.29)

$$ {\displaystyle \begin{array}{l}{R}_{\mathrm{ct}}=\frac{1}{2F}\frac{b_4}{a_4};{R}_{\mathrm{p}}=\frac{1}{2F}\left(\frac{b_2}{a_2}-\frac{b_4}{a_4}\right);{C}_{\mathrm{p}}=2F\frac{b_3}{b_1}\frac{1}{\left(\frac{b_2}{a_2}-\frac{b_4}{a_4}\right)};\\ {}\sigma =\frac{1}{2F}\frac{b_1}{a_2};{b}_1=\frac{a_2{b}_3}{a_4}\end{array}} $$

(6.29)

p. 157, Fig. 7.1, there is: “continuous line Langmuir isotherm, g = 1”

should be: “continuous line Langmuir isotherm, g = 0”

Corrected figure:

p. 160, Eqs. (7.19)–(7.20)

$$ {\displaystyle \begin{array}{c}{k}_3={k}_3^0{\Gamma}_{\infty}^2\\ {}{k}_{-3}={k}_{-3}^0{\Gamma}_{\infty}^2{a}_{{\mathrm{H}}_2}^{\ast}\end{array}} $$

(7.21)

p. 161, Eq. (7.23)

$$ {v}_2={v}_2^0\left[\begin{array}{l}\left(\frac{\theta_{\mathrm{H}}}{\theta_{\mathrm{H}}^{\ast }}\right)\left(\frac{a_{{\mathrm{H}}_2\mathrm{O}}}{a_{{\mathrm{H}}_2\mathrm{O}}^{\ast }}\right)\exp \left(-{\beta}_2f\;\eta \right)\\ {}-\left(\frac{1-{\theta}_{\mathrm{H}}}{1-{\theta}_{\mathrm{H}}^{\ast }}\right)\left(\frac{a_{{\mathrm{H}}_2}}{a_{{\mathrm{H}}_2}^{\ast }}\right)\left(\frac{a_{{\mathrm{OH}}^{-}}}{a_{{\mathrm{OH}}^{-}}^{\ast }}\right)\exp \left[\left(1-{\beta}_2\right)f\;\eta \right]\end{array}\right] $$

(7.23)

p. 161, Eq. (7.27)

$$ {v}_3^0=\frac{k_3^0{k}_{-3}^0{\Gamma}_{\infty}^2{a}_{{\mathrm{H}}_2}^{\ast }}{{\left(\sqrt{k_3^0}+\sqrt{k_{-3}^0{a}_{{\mathrm{H}}_2}^{\ast }}\right)}^2}=\frac{k_3{k}_{-3}}{{\left(\sqrt{k_3}+\sqrt{k_{-3}}\right)}^2} $$

(7.27)

p. 161, Eq, (7.30)

$$ \frac{k_1{k}_2}{k_{-1}{k}_{-2}}=\frac{k_1^2{k}_3}{k_{-1}^2{k}_{-3}}=\frac{k_{-2}^2{k}_3}{k_2^2{k}_{-3}}=1 $$

(7.30)

p. 193, Eq. (8.28)

$$ \hat{C}=\frac{1}{j\omega \left({\hat{Z}}_{\mathrm{tot}}-{R}_{\mathrm{s}}\right)}{C}_{\mathrm{d}1}+\frac{1}{\frac{1}{C_{\mathrm{ad}}}+{\sigma}_{\mathrm{ad}}\sqrt{j\omega}+{R}_{\mathrm{ad}}} $$

(8.28)

p. 205, l. 8

$$ {\hat{\mathrm{Z}}}_s=1/\left( j\omega {C}_{dl}2\pi rl\right) $$

p. 207, Fig. 9.3, legend.

$$ {C}_{\mathrm{dl}}=20\;\upmu \mathrm{F}\;{\mathrm{cm}}^{-2} $$

p. 338

There is:

$$ \mathrm{Hg}\mid {\mathrm{Hg}}_2{\mathrm{SO}}_4\mid {{\mathrm{SO}}_2}^{2\hbox{-} } $$

Should be:

$$ \mathrm{Hg}\mid {\mathrm{Hg}}_2{\mathrm{SO}}_4\mid {{\mathrm{SO}}_4}^{2\hbox{-} } $$