On the Lark-Horovitz equation for ion selective membrane electrodes and its derivation

Abstract

The principles of non-equilibrium thermodynamics are used to allow for entropy production due to the current transfer across regions of varying composition in electrochemical cells containing a membrane between electrolyte solutions and, optionally, a liquid junction. By this approach, a general equation is derived for the response of a cell as a whole, rather than for separate contributions to it. The equation is applied to cells with solid-electrolyte membranes of different types, as exemplified by alkali aluminosilicate glasses responsive to alkali metal cations, the pre-conditioned silicate glasses responsive to hydrogen ions, and lanthanum trifluoride responding to fluoride ions. Among the more significant results of the applications is a generalized form of the Lark-Horovitz equation for two ionic charge carriers in a sublattice of opposite charge sign. However, its derivation is circumscribed by the conditions of a uniform charge density of the sublattice and equal charge numbers and coordination numbers of the carriers. The degree to which these conditions are met by specific membranes is discussed from the structural chemistry point of view. As a special feature of hydrogen ion-responsive silicate glasses accounting for the extremely fast response, the concept of a pH-sensitive silica network is introduced on the basis of the previously found interrelationships between the bond lengths Si–OH, Si–OSi, and Si–NBO in hydrous alkali silicates, where NBO means a non-bridging oxygen coordinated by hydrogen bonds and/or alkali cations.

Appendices

Appendices

Appendix A

On substituting I _M from Eq. 34 into Eq. 30, it follows

$$ -\frac{F\mathcal{E}}{RT}-{\mathcal{I}}_{\mathcal{L}}={z}_{\mathrm{X}}^{-1}\left(\ln \frac{m_{\mathrm{X}}^{\upbeta}}{m_{\mathrm{X}}^{\upeta}}+\ln \frac{\gamma_{\mathrm{X}}^{\upbeta}}{\gamma_{\mathrm{X}}^{\upeta}}\right)+{z}_{\mathrm{A}}^{-1}\ln \frac{\gamma_{\mathrm{A}}^{\updelta}}{\gamma_{\mathrm{A}}^{\upbeta}}+{z}_{\mathrm{M}}^{-1}\left(\ln \frac{m_{\mathrm{M}}^{\upeta}}{m_{\mathrm{M}}^{\updelta}}+\ln \frac{\gamma_{\mathrm{M}}^{\upeta}}{\gamma_{\mathrm{M}}^{\updelta}}\right)={\left({z}_{\mathrm{X}}^{-1}\ln \frac{m_{\mathrm{X}}^{\upbeta}}{m_{\mathrm{X}}^{\upeta}}+{z}_{\mathrm{M}}^{-1}\ln \frac{m_{\mathrm{M}}^{\upeta}}{m_{\mathrm{M}}^{\updelta}}\right)}_1+{\left({z}_{\mathrm{A}}^{-1}\ln \frac{\gamma_{\mathrm{A}}^{\updelta}}{\gamma_{\mathrm{A}}^{\upbeta}}+{z}_{\mathrm{X}}^{-1}\ln \frac{\gamma_{\mathrm{X}}^{\upbeta}}{\gamma_{\mathrm{X}}^{\upeta}}+{z}_{\mathrm{M}}^{-1}\ln \frac{\gamma_{\mathrm{M}}^{\upeta}}{\gamma_{\mathrm{M}}^{\updelta}}\right)}_2 $$

(A1)

Let us introduce stoichiometric numbers for electroneutral combinations of ions, denoted by + and – in subscript:

$$ {\nu}_{+}{z}_{+}+{\nu}_{-}{z}_{-}=0 $$

(A2)

Let us also introduce the mean activity coefficients defined as

$$ \nu \kern0.2em \ln {\gamma}_{\pm }=\left({\nu}_{+}+{\nu}_{-}\right)\ln {\gamma}_{\pm }={\nu}_{+}\ln {\gamma}_{+}+{\nu}_{-}\ln {\gamma}_{-} $$

(A3)

Note that this familiar expression can be transformed with the help of Eq. A2 to

$$ \left({z}_{+}^{-1}-{z}_{-}^{-1}\right)\ln {\gamma}_{\pm }={z}_{+}^{-1}\ln {\gamma}_{+}-{z}_{-}^{-1}\ln {\gamma}_{-} $$

(A4)

Hence, the content of the round brackets indexed by 2 in Eq. A1 transforms as follows

$$ {\left(\right)}_2={z}_{\mathrm{X}}^{-1}\ln {\gamma}_{\mathrm{X}}^{\upbeta}-{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\upbeta}+\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{X}}^{\upeta}-\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{A}}^{\updelta} $$

(A5)

The remaining single ion activity coefficients are those of ions of the same charge sign in the solution β. Suppose that they are counterbalanced by ions K in that solution. Then Eq. A5 can be transformed further

$$ {\left(\right)}_2={z}_{\mathrm{X}}^{-1}\ln {\gamma}_{\mathrm{X}}^{\upbeta}-{z}_{\mathrm{K}}^{-1}\ln {\gamma}_{\mathrm{K}}^{\upbeta}+{z}_{\mathrm{K}}^{-1}\ln {\gamma}_{\mathrm{K}}^{\upbeta}-{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\upbeta}+\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{X}}^{\upeta}-\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{A}}^{\updelta}=-\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{X}}^{\upbeta}+\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{A}}^{\upbeta}+\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{X}}^{\upeta}-\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{A}}^{\updelta} $$

(A6)

On substituting this content back into Eq. A1 and doing some additional manipulations, we obtain

$$ \mathcal{E}=\frac{RT}{F}\left({z}_{\mathrm{M}}^{-1}\ln \frac{m_{\mathrm{M}}^{\updelta}}{m^{\ominus }}+\frac{\nu_{\mathrm{M}\mathrm{A}}}{z_{\mathrm{M}}{\nu}_{\mathrm{M}}}\ln {\gamma}_{\mathrm{M}\mathrm{A}}^{\updelta}\right)+{\mathcal{E}}_0-\frac{RT}{F}{\mathcal{I}}_{\mathcal{L}} $$

(A7)

where

$$ {\mathcal{E}}_0=\frac{RT}{F}\left[{z}_{\mathrm{M}}^{-1}\ln \frac{m^{\ominus }}{m_{\mathrm{M}}^{\upeta}}+{z}_{\mathrm{X}}^{-1}\ln \frac{m_{\mathrm{X}}^{\upeta}}{m_{\mathrm{X}}^{\upbeta}}+\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{X}}^{\upbeta}-\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{A}}^{\upbeta}-\left({z}_{\mathrm{M}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{M}\mathrm{X}}^{\upeta}\right] $$

(A8)

Equation A7 is Eq. 35 in the main text.

Appendix B

On introducing a temporary notation r = u _L/u _M and using the notations of Eqs. 47 and 48, we have for the transport numbers by Eq. 43

$$ {t}_{\mathrm{M}}=\frac{c_{\mathrm{M}}}{c_{\mathrm{M}}+r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)};\kern3em {t}_{\mathrm{L}}=1-{t}_{\mathrm{M}}=\frac{r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)}{c_{\mathrm{M}}+r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)} $$

(B1)

Then Eq. 41 becomes

$$ {\displaystyle \begin{array}{l}{\mathcal{I}}_{\mathcal{M}}=\kern0.5em {z}^{-1}\int \kern0.66em \left[\frac{c_{\mathrm{M}}\mathrm{d}\kern0.1em \ln {c}_{\mathrm{M}}}{c_{\mathrm{M}}+r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)}+\frac{r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)\mathrm{d}\kern0.1em \ln \left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)}{c_{\mathrm{M}}+r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)}\right]\\ {}={z}^{-1}\int \kern0.36em \frac{\mathrm{d}\kern0.1em \left({c}_{\mathrm{M}}+r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)\right)}{c_{\mathrm{M}}+r\left({c}_{\mathrm{T}}-{c}_{\mathrm{M}}\right)}\\ {}={z}^{-1}\ln \frac{{\left({c}_{\mathrm{M}}+{rc}_{\mathrm{L}}\right)}^{\upvarepsilon \upeta}}{{\left({c}_{\mathrm{M}}+{rc}_{\mathrm{L}}\right)}^{\updelta \upvarepsilon}}\end{array}} $$

(B2)

Using Eqs. 24 and 49, the conditions of phase equilibria at the membrane boundaries can be written as

$$ {\mu}_i^{\ominus }+ RT\;\ln \left({\gamma}_i{m}_i/{m}^{\ominus}\right)={\mu}_{i\kern0.2em \left(\mathrm{R}\right)}^{\ast }+ RT\;\ln \left({c}_i/{c}_{\mathrm{T}}\right) $$

(B3)

From this, it follows

$$ {c}_i={k}_i{\gamma}_i{m}_i{c}_{\mathrm{T}}/{m}^{\ominus } $$

(B4)

where introduced is a temporary notation

$$ {k}_i=\exp \left[\left({\mu}_i^{\ominus }-{\mu}_{i\kern0.2em \left(\mathrm{R}\right)}^{\ast}\right)/ RT\right] $$

(B5)

Substituting c _i of Eq. B4 into Eq. B2 gives

$$ {\mathcal{I}}_{\mathcal{M}}=\kern0.5em {z}^{-1}\ln \frac{{\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\upeta}}{{\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\updelta}} $$

(B5)

where B is another temporary notation

$$ B=r\kern0.1em {k}_{\mathrm{L}}/{k}_{\mathrm{M}} $$

(B6)

Now we substitute I _M of Eq. B5 into Eq. 30 and proceed to transform it:

$$ {\displaystyle \begin{array}{l}-\frac{F\mathcal{E}}{RT}-{\mathcal{I}}_{\mathcal{L}}={z}_{\mathrm{X}}^{-1}\left(\ln \frac{m_{\mathrm{X}}^{\upbeta}}{m_{\mathrm{X}}^{\upeta}}+\ln \frac{\gamma_{\mathrm{X}}^{\upbeta}}{\gamma_{\mathrm{X}}^{\upeta}}\right)+{z}_{\mathrm{A}}^{-1}\ln \frac{\gamma_{\mathrm{A}}^{\updelta}}{\gamma_{\mathrm{A}}^{\upbeta}}+{z}_{\mathrm{M}}^{-1}\ln \frac{{\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\upeta}}{{\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\updelta}}\\ {}={z}_{\mathrm{X}}^{-1}\left(\ln {m}_{\mathrm{X}}^{\upbeta}-\ln {m}_{\mathrm{X}}^{\upeta}\right)+{\left({z}_{\mathrm{X}}^{-1}\ln {\gamma}_{\mathrm{X}}^{\upbeta}-{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\upbeta}\right)}_1-{\left[{z}_{\mathrm{X}}^{-1}\ln {\gamma}_{\mathrm{X}}^{\upeta}-{z}_{\mathrm{M}}^{-1}\ln {\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\upeta}\right]}_2\\ {}+{\left[{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\updelta}-{z}_{\mathrm{M}}^{-1}\ln {\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\updelta}\right]}_3\end{array}} $$

(B7)

The content of the brackets indexed by 1 is transformed in the same way as in Appendix A, Eq. A6

$$ {\displaystyle \begin{array}{l}{\left(\right)}_1={z}_{\mathrm{X}}^{-1}\ln {\gamma}_{\mathrm{X}}^{\upbeta}-{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\upbeta}\\ {}={z}_{\mathrm{X}}^{-1}\ln {\gamma}_{\mathrm{X}}^{\upbeta}-{z}_{\mathrm{K}}^{-1}\ln {\gamma}_{\mathrm{K}}^{\upbeta}+{z}_{\mathrm{K}}^{-1}\ln {\gamma}_{\mathrm{K}}^{\upbeta}-{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\upbeta}\\ {}=-\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{X}}^{\upbeta}+\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{A}}^{\upbeta}\end{array}} $$

(B8)

The contents of brackets indexed by 2 and 3 are similar. Here is the transformation of the brackets 3:

$$ {\displaystyle \begin{array}{l}{\left(\right)}_3={z}_{\mathrm{M}}^{-1}\left[{z}_{\mathrm{M}}{z}_{\mathrm{A}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\updelta}-\ln {\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\updelta}\right]\\ {}=-{z}_{\mathrm{M}}^{-1}\left[{\nu}_{\mathrm{A}}{\nu}_{\mathrm{M}}^{-1}\ln {\gamma}_{\mathrm{A}}^{\updelta}+\ln {\left({\gamma}_{\mathrm{M}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{m}_{\mathrm{L}}\right)}^{\updelta}\right]\\ {}=-{z}_{\mathrm{M}}^{-1}\ln {\left({\gamma}_{\mathrm{M}}{\gamma}_{\mathrm{A}}^{\nu_{\mathrm{A}}/{\nu}_{\mathrm{M}}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}}{\gamma}_{\mathrm{A}}^{\nu_{\mathrm{A}}/{\nu}_{\mathrm{M}}}{m}_{\mathrm{L}}\right)}^{\updelta}\\ {}=-{z}_{\mathrm{M}}^{-1}\ln {\left({\gamma}_{\mathrm{M}\mathrm{A}}^{\nu_{\mathrm{M}\mathrm{A}}/{\nu}_{\mathrm{M}}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}\mathrm{A}}^{\nu_{\mathrm{L}\mathrm{A}}/{\nu}_{\mathrm{L}}}{m}_{\mathrm{L}}\right)}^{\updelta}\end{array}} $$

(B9)

Note that ν _M = ν _L as follows from Eq. 47 in the main text.

On substituting the transformed brackets back into Eq. B7, we arrive at the following

$$ \mathcal{E}=\frac{RT}{z_{\mathrm{M}}F}\ln \frac{{\left({\gamma}_{\mathrm{M}\mathrm{A}}^{\nu_{\mathrm{M}\mathrm{A}}/{\nu}_{\mathrm{M}}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}\mathrm{A}}^{\nu_{\mathrm{L}\mathrm{A}}/{\nu}_{\mathrm{L}}}{m}_{\mathrm{L}}\right)}^{\updelta}}{{\left({\gamma}_{\mathrm{M}\mathrm{A}}^{\nu_{\mathrm{M}\mathrm{A}}/{\nu}_{\mathrm{M}}}{m}_{\mathrm{M}}+B{\gamma}_{\mathrm{L}\mathrm{A}}^{\nu_{\mathrm{L}\mathrm{A}}/{\nu}_{\mathrm{L}}}{m}_{\mathrm{L}}\right)}^{\upeta}}+{\mathcal{E}}_0-\frac{RT}{F}{\mathcal{I}}_{\mathcal{L}} $$

(B10)

where

$$ {\mathcal{E}}_0=\frac{RT}{F}\left[{z}_{\mathrm{X}}^{-1}\ln \frac{m_{\mathrm{X}}^{\upeta}}{m_{\mathrm{X}}^{\upbeta}}+\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{X}}^{\upbeta}+\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{A}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{A}}^{\upbeta}\right] $$

(B11)

Equation B10 is Eq. 50 of the main text.

Appendix C

On substituting I _M from Eq. 54 into Eq. 30, it follows

$$ -\frac{F\mathcal{E}}{RT}-{\mathcal{I}}_{\mathcal{L}}={z}_{\mathrm{X}}^{-1}\left(\ln \frac{m_{\mathrm{X}}^{\upbeta}}{m_{\mathrm{X}}^{\upeta}}+\ln \frac{\gamma_{\mathrm{X}}^{\upbeta}}{\gamma_{\mathrm{X}}^{\upeta}}\right)+{z}_{\mathrm{A}}^{-1}\ln \frac{\gamma_{\mathrm{A}}^{\updelta}}{\gamma_{\mathrm{A}}^{\upbeta}}+{z}_{\mathrm{Y}}^{-1}\left(\ln \frac{m_{\mathrm{Y}}^{\upeta}}{m_{\mathrm{Y}}^{\updelta}}+\ln \frac{\gamma_{\mathrm{Y}}^{\upeta}}{\gamma_{\mathrm{Y}}^{\updelta}}\right) $$

(C1)

Applying algebraic transformations similar to Appendix A gives (cf. Eq. A7)

$$ \mathcal{E}=\frac{RT}{F}\left({z}_{\mathrm{Y}}^{-1}\ln \frac{m_{\mathrm{Y}}^{\updelta}}{m^{\ominus }}+\frac{\nu_{\mathrm{Y}\mathrm{A}}}{z_{\mathrm{Y}}{\nu}_{\mathrm{Y}}}\ln {\gamma}_{\mathrm{Y}\mathrm{A}}^{\updelta}\right)+{\mathcal{E}}_0-\frac{RT}{F}{\mathcal{I}}_{\mathcal{L}} $$

(C2)

where

$$ {\mathcal{E}}_0=\frac{RT}{F}\left[{z}_{\mathrm{Y}}^{-1}\ln \frac{m^{\ominus }}{m_{\mathrm{Y}}^{\upeta}}+{z}_{\mathrm{X}}^{-1}\ln \frac{m_{\mathrm{X}}^{\upeta}}{m_{\mathrm{X}}^{\upbeta}}+\left({z}_{\mathrm{A}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{A}\mathrm{X}}^{\upbeta}-\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{X}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{X}}^{\upeta}+\left({z}_{\mathrm{K}}^{-1}-{z}_{\mathrm{Y}}^{-1}\right)\ln {\gamma}_{\mathrm{K}\mathrm{Y}}^{\upeta}\right] $$

(C3)

Equation C2 is Eq. 55 in the main text.