Tuesday, October 18, 2022

A model for the diffusion coefficient for dissolved oxygen in NaCl solutions

From Fick’s second law of diffusion1, the diffusion coefficient is defined as the proportionality constant between the molar flux of the diffusing species and the concentration gradient, as shown below, \begin{equation} \label{eq:eq18} J=-D \nabla C \end{equation} where, $$\begin{eqnarray} J \equiv flux \nonumber \\ \nabla c \equiv concentration\;gradient \nonumber \\ D \equiv diffusion\;coefficient \nonumber \end{eqnarray}$$ In the Einstein relation, the diffusion coefficient of a particle undergoing Brownian motion is given by, \begin{equation} \label{eq:eq19} D=\frac{k_B T}{\xi} \end{equation} ξ is defined as a friction factor of the diffusing particles, k_B is Boltzmann’s constant, and T is the absolute temperature. For diffusion of spherical particles through a liquid, equation (B2) can be rewritten as the Stokes-Einstein equation2, (B3). D=(k_B T)/6πηr (B3) The friction factor has now been replaced by the fluid viscosity, η, and a relationship with the particle radius, r. For small solute molecules in water, Wilke and Chang3 proposed an empirical change to equation (B3), resulting in equation (B4). D=K (T√ϕM)/(ηV^0.6 ) (B4) Where K is a fitting coefficient, ϕ is an empirical parameter to account for association between solvent molecules, Mis the molecular mass of the solvent, and V is the molar volume of the solute. For water, the best agreement with diffusion coefficients suggested that ϕ=2.6. While there is a temperature dependence for the diffusion coefficient included in equation (B4), the viscosity of the solution also changes as a function of temperature. This temperature dependence has two principal effects4. The first effect can be captured with an Arrhenius expression, equation (B5). η_0=C_1 e^(C_2/T ) (B5) C_1 and C_2 are fitting coefficients. The second effect occurs because adding solute molecules to a solvent can change the solvent structure and thereby influence its viscosity. From, Jones and Dole5, the effect of the solute on viscosity can be estimated using equation (B6). η_1=1+A'√c+B'c (B6) A' is a parameter to estimate ion-ion interactions6 and B', known as the Jones-Dole B coefficient7, captures solvent-solute interactions. The viscosity of the solvent is then estimated from equation (B7). η=η_0 η_1 (B7) In this approach, then, η_0 from equation (B5) must be unitless, with η_1in units of mPa s. Using equations (B4), (B5), (B6), and (B7), and fitting to diffusion coefficient data obtained from the literature8, results in the plot shown in Figure B1. Figure B1. Plots of dissolved O2 diffusion coefficient data at various temperatures and chloride concentrations (symbols) overlaid with estimated diffusion coefficients (solid lines). The temperature dependence of each of the parameters from equations (B4) - (B7) can then be estimated by fitting them to a Padé approximant model, as given in equation (B7). p_i=(b_1+b_2 x)/(1.0+b_3 x) (B7) p_i indicates the relevant parameter and b_iare the fitting coefficients. The temperature-dependent model fits for these parameters are shown in Figure B2. a. b. c. d. Figure B2. Plots of the temperature dependence of parameters a. K, b.A', c.B', and d. C_1 and C_2. Thus, the diffusion coefficient for dissolved O2 can be calculated for other temperatures and chloride concentrations. An estimate of the diffusion coefficient, compared with several literature values9, is shown in Figure B3. Figure B3. Plot of the modeled dissolved O2 diffusion coefficient at T = 25oC for a range of Cl- concentrations from 0.0 M to 5.0 M NaCl (solid line) compared with experimental data (+ symbols). 


  1. Aifantis, E. C., On the problem of diffusion in solids. Acta Mechanica 1980, 37 (3), 265-296.
  2. Einstein, A., Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 1905, 322 (8), 549-560.
  3. Wilke, C. R.; Chang, P., Correlation of diffusion coefficients in dilute solutions. AIChE Journal 1955, 1 (2), 264-270.
  4. Reinsberg, P. H.; Bawol, P. P.; Thome, E.; Baltruschat, H., Fast and Simultaneous Determination of Gas Diffusivities and Solubilities in Liquids Employing a Thin-Layer Cell Coupled to a Mass Spectrometer, Part II: Proof of Concept and Experimental Results. Analytical Chemistry 2018, 90 (24), 14150-14155.
  5. Jones, G.; Dole, M., THE VISCOSITY OF AQUEOUS SOLUTIONS OF STRONG ELECTROLYTES WITH SPECIAL REFERENCE TO BARIUM CHLORIDE. Journal of the American Chemical Society 1929, 51 (10), 2950-2964.
  6. Falkenhagen, H.; Vernon, E. L., LXII. The viscosity of strong electrolyte solutions according to electrostatic theory. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 1932, 14 (92), 537-565.
  7. Jenkins, H. D. B.; Marcus, Y., Viscosity B-Coefficients of Ions in Solution. Chemical Reviews 1995, 95 (8), 2695-2724.
  8. van Stroe, A. J.; Janssen, L. J. J., Determination of the diffusion coefficient of oxygen in sodium chloride solutions with a transient pulse technique. Analytica Chimica Acta 1993, 279 (2), 213-219.
  9. Ho, C. S.; Ju, L.-K.; Baddour, R. F.; Wang, D. I. C., Simultaneous measurement of oxygen diffusion coefficients and solubilities in electrolyte solutions with a polarographic oxygen electrode. Chemical Engineering Science 1988, 43 (11), 3093-3107.

Sunday, October 16, 2022

Derivation of the mixed activation and transport-limited governing equations for first order electrochemical reactions

It took me awhile to work out all the steps to deriving the Koutecky-Levich equation so I wanted to save this derivation where it would be easily accessible. The general form of the Butler-Volmer equation for outer-sphere electron transfer reactions is given in the following: \begin{equation}\label{eq:eq1} i=i_{0} \left\{ \frac{c_{a}(0,t)}{c_(a,b)} e^{\frac{\alpha_{a}zF}{RT}}- \frac{c_c(0,t)}{c_(c,b)}e^{\frac{-\alpha_c zF}{RT}}\right\} ^{(a)} \end{equation} where \begin{equation} c_i(0,t) \end{equation} indicates the concentration of the reactant at the electrolyte-electrode interface and \begin{equation} c_{(i,b)} \end{equation} indicates the bulk concentration of the reactant. For concentration-gradient mediated transport to the electrolyte-electrode interface, the flux, j, of the reacting species in the electrolyte is given by, \begin{equation} \label{eq:eq2} j=-\frac{i}{zF} \end{equation} Then, from Fick’s first law, \begin{equation} \label{eq:eq3} |i|=zFD \frac{\partial c}{\partial x} \end{equation} And, using the first two terms in the Taylor series expansion for the concentration, we can rewrite Fick’s first law as, \begin{equation} \label{eq:eq4} |i|=zFD \frac{c_{c,b}- c_c(0,t)}{x_b-x_0}^{(b)} \end{equation} where we assign the distance, \begin{equation} x_b-x_0 = \delta \end{equation}. In the limit that, \begin{equation} c_c (0,t)=0 \end{equation}, * becomes, \begin{equation} \label{eq:eq5} |i_L | = zFD \frac{c_{c,b}}{\delta} ^{(c)} \end{equation} And, dividing equation b by c results in, \begin{equation} \label{eq:eq6} \frac{c_c (0,t)}{c_b} = 1-\frac{|i|}{|i_L |} ^{(d)} \end{equation} Focusing on just the cathodic term from a, and substituting d, results in the following, \begin{equation} \label{eq:eq7} i= -i_0\left\{\left ( 1-\frac{|i|}{|i_L | } \right ) e^{\frac{-\alpha _c zF}{RT}} \right\} \end{equation} This equation is transcendental in i, but an analytical solution can be obtained, as follows: \begin{equation} \label{eq:eq8} i = \left\{\left ( 1-\frac{|i|}{|i_L|} \right ) i_{act}\right\} \end{equation} \begin{equation} \label{eq:eq9} i_{act} = -i_0 e^{\frac{-\alpha_c zF}{RT}} (e) \end{equation} \begin{equation} \label{eq:eq10} 0 = i_{act} -\frac{|i|i_{act}}{|i_L |} -|i| \end{equation} \begin{equation} \label{eq:eq11} 0=|i|\left [\frac{i_{act}}{|i|} - \frac{i_{act}}{|i_L|}-1 \right ] \end{equation} \begin{equation} \label{eq:eq12} 0=\frac{i_{act}}{|i|} - \frac{i_{act}}{|i_L|}-1 \end{equation} \begin{equation} \label{eq:eq13} 1=\frac{i_{act}}{|i|} - \frac{i_{act}}{|i_L|} \end{equation} \begin{equation} \label{eq:eq14} \frac{1}{i_{act}} = \frac{1}{|i|} - \frac{1}{|i_L|} \end{equation} \begin{equation} \label{eq:eq15} \frac{1}{i_{act}} + \frac{1}{|i_L|} = \frac{1}{|i|} \end{equation} \begin{equation} \label{eq:eq16} \left [ \frac{1}{i_{act}} + \frac{1}{|i_L|} \right ]^{-1}=|i| \end{equation} \begin{equation} \label{eq:eq17} |i|=\frac{|i_{act} ||i_L |}{|i_{act} |+|i_L |} ^{(f)} \end{equation} Where f is the Koutecky-Levich equation and b, e, and f provide expressions that can be used to calculate and model transport-limited kinetics, activation kinetics, and mixed kinetics, respectively.

A model for the diffusion coefficient for dissolved oxygen in NaCl solutions

From Fick’s second law of diffusion 1 , the diffusion coefficient is defined as the proportionality constant between the molar flux of the d...