The moving boundary method

Abstract

The equations of the preceding section — no explicit considerations of the electrodes — have their most important application in conjunction with the moving boundary method which serves to measure transport numbers¹⁾. In a typical experiment we have two different cations and only one anion. In order to remain within the general framework of the main treatment we consider the special constituents Na, Li, and Cl. The replacement of K by Li is necessary for technical reasons as will be seen later. Now the set of eqs.(112)–(114) only contains two equations: the third one, eq. (114) is no longer necessary. Thus we have

$$\frac{{\partial {Q_s}}}{{\partial t}} = \mathop \Sigma \limits_{l = s,Na} {D_{sl}}\frac{{{\partial ^2}{Q_l}}}{{\partial {x^2}}} - \frac{{{j_2}}}{F}\left[ {({M_{Na}} + {M_{Cl}})\frac{{d{t_{Na}}}}{{dx}} + ({M_{Li}} + {M_{Cl}})\frac{{d{t_{Li}}}}{{dx}}} \right]$$

((125))

and

$$\begin{array}{l} \frac{{\partial {Q_{Na}}}}{{\partial t}} = \mathop \Sigma \limits_{l = s,Na} {D_{Nal}}\frac{{{\partial ^2}{Q_l}}}{{\partial {x^2}}} - \frac{{{j_2}}}{F}{M_{Na}}\frac{{d{t_{Na}}}}{{dx}}\\ \frac{{\partial {Q_W}}}{{\partial t}} = \mathop \Sigma \limits_{l = s,Na} {D_{Wl}}\frac{{{\partial ^2}{Q_W}}}{{\partial {x^2}}} \end{array}$$

((126))

where the concentration dependence of the diffusion coefficients has been neglected. The concentration distributions are sketched in Fig.7. The purpose of this arrangement is the measurement of the transport number of Na. Of course this is the transport number in the binary homogeneous electrolyte NaCl + water which we may denote by t_Na. To begin with, let us consider the first term on the right-hand side of eq.(126) which is − divj _Na .