Results

The measured solution conductivities $\sigma_{\mbox{\small exp}}$_exp are shown in Table 2, along with the estimated conductivities $\sigma_{\mbox{\small calc}}$_calc calculated from Eqn. 2. Within the table, the results are divided among the three molar ratios. Also shown in Table 2 are the coefficients of variation $\eta$:

$$\eta = \frac{\sigma_{\mbox{\small calc}} - \sigma_{\mbox{\small exp}}}{\sigma_{\mbox{\small exp}}}$$ (8)

Because the coefficients G_i were not optimized for these three ionic species, all the estimated values lie below the measured values. Optimizing the G may not be warranted because the empirical relation in Eqn. 3 is a coarse approximation. Moreover, the present error is already less than 8 % over the entire concentration range.

Table: 2 Measured solution conductivities $\sigma_{\mbox{\small exp}}$_exp, calculated solution conductivities $\sigma_{\mbox{\small calc}}$_calc, and the coefficient of variation $\eta$. The uncertainty in $\sigma_{\mbox{\small exp}}$_exp is approximately 0.2 % (see text).

[K$^+$](mol/l)	[Na$^+$] (mol/l)	$\sigma_{\mbox{\small exp}}$exp (S/m)	$\sigma_{\mbox{\small calc}}$calc (S/m)	$\eta$
0.125	0.03125	3.707	3.591	-0.031
0.250	0.06250	7.133	6.796	-0.047
0.500	0.12500	13.56	12.64	-0.068
1.000	0.25000	24.78	22.99	-0.072
0.125	0.0625	4.346	4.201	-0.033
0.250	0.1250	8.330	7.913	-0.050
0.500	0.2500	15.60	14.63	-0.062
1.000	0.5000	27.61	26.45	-0.042
0.125	0.125	5.642	5.387	-0.045
0.250	0.250	10.70	10.07	-0.059
0.500	0.500	19.57	18.45	-0.057
1.000	1.000	33.50	33.02	-0.014

The performance of Eqn. 2 is relatively uniform over the range of ionic strengths investigated. The data from Table 2 are plotted in Fig. 2(a) (filled symbols) as a function of the solution ionic strength I_M. The predictions from Eqn. 2 are shown as solid curves, one for each of the potassium to sodium ratios. The three curves are nearly collinear, as are the measured values.

For comparison purposes, also shown in Fig. 2(a) are estimates that neglect the concentration dependence of the equivalent conductivity:

$$\lambda_i = \lambda^o_i$$ (9)

Figure 2: Measured and predicted solution conductivity $\sigma$ as a function of molar ionic strength I_M: (a) Comparison among measured conductivity (filled symbols), estimated conductivity (solid curve) from Eqn. 3, and estimated conductivity (dashed curve) from Eqn. 9; and (b) Coefficient of variation $\eta$ between estimated conductivity and measured conductivity.

This idealized approximation is mathematically equivalent to setting all the G_i coefficients to zero in Eqn. 3. These approximations are shown as dashed curves in Fig. 2(a), one curve for each of the three potassium to sodium ratios. At an ionic strength of 1.0 mol/L, the idealized approximation differs by 36 % from the experimental data, and differs by 55 % at 2.0 mol/L.

The graph of the coefficient of variation $\eta$ plotted in Fig. 2(b) shows that the relative error is fairly constant over the entire range of ionic strengths. This suggests that small extrapolations of either the ratios or concentrations beyond the parameter space investigated here should not introduce extensive uncertainty.