Surface complexation models are very successful in describing acid-base properties of minerals in electrolyte solutions. Among these models, the constant capacitance model (CCM) was originally restricted to high ionic strength as a limiting case of a more comprehensive framework. More recently it has been applied to lower ionic strength as well. The present paper shows that a high Delta pK(a) along with a relatively low capacitance value allows the description of the nonlinear charging curves obtained at low ionic strength. However, it becomes evident that the capacitance values are in those cases not in agreement with the more comprehensive picture of the solid-electrolyte interface. With the constant capacitance in the CCM taken as the overall capacitance in that picture it is concluded that the capacitance values obtained in this study from fitting literature surface charge data as well as many published CCM capacitance values are too high to be physically reasonable. Even for situations promoting lower overall capacitance values it had to be concluded that the CCM should be considered as a data fitting model without mechanistic background. Experimentally, the presence of a diffuse layer is at low ionic strength clearly demonstrated in published force measurements between mineral surfaces. In rare cases such measurements indicate that even at 0.1 M the use of a CCM is not at all in agreement with experimental evidence. An exception may be cases where linear charging curves are experimentally found, since such examples (typically found at high ionic strength) may a priori justify the use of a constant capacitance. This exception is allowed accepting linear surface potential-pH relationships reported in the literature.