The Parallel-Plate Model of Frumkin

The Frumkin isotherm is based on the assumption that at any value of the coverage, the mterface can be viewed as two capacitors connected in parallel. It follows immediately from this assumption that the charge ge corresponding to a given value of 0 can be written as  

The derivation of the Frumkin isotherm is rather involved and is not given here. We note only that it is based on calculating the difference in electrostatic energy of charging the double-layer capacitor with and without the adsorbed species. The final result is as follows  

All the quantities that appear in the exponent of Eq. (13.23) can be measured, at least in principle, thus permitting the isotherm to be tested by comparison with experiment. This is a great advantage of any theory, inasmuch as it does not contain any adjustable parameters. The disadvantage of the Frumkin model is that it makes no attempt to explain the observed phenomena on the molecular level. Thus, the values of (Co—Cl) and Fn are taken as such it is not explained why they have their observed values or how these values depend on molecular size, on the orientation of the molecules at the interface and on the nature of the interactions with the surface. 

The assumption underlying the derivation of the Frumkin isotherm is tantamount to assuming that the surface charge density is a linear function of coverage at constant potential, as seen in Eq. (13.22). This is by no means generally correct, although itmay constitute a good approximation in many cases. 

An approximate form of Eq. (13.24) is sometimes used to determine coverage on solid electrodes. Differentiating the charge in Eq. (13.22) with respect to potential, one has 

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