W C Maskell
where p A .pB and p c are the partial pressures of A. B and C respectively adjacent to the electrode surface; p i , p i and p & refer to pressures in the bulk; 8,is fractional surface coverage by vacancies; k , , k3 and k5 are the mass transfer coefficients between the bulk gas and boundary layer; k,. kBand kc are the adsorption rate constants for the subscript species while k2,k4 and kg are the desorption rate constants: kR and kD are the forward and reverse rate constants for the surface reaction.
The difficulty of the task is now apparent in this considerably simplified model. For example, only reactions between adsorbed gas atoms or molecules is considered whereas reaction between an adsorbed species and a gaseous species also is also possible. Further, only chemical reaction between A and B is allowed for and electrochemical redox reactions are ignored. Notwithstanding this, the analysis can provide a valuable insight into the processes occurring at the electrode-electrolyte interface.
The rate constants in equations (21)-(26) are related by the equilibrium constant, Keq.for the reaction
In principle, at this stage, if the rate constants, mass transfer coefficients and the equilibrium constant are known then the surface coverages BA, Os and Bc may be determined for given bulk gas pressures. These theta values then determine the electrochemical potential of the electrode or the sensor EMF,as shown below. In the relatively simple scheme being considered the observed voltage is a consequence of the following two reactions
where prepresents a vacant surface site. The observed voltage is a mixed potential, as shown in figure 3. The anodic current of one reaction is equal in magnitude to the cathodic current of the other so that the net current is zero. In the special case where the reactants A and B are in thermodynamic equilibrium with the product then E , =E: =Emlx An. alternative and equivalent scheme is to invoke reaction rate theory (Anderson and Graves
Figure 3. Formation of a mixed potential betkeen reactions (28) and (29) at E m l x e d where the net current IS zero, i.e. i28 + i29 =o.
1981): which leads to an expression for the mixed potential y :
where U' and A , are the surface concentrations of V i ' and A; respectively: taking these as identical on the sample and reference sides of the electrolyte the potential of the reference y' is given by
and the cell EMF is AV=p-v'
2 . 4 . 2 . Application of the model to oxygert sensing. Combustion systems and in particular the exhaust of internal combustion engines contain a variety of hydrocarbons (Fleming 1977). CO and H 2 comprise less than a few per cent of the total exhaust gas but they are ten times more abundant than all other hydrocarbons. Fleming gives a number of reasons why CO? rather than H2, has the principal influence on cell EMF. Also Haaland (1980) found H 2 to be very readily oxidised with evidence of substantial homogeneous oxidation at 700 O C . Thus most work has considered the reaction
C O + to2 f CO: to be the most important in determining the EMF of oxygen
sensors when exposed to reactive gases.
techniques to solve equations (2 1)-(32) involving substituting selected values for the unknown rate constants. In all cases k9 and k j o were set equal to zero on the basis of insufficient knowledge of their relative values: Heyne (1978) in a much simpler treatment also did this to facilitate the analysis. In effect it is equivalent to ignoring reaction (29) so that the potential 9 is no longer a mixed potential but is controlled totally by the oxygen reaction (28). This might be expected to lead to considerable errors for 0 2 / C 0 ratios of less than 0.5 where reaction (29) should be dominant. Hence, the predicted characteristics in the rich region must be viewed with caution when employing this assumption.
Some of the results of the numerical analysis of Anderson and Graves are presented below. Figure 4 shows the situation where mass transport is fast. As expected (curve A), when there is negligible surface reaction the sensor detects the total bulk oxygen and does not respond to changes in the 0 2 / C 0 ratio: this is the equivalent of sensing C H 4 / 0 2 on Ag. In curve C the surface reaction is fast and the behaviour is similar to the theoretical curve D, which corresponds to thermodynamic equilibrium.
Figure 5 shows the influence of mass transfer rate. In curve A mass transfer is fast but different adsorption rates of CO and O2 are assumed: the deviation from the theoretical equilibrium curve is substantial as was predicted by Heyne's analysis (1978). Curves B and C represent progressively slower mass transfer and the result is now close to the theoretical equilibrium curve. This illustrates very well the important finding that for gases with similar transfer coefficients (such as 0: and CO) the position of the potential step should correspond reasonably well with the theoretical equilibrium result, provided mass transfer rates are slow.
However, the analysis predicts substantial deviations from the theoretical equilibrium curve when mass transfer is slow and