Bayesian methods stem from the principle of linking prior probability and conditional probability (likelihood) to posterior probability via Bayes' rule. The posterior probability is an updated (improved) version of the prior probability of an event, through the likelihood of finding empirical evidence if the underlying assumptions (hypothesis) are valid. In the absence of a frequency distribution for the prior probability, Bayesian methods have been found more satisfactory than distribution-based techniques. The paper illustrates the utility of Bayes' rule in the analysis of electrocatalytic reactor performance by means of four numerical examples involving a catalytic oxygen cathode, hydrogen evolution on a synthetic metal, the reliability of a device testing the quality of an electrocatalyst, and the range of Tafel slopes exhibited by an electrocatalyst.

In a comprehensive overall treatment of the subject matter, Bockris and Khan [

This paper was written with this dichotomy in mind, from the vantage point of the electrochemical engineer, whose responsibilities dealing with production quota, the possibility of (temporary) reactor breakdown, safety, and environmental considerations reach well beyond purely scientific quantities. Major tools for dealing with these responsibilities are provided by probability-based (e.g., statistical) methods. Bayes’ rule is one such tool, whose specific application to scenarios with EC is the subject of this article.

Following a concise definition [

Specific exploratory applications to electrochemical processes and technology at various levels of complexity are relatively recent [

Table

Postulated distribution pattern of 137 failure occurrences over a fixed time period in three independently operating (hypothetical) electrolytic plants using identical catalytic oxygen cathodes (Application 1).

Source of failure | Number of cathode failures over a fixed period of operation | ||
---|---|---|---|

Plant 1 (_{1} | Plant 2 ( | Plant 3 ( | |

_{1}^{(1)} | 12 | 14 | 10 |

_{2}^{(2)} | 7 | 8 | 9 |

_{3}^{(3)} | 9 | 5 | 7 |

_{4}^{(4)} | 6 | 4 | 5 |

_{5}^{(5)} | 14 | 11 | 16 |

Total number of failures | 48 | 42 | 47 |

^{(1)} Electrolytic carbon type P33: specific surface area is less than stipulated 1000 m^{2}/g.

^{ (2)} Electrolytic carbon type P33: specific pore volume is less than stipulated 2.3 cm^{3}/g.

^{ (3)} PTFE binder content in electrode layer is less than stipulated 10%.

^{ (4)} CoTAA (dibenzotetraazaannulen cobalt II) catalyst content on carbon is less than stipulated 15%.

^{ (5)} Careless stack assemblage and general operation.

The complete set of probabilities computed via (

Source-of-failure events | _{j}_{k} | ||
---|---|---|---|

_{1} | 0.3333 | 0.3889 | 0.2778 |

_{2} | 0.2917 | 0.3333 | 0.3750 |

_{3} | 0.4286 | 0.2381 | 0.3333 |

_{4} | 0.4000 | 0.2667 | 0.3333 |

_{5} | 0.3415 | 0.2683 | 0.3902 |

A recently developed electrocatalyst for a hydrogen evolution process, made up of certain synthetic metals, is expected to possess an exchange current ^{2} at design operating conditions in a pilot scale electrolyzer. Inspection of Trasatti’s [^{2} electric charge per unit area, then there should exist an a priori chance that a catalyst-carrying electrode (CCE), selected randomly from a lot of identically prepared specimens, can sustain its catalytic activity, at an acceptable level, up to the passage of ^{2}. During the

The set of events of interest here, involving a randomly selected CCE, is defined as follows:

Consequently, the stipulations can be expressed in terms of their probabilities as follows:

The effect of prior probability

20 | 67.4 | 2.5 |

40 | 84.6 | 6.3 |

60 | 92.5 | 13.2 |

70 | 95.1 | 19.1 |

80 | 97.1 | 28.8 |

90 | 98.7 | 47.6 |

_{2}_{1}

_{1}_{1 }

A device for testing defects in a certain electrocatalyst (EC) is envisaged to be advertised by the catalyst producer, claiming that it is 97% reliable if the EC is defective, and 99% reliable when it is flawless. Independently from any testing device and based upon earlier experience, 4% of said EC may be expected to be defective upon delivery. In order to ascertain the true reliability of the device, Bayes’ rule is applied to basic event set

The probabilities of events to be computed via Bayes’ rule, shown in Table

Probabilities of flawlessness/defectiveness expected from an EC tester (Application No. 3).

Event | Bayes’ rule | Event probability |
---|---|---|

EC tested defective, but found flawless | 0.1983 | |

EC tested flawless, and found flawless | 0.9987 | |

EC tested flawless, but found defective | 0.0013 | |

EC tested defective, and found defective | 0.8017 |

This example is motivated by an experimental study of Pt:Mo dispersed catalysts (PMDCs) for the electro-oxidation of methanol in acid medium [

The research team is assumed to report that 96% of the new PMDC possess the claimed Tafel slope ranges; Bayes’ theorem yields

Perhaps the most striking feature of Bayes’ rule is the amount of information that can be gleaned from a few uncomplicated probability ratios (the fact that Bayesian methods are at present more than two hundred years old is equally impressive). Within the Bayesian framework, a prior event probability is updated to a posterior probability of that event by means of a likelihood. The latter provides the (conditional) probability of corroborating the a-priori stated hypothesis; this aspect is numerically illustrated in the Appendix.

The examples presented in this paper provide a small “window” to the realm of Bayesian methods whose further exploration in electrochemical science and engineering requires further work. Bayes’ rule is just one of many other mathematical devices of applied probability theory with potential interest to the field.

Let AB and BA denote the combined event of both events A and B occurring, the order of occurrence being immaterial. The veracity of the statement

Utilities provided by the University of Waterloo and the Natural Sciences and Engineering Research Council of Canada (NSERC) are gratefully acknowledged.