Use and misuse of logistic equations for modeling chemical kinetics Alan K. Burnham 1 Received: 11 March 2015 / Accepted: 17 June 2015 Ó Akade ´miai Kiado ´, Budapest, Hungary 2015 Abstract A generalized logistic function has been pro- posed as a kinetic analysis method that is superior to tra- ditional methods. In fact, the parameter s in the generalized logistic function has an effect on the reaction profile similar to the parameter n in the extended Prout–Tompkins model for autocatalytic reactions. Furthermore, the comparisons made in some papers to traditional methods were made by using the discredited method of determining kinetic parameters from a single heating rate, so they are mis- leading compared to proper kinetic analysis methods that simultaneously analyze multiple thermal histories. In addition, the current implementation of the generalized logistic function of fitting each experiment individually is prone to introduce errors in the kinetic parameters. Guid- ance is given on how the generalized logistic function might be used for proper chemical kinetic analysis. Keywords Kinetic analysis  Thermal analysis  Logistic function  Generalized logistic equation  Prout–Tomkins model  Autocatalytic reactions Introduction The application of the S-shaped (sigmoidal or logistic) curve to describe population growth dates back nearly 180 years [1] and is a standard topic in elementary biology. The logistic curve is given by ln a= 1  a ð Þ ½ ¼ kt þ c; ð1Þ where a is the population, t is time, k is a growth constant, and c is a constant. Although this form is also frequently observed for chemical reactions, use of the logistic curve is not com- monly treated in elementary chemistry even though it has a long history of application in chemical kinetics. Despite several earlier uses of the autocatalytic logistic reaction model, this approach is often referred to as the Prout– Tompkins model [2, 3], which in differential form is da=dt ¼ k 1  a ð Þa: ð2Þ In this case, a is the fraction reacted (growth of product) and k is the reaction rate constant. More than a century ago, Lewis [4] used the auto- catalytic logistic equation to describe the thermal decomposition of silver oxide. Austin and Rickett [5] used the logarithmic form of the logistic equation to characterize the thermal decomposition of austenite. Akulov [6] in the Russian literature and Young [7] in the English literature used an extended form of the auto- catalytic equation with reaction orders to model chemical reactions: da=dt ¼ ka m 1  a ð Þ n ; ð3Þ where n is the traditional reaction order and m is a ‘‘growth’’ parameter than can be correlated with the growth dimensionality of the geometric JMAEK nucleation- growth model [8, 9], although the precise relationship is more complicated [10, 11] and will not be detailed here. Variations of this form have been used in many papers in the kinetics literature. Prout and Tompkins used the linear [2] and logarithmic [12] forms of the logistic equation to describe the thermal decomposition of potassium and silver permanganate, respectively. The logarithmic form is: & Alan K. Burnham akburnham@yahoo.com 1 4221 Findlay Way, Livermore, CA 94550, USA 123 J Therm Anal Calorim DOI 10.1007/s10973-015-4879-3