### Unlimited growth: Applications of exponential growth models

### First-order chemical reaction kinetics

Hydrolysis of ethyl acetate The hydrolysis of ethyl acetate in an excess of water into acetic acid and ethanol \[\mathrm{CH}_{3}\mathrm{COOC}_{2}\mathrm{H}_{5}+\mathrm{H}_2\mathrm{O}\longrightarrow \mathrm{CH}_{3}\mathrm{COOH}+\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH} \] is an example of a chemical reaction that proceeds according to first-order reaction kinetics. Translated in terms of a differential equation, this means: \[\frac{\dd C}{\dd t}=-k\cdot C\] where \(C\) represents the concentration of ethyl acetate and \(k\) is the reaction rate constant. The solution of this differential equation is simple: \[C(t)=C_0\cdot e^{-k\cdot t}\] where \(C_0\) is the initial concentration at time \(t=0\).

Hydrolysis of sucrose The hydrolysis of sucrose to glucose and fructose \[\mathrm{C}_{12}\mathrm{H}_{22}\mathrm{O}_{11}+\mathrm{H}_2\mathrm{O}\longrightarrow \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6} (\mathrm{glucose}) + \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6} (\mathrm{fructose}) \] is an example of a reaction that proceeds according to first-order reaction kinetics. Because the sugars all cause their own specific rotation of polarized light, the change in concentration in the course of time ca nbe easily followed with a polarimeter. Ludwig Wilhelmy discovered in this way in 1850 that the rate of conversion is at any time directly proportional to the concentration of the amount sucrose still remainng at that time.