Single molecules of alkaline phosphatase are captured in a capillary filled with a fluorogenic substrate. During incubation, each enzyme molecule creates a pool of fluorescent product. After incubation, the product is swept through a high-sensitivity laser-induced fluorescence detector; the area of the peak provides a precise measure of the activity of each molecule. Three studies are performed on captured enzyme molecules. In the first study, replicate incubations are performed on the same molecule at constant temperature; the amount of product increases linearly with incubation time. Single enzyme molecules show a range of activity; the most active molecules have over a 10-fold higher activity than the least active molecules. In the second study, replicate incubations are performed on the same molecule at successively higher temperatures. The activation energy of the reaction catalyzed by a single molecule is determined with high precision. Single enzyme molecules show a range of activation energy; microheterogeneity extends to thermodynamic properties of catalysis. The average activation energy is within experimental error of the activation energy obtained from analysis of a bulk sample. These results are consistent with the first postulate of statistical thermodynamics: a thermodynamic property obtained from the time average of an individual molecule is identical to that produced by an ensemble average over a large number of molecules. In the third study, the activity of single enzyme molecules is measured after partial heat denaturation. The number of active molecules decreases in proportion to the extent of denaturation. However, the activity of the surviving molecules is experimentally indistinguishable from the activity of control enzyme. Thermal denaturation of alkaline phosphatase is a catastrophic process, wherein the molecule undergoes irreversible conversion to an inactive form.