Some bacteria are resistant to the antibiotic penicillin...

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of $3 \times 10^{4} {g} / {mol}$ that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg $(0.15 \times 10^{-6} {g})$ of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

Key Concepts

Reaction Order

This concept involves determining how the rate of a reaction depends on the concentration of each reactant. In enzyme?catalyzed reactions, especially at low substrate concentrations, the rate law can often be written as a product of the concentrations of the enzyme and the substrate. In this case, the rate is first order with respect to the enzyme and, under low substrate conditions, also first order with respect to the substrate, giving an overall second?order reaction.

Rate Constant

The rate constant (k) is the proportionality factor in the rate law that relates the reaction rate to the concentrations of the reactants. Its units depend on the overall order of the reaction; for a second?order reaction (first order in each reactant) the units are typically liter per mole per time. Calculating the rate constant requires accurate conversion of all given quantities to consistent units.

Enzyme Kinetics at Low Substrate Concentration

In enzyme kinetics, particularly under low substrate conditions, the reaction velocity is proportional to the substrate concentration because the enzyme active sites are not saturated. This situation simplifies the kinetics, allowing the reaction to be described by a simple rate law where the reaction is first order in the substrate, which is crucial for determining kinetic parameters.

Molar Conversions

Converting a given mass of enzyme (or any reactant) to molar concentration is a key step in kinetic calculations. This involves using the molecular weight to change units from grams to moles, and then using the volume of the solution to express the concentration in molarity. Accurate conversion is essential for applying the rate law and calculating the rate constant.

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