Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of \(3 \times 10^{4} \mathrm{g} / \mathrm{mol}\) 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 law that first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: \(1.0 \mathrm{L}\) of a solution containing \(0.15 \mu \mathrm{g}\left(0.15 \times 10^{-6} \mathrm{g}\right)\) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant. $$\begin{array}{|c|c|} \hline \text { [Penicillin] (M) } & \text { Rate \(\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{min}^{-1}\right)\) } \\ \hline 2.0 \times 10^{-6} & 1.0 \times 10^{-10} \\ \hline 3.0 \times 10^{-6} & 1.5 \times 10^{-10} \\ \hline 4.0 \times 10^{-6} & 2.0 \times 10^{-10} \\\ \hline \end{array}$$

The reaction order is a critical concept in chemical kinetics, representing the power to which the concentration of a reactant is raised in the rate law equation. Understanding reaction order is key to predicting how the concentration of reactants will affect the rate of the reaction.

For instance, in our exercise dealing with penicillinase and penicillin, the reaction order with respect to penicillin needed to be established. By comparing the rate of the reaction under different concentrations of penicillin while keeping the enzyme concentration constant, we can deduce the reaction order. If doubling the concentration of penicillin leads to a doubling of the reaction rate, the reaction is first-order with respect to penicillin. This linear relationship indicates that the rate directly depends on the penicillin concentration.

In contrast, if the rate had quadrupled upon doubling the concentration of penicillin, it would suggest a second-order relationship with respect to penicillin. Reaction orders can be zero, meaning the rate is independent of the concentration of that reactant, or even fractional, indicating a more complex relationship.

The rate law, or rate equation, expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. A general form of the rate law might look like this: \[ \text{Rate} = k[\text{A}]^m[\text{B}]^n \] where \( k \) is the rate constant, \( m \) and \( n \) are the reaction orders with respect to reactants A and B, respectively.

In our enzyme-catalyzed reaction example, the rate law was given to be first-order with respect to the catalyst (penicillinase) and to also involve the concentration of penicillin. Therefore, by arranging the given rate data and penicillin concentrations, we can derive the specific rate law for the reaction. Through the process of determining the reaction order with respect to penicillin, as shown in the exercise, we clarified that the rate law reflects that direct relationship.

The understanding of rate law is not only useful for predicting how the rate will change with concentrations in small-scale laboratory reactions but also for designing commercial-scale processes where reaction rate controls productivity.

The rate constant \( k \) is a proportionality constant in the rate law equation that is specific to a particular reaction at a given temperature. The magnitude of the rate constant gives valuable insight into the rate of the reaction; a larger value of \( k \) indicates a faster reaction under the same concentration conditions.

In the penicillinase activity example, once the reaction order with respect to penicillin was determined to be first-order, we could calculate the rate constant using the rate law. The rate constant reflects not just reaction speed but also includes factors such as temperature or the presence of a catalyst.

It's important for students to comprehend that the rate constant does not change with the concentration of reactants but might change with temperature. Therefore, in experiments or industrial settings, maintaining consistent temperatures is crucial for ensuring the accuracy and reproducibility of reaction rates.