Question

The rate law for a certain enzymatic reaction is zero order with respect to the substrate. The rate constant for the reaction is \(6.4 \times 10^{2} M \mathrm{~s}^{-1}\). If the initial concentration of the substrate is \(0.275 \mathrm{~mol} \mathrm{~L}^{-1}\), what is the initial rate of the reaction?

Short answer

The initial rate of the reaction is \(6.4 \times 10^{2} M s^{-1}\).

Step-by-step solution

Understanding Zero-Order Reaction

In a zero-order reaction, the rate of the reaction is independent of the concentration of the reactants. This means that the rate of reaction remains constant over time until the reactant is depleted. The general form of the rate law for a zero-order reaction is: Rate = k, where k is the rate constant.

Identifying the Given Parameters

For this particular enzymatic reaction, we are given the rate constant (k) as \(6.4 \times 10^2 \, M\,s^{-1}\) and the initial concentration of the substrate as \(0.275 \, mol \, L^{-1}\). In a zero-order reaction, the rate is independent of the substrate concentration. Thus, we do not use the initial concentration of the substrate to calculate the initial rate.

Calculating the Initial Rate of the Reaction

Since the reaction is zero-order with respect to the substrate, the initial rate of the reaction is equal to the rate constant. Therefore, the initial rate is: Initial Rate = k = \(6.4 \times 10^2 \, M\,s^{-1}\).

Key concepts

Reaction Rate

The concept of reaction rate is fundamental to understanding how chemical reactions occur over time. It is essentially a measure of the speed at which reactants are converted into products in a chemical process. When considering reactions in a chemical or biological setting, it's important to note that the rate can be affected by various factors such as temperature, pressure, concentration of reactants, and the presence of a catalyst.

In the context of a zero-order reaction, like the one in our exercise, the rate is unique because it does not depend on the concentration of the reactants. This characteristic means that as long as the reactant is present, the reaction will proceed at a constant rate. The reaction will only stop when the reactant has been completely consumed. Understanding this allows us to predict how the concentration of the reactant will decrease over time, which is vital in applications such as pharmaceuticals, where the consistent release of a drug is necessary.

Rate Law

The rate law, or rate equation, expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It can take many forms, depending on the order of the reaction with respect to each reactant.

For a general reaction, where the order may be first, second, or any other integer or fraction, the rate law can be written as: \[ \text{Rate} = k[A]^{n}[B]^{m} \] where \(k\) is the rate constant, \([A]\) and \([B]\) are the concentrations of the reactants, and \(n\) and \(m\) represent their respective reaction orders. This equation shows that the rate of reaction is proportional to the product of the concentrations raised to their respective orders. However, in the special case of a zero-order reaction, the rate is independent of the concentration of reactants, making the rate law simply: \[ \text{Rate} = k \]This simplification is crucial for understanding reactions that are not influenced by changes in reactant concentration, such as some enzymatic and surface catalyzed reactions.

Rate Constant

The rate constant is an integral part of the rate law and dictates the speed of the reaction under specific conditions. The value of the rate constant, represented by the symbol \(k\), changes with temperature and the presence of a catalyst, but it is independent of the concentrations of the reactants.

In our zero-order reaction exercise, the rate constant is given as \(6.4 \times 10^2 \, \text{M} \, \text{s}^{-1}\). This value tells us the rate at which the reaction is proceeding. A higher rate constant means a faster reaction. For zero-order reactions, since the rate is equivalent to the rate constant, knowing the value of \(k\) gives us a direct understanding of the reaction speed without needing to consider the concentration of the reactant.

Enzymatic Reaction Kinetics

When we delve into enzymatic reaction kinetics, we study how enzymes catalyze biochemical reactions. Enzymes are remarkable because they can greatly enhance the rate of reactions without undergoing permanent chemical changes themselves. They accomplish this by lowering the activation energy required for a reaction to proceed.

Enzymatic reactions often follow complex kinetics that can be zero-order, first-order, or even follow Michaelis-Menten kinetics depending on conditions like substrate concentration and enzyme activity. Our zero-order example assumes that the enzyme is saturated with substrate, which means adding more substrate will not increase the reaction rate further. This is often seen in biological systems where high substrate concentrations ensure that the enzyme is working at a constant rate, a situation vital for maintaining regulated levels of important biochemicals in an organism.