Enzyme \(E\) catalyzes the decomposition of substrate \(A\). To see whether substance \(B\) acts as inhibitor we make two kinetic runs in a batch reactor, one with B present, the other without B. From the data recorded below (a) Find a rate equation to represent the decomposition of A. (b) What is the role of \(B\) in this decomposition? (c) Suggest a mechanism for the reaction. $$\begin{aligned} &\operatorname{Run} 1 . C_{\mathrm{A} 0}=600 \mathrm{mol} / \mathrm{m}^{3}, C_{\mathrm{E} 0}=8 \mathrm{gm} / \mathrm{m}^{3}, \text { no } \mathrm{B} \text { present }\\\ &\begin{array}{c|rrrr} C_{\mathrm{A}} & 350 & 160 & 40 & 10 \\ t, \mathrm{hr} & 1 & 2 & 3 & 4 \end{array} \end{aligned}$$ $$\begin{aligned} &\text {Run } 2 . C_{\mathrm{A} 0}=800 \mathrm{mol} / \mathrm{m}^{3}, C_{\mathrm{E} 0}=8 \mathrm{gm} / \mathrm{m}^{3}, C_{\mathrm{B}}=C_{\mathrm{B} 0}=100 \mathrm{mol} / \mathrm{m}^{3}\\\ &\begin{array}{c|rrrrr} C_{\mathrm{A}} & 560 & 340 & 180 & 80 & 30 \\ t, \mathrm{hr} & 1 & 2 & 3 & 4 & 5 \end{array} \end{aligned}$$

Step by step solution

01

Identify the Reaction Order

The rate of decomposition of A should align with a certain reaction order. Generally, these orders are zero, first or second. Analyze the given concentration-time data from Run 1 to determine the reaction order. For a zero-order reaction, concentration decreases linearly with time. For a first-order reaction, the natural logarithm of concentration vs. time is linear. For a second-order reaction, the inverse of the concentration vs. time is linear. Check which relationship fits the data from Run 1 the best.

02

Plot Data and Determine Equation

Using graphing software or plotting by hand, create graphs for zero, first, and second-order reactions based on the concentration and time data given for Run 1. Fit the appropriate relationships to the data points, and determine the rate equation by identifying the slope of the line, which represents the rate constant.

03

Analyze the Impact of Substance B

Compare the data from Run 1 and Run 2 (the latter having substance B present). Note the differences in the concentration of A over time when B is present. If the reaction rate continually decreases in the presence of B, substance B acts as an inhibitor.

04

Suggest a Mechanism for the Reaction

Propose a reaction mechanism based on the observed kinetics and the role of B. A possible mechanism could involve the enzyme E binding with substrate A to form a complex, which then decomposes to products, with B potentially binding to the enzyme or to the enzyme-substrate complex as an inhibitor.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reaction Mechanism

Understanding the sequence of events during a chemical reaction is critical for grasping how a reaction occurs on a molecular level. This sequence is referred to as the reaction mechanism. Enzyme-catalyzed reactions often follow a series of steps, beginning with the binding of a substrate to the active site of an enzyme, followed by a series of changes that lead to the product. In the exercise provided, the enzyme E catalyzes the decomposition of substrate A, where a possible mechanism includes the formation of an enzyme-substrate complex.

With further analysis, if a substance such as B alters the rate of the reaction, we may hypothesize that it interacts with the enzyme or the enzyme-substrate complex, possibly by occupying the active site or by changing the shape of the enzyme or complex. This affects the overall efficiency and sequence of the reaction, showcasing the intricacy of enzyme catalysis and the importance of each step in the reaction mechanism.

Reaction Order Determination

The reaction order is a key aspect in chemical kinetics that refers to the power to which the concentration of a reactant is raised in the rate equation. Determining the reaction order is typically done by examining how the concentration of reactants changes over time. In the provided exercise, the step-by-step approach involves analyzing the decrements in substrate A's concentration across time intervals to surmise whether the reaction follows a zeroth, first, or second-order kinetics.

The type of graph that gives a straight line indicates the order of the reaction: a straight line for concentration versus time suggests zero-order, for natural log of concentration versus time suggests first-order, and for inverse concentration versus time suggests second-order. Establishing the order is a crucial step towards deciphering the rate law of the reaction and is pivotal in understanding how the reaction progresses.

Chemical Kinetics

The field of chemical kinetics deals with the speed or rate at which chemical reactions occur. It is not only concerned with how fast a reaction progresses but also with the different factors that affect this rate, such as temperature, pressure, concentration, and the presence of catalysts or inhibitors. The concentration-time data in the exercise illustrates a practical application of chemical kinetics: comparing rates of decomposition of substrate A with and without an inhibitor.

Kinetics offers insight into the stability of compounds and reaction conditions needed to synthesize products effectively. By understanding kinetics, chemists can optimize processes, reducing the time and resources required for chemical reactions, which is paramount in diverse areas such as pharmaceuticals, material science, and environmental chemistry.

Rate Equation

The rate equation, or rate law, is an expression that links the rate of a chemical reaction to the concentration of its reactants. For a reaction where the concentration of reactant A decreases over time, the rate equation can take on different forms based on the reaction order. In the exercise, once the reaction order is determined, the appropriate rate equation is formulated with a rate constant and the concentrations of the reactants raised to their respective order exponent.

For example, a first-order reaction rate law might appear as \( rate = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of reactant A. This equation becomes a key tool in predicting the concentration of reactants at any given time and assessing the impact of altering reaction conditions, such as the presence of inhibitors.

Reaction Inhibitor

A reaction inhibitor is a substance that decreases the rate of a chemical reaction. In enzymatic reactions, inhibitors can function by binding to the active site of the enzyme, preventing substrate interaction, or by binding to a different site, altering the enzyme's configuration. The exercise addresses the presence of substance B as a potential inhibitor in the decomposition of substrate A by comparing the kinetics of the reaction with and without B.

An inhibitor, such as B, can be either reversible or irreversible, impacting the kinetics temporarily or permanently, respectively. The study of inhibitors is essential in biochemistry and pharmacology, as they are used to regulate enzymes' activity within metabolic pathways and as drugs to block harmful enzyme activity in diseases.