Question

You and a coworker have developed a molecule that has shown potential as cobra antivenin (AV). This antivenin works by binding to the venom (V), thereby rendering it nontoxic. This reaction can be described by the rate law $$\text { Rate }=k[\mathrm{AV}]^{1}[\mathrm{V}]^{1}$$ You have been given the following data from your coworker: $$[\mathrm{V}]_{0}=0.20 \space\mathrm{M}$$ $$[\mathrm{AV}]_{0}=1.0 \times 10^{-4} \space\mathrm{M}$$A plot of \(\ln [\mathrm{AV}]\) versus \(t\) (s) gives a straight line with a slope of \(-0.32 \mathrm{s}^{-1} .\) What is the value of the rate constant \((k)\) for this reaction?

Short answer

The value of the rate constant (k) for this reaction is 0.32 s⁻¹.

Step-by-step solution

Isolate the rate constant in the rate law

First, let's isolate the rate constant (k) in the rate law: \(\text{Rate } = k[\mathrm{AV}]^{1}[\mathrm{V}]^{1}\) Since the Reaction Order for both reactants is 1, this reaction is a first-order reaction.

Write the integrated rate law

For a first-order reaction, the integrated rate law is given as: \(\ln [\mathrm{AV}] = -kt + \ln [\mathrm{AV_0}]\) Now, we know that the slope of the plot of \(\ln [\mathrm{AV}]\) versus time t is -0.32 s⁻¹. This slope is equal to the rate constant (-k) for a first-order reaction: \(-k = -0.32 \space s^{-1}\)

Calculate the value of k

Solve for the rate constant k: \(k = 0.32 \space s^{-1}\) The value of the rate constant (k) for this reaction is 0.32 s⁻¹.

Key concepts

Understanding Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical reactions proceed and the factors that influence these rates. It's crucial for predicting reaction behavior and designing processes that provide the desired products efficiently. The concept entails analyzing how various conditions, such as concentration of reactants, temperature, and the presence of a catalyst, can influence the speed of a reaction.

In the context of the exercise, we're looking at how quickly an antivenin (AV) neutralizes venom (V) within the body. Understanding kinetics allows researchers to design this antivenin so that it reacts with the venom at an appropriate and effective rate. For both safety and efficacy, the rate at which the antivenin binds to the venom is just as important as the binding strength.

Without a proper understanding of chemical kinetics, developing effective pharmaceutical treatments like the cobra antivenin described in our exercise would be a hit-or-miss process. Thanks to kinetics, we can ensure that treatments work quickly enough to counteract toxins and save lives.

First-Order Reaction Dynamics

A first-order reaction is a type of chemical reaction where the rate is directly proportional to the concentration of one of the reactants. In mathematical terms, for a reaction where a reactant A transforms into products, the rate can be expressed as \(\text{Rate} = k[A]\), where \(k\) is the rate constant.

These reactions are characterized by their linear relationship between the natural logarithm of the reactant concentration and time, also known as their linear integrated rate law. This is significant for pharmaceuticals such as the antivenin from the exercise. In a first-order reaction, the rate of reaction doesn't depend on the concentration of the venom; it only depends on the concentration of AV. The implication is that the antivenin can still work effectively even as the venom concentration decreases, which is precisely what's needed in a real-world scenario where venom is being neutralized in the body.

Understanding this concept is vital for pharmacologists who need to predict how quickly a medicine will act in the body, allowing them to design dosing schedules that maintain the drug's effectiveness over time.

Integrated Rate Law for First-Order Reactions

The integrated rate law for a first-order reaction gives us a way to connect concentrations of reactants to the time elapsed during the reaction. For a first-order reaction, this law can be mathematically expressed as \(\ln [A] = -kt + \ln [A_0]\), where \(\ln\) is the natural logarithm, \(k\) is the rate constant, \(t\) is the time, and \(A_0\) is the initial concentration of the reactant A.

In our exercise, a plot of \(\ln [AV]\) against time gives a straight line, indicating that the relationship between the concentration of antivenin and time is indeed logarithmic, as expected for a first-order reaction. The slope of this line corresponds to \( -k \), which provides a direct method for determining the rate constant. The rate constant, \(k\), encapsulates all the effects of molecular properties and temperature on the reaction rate. It's a critical value that helps pharmacologists and chemists understand how fast a reaction occurs under a given set of conditions.

By calculating the rate constant, we can predict how long it will take for a certain percentage of the antivenin to react with the venom. This information is fundamental for developing effective treatment protocols and understanding the kinetics of the life-saving reaction.