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: $$ \begin{aligned} [\mathrm{V}]_{0} &=0.20 \mathrm{M} \\ [\mathrm{AV}]_{0} &=1.0 \times 10^{-4} \mathrm{M} \end{aligned} $$ 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 rate constant (k) for this reaction is \(0.32 \text{ s}^{-1}\).

Step-by-step solution

Evaluate the rate law

The rate law for the reaction is given by: \[ \text { Rate }=k[\mathrm{AV}]^{1}[\mathrm{V}]^{1} \] We also know the initial concentrations of V and AV, so we should be able to use the rate law to find the value of k.

Determine the initial rate

Using the given initial concentrations of AV and V, we can calculate the initial rate of the reaction: \[ \text { Rate } = k[\mathrm{AV}]_{0}[V_{0}] \] We are given: \[\begin{aligned} [\mathrm{V}]_{0} &= 0.20 \mathrm{M} \\ [\mathrm{AV}]_{0} &= 1.0 \times 10^{-4} \mathrm{M} \end{aligned}\] Substitute these values into the initial rate equation: \[ \text { Rate } = k(1.0 \times 10^{-4})(0.20) \]

Use the slope to relate rate and concentrations

A plot of \(\ln [\mathrm{AV}]\) versus \(t\) (s) gives a straight line with a slope of \(-0.32 \mathrm{~s}^{-1}\). This slope is related to the rate constant (k) in the following manner: \[ -\frac{d[\mathrm{AV}]}{dt} = -0.32[\mathrm{AV}][\mathrm{V}] \] At the start of the reaction, we can use the initial concentrations of AV and V to find the initial rate: \[ -0.32[\mathrm{AV}]_{0}[\mathrm{V}]_{0} = -\frac{d[\mathrm{AV}]}{dt} \] Substitute the initial concentrations of AV and V into the equation: \[ -0.32(1.0 \times 10^{-4})(0.20) = -\frac{d[\mathrm{AV}]}{dt} \]

Solve for the rate constant (k)

We now have two equations with the same variable (Rate or \(\frac{d[\mathrm{AV}]}{dt}\)). We can set them equal and solve for k. \[ k(1.0 \times 10^{-4})(0.20) = 0.32(1.0 \times 10^{-4})(0.20) \] Solving for k, we get: \[ k = 0.32 \] Therefore, the rate constant (k) for this reaction is 0.32 s⁻¹.

Key concepts

Chemical Kinetics

Chemical kinetics is a branch of physical chemistry that studies the rates of chemical processes. The speed, or rate, at which a chemical reaction proceeds is influenced by various factors such as temperature, concentration of the reactants, and the presence of catalysts. In the context of developing an antivenin for cobra venom, understanding the chemical kinetics involved is crucial to determine how quickly the antivenin neutralizes the venom.

A key component in chemical kinetics is the rate constant, symbolized as k, which provides insight into the speed of the reaction under specific conditions. It is unique for every chemical reaction and can be influenced by environmental factors. Determining the rate constant is pivotal for evaluating the efficacy of treatments like antivenins, as it can predict how swiftly the treatment will act within the body.

Reaction Rate Law

The reaction rate law describes the relationship between the rate of a chemical reaction and the concentration of the reactants. In the exercise, the rate law takes the form \[ \text{Rate} = k[\mathrm{AV}]^{1}[\mathrm{V}]^{1} \] where the exponents denote the order of the reaction with respect to each reactant. The rate law is an equation that allows us to calculate the rate of reaction given the concentrations of the reactants and the rate constant.

For the antivenin (AV) and venom (V) system, both AV and V are first-order reactants, meaning the rate of the reaction is directly proportional to the product of their concentrations. By understanding the rate law, we can manipulate the concentrations of AV and V for optimal reaction rates, which is especially important in time-sensitive applications like treating a venomous snakebite.

Initial Concentrations

Initial concentrations refer to the amounts of reactants present at the start of a reaction. They are usually denoted by \( [\mathrm{Reactant}]_{0} \) and are vital components in the reaction rate formula. In our exercise, the initial concentrations are: \[ [\mathrm{V}]_{0} = 0.20 \mathrm{M} \] \[ [\mathrm{AV}]_{0} = 1.0 \times 10^{-4} \mathrm{M} \]

Understanding the initial concentrations allows us to predict the course of the reaction. For instance, in the production of antivenins, initial concentrations help in determining the required dosages for effective neutralization of venom. The optimal dosage directly relates to the rate at which the antivenin can bind and neutralize the venom and is a critical aspect in creating effective and safe treatments.

Antivenin Mechanisms

Antivenin mechanisms involve the interaction between an antivenin and a venom to mitigate the venom's toxic effects. Specifically, the antivenin molecules bind to the venom molecules to render them harmless. The rate at which these interactions occur is of great importance in emergency situations where rapid response is essential.

Understanding the kinetics behind antivenin action allows medical professionals to administer the correct dosage and potentially develop more efficient antivenins. By manipulating the binding rates through the influence of factors such as temperature, pH, and the presence of other substances, researchers can improve the performance of these life-saving treatments. The culmination of this knowledge is a powerful tool in increasing the survival rate of venomous animal encounters.