Question

The first-order integrated rate law for a reaction \(\mathrm{A} \longrightarrow\) products is derived from the rate law using calculus. \begin{equation} \begin{aligned} \text { Rate } &=k[\mathrm{A}] \quad \text { (first-order rate law) } \\ \text { Rate } &=\frac{d[\mathrm{A}]}{d t} \\\ \frac{d[\mathrm{A}]}{d t} &=-k[\mathrm{A}] \end{aligned} \end{equation} The equation just given is a first-order, separable differential equa- tion that can be solved by separating the variables and integrating: \begin{equation} \begin{aligned} \frac{d[\mathrm{A}]}{[\mathrm{A}]} &=-k d t \\\ \int_{[\mathrm{A}]_{0}}^{[\mathrm{A]}} \frac{d[\mathrm{A}]}{[\mathrm{A}]} &=-\int_{0}^{t} k d t \end{aligned} \end{equation} In the integral just given, \([\mathrm{A}]_{0}\) is the initial concentration of \(\mathrm{A} . \mathrm{We}\) then evaluate the integral: \begin{equation} \begin{aligned}[\ln [\mathrm{A}]]_{[\mathrm{A}]_{0}}^{[\mathrm{Al}} &=-k[t]_{0}^{t} \\ \ln [\mathrm{A}]-\ln [\mathrm{A}]_{0} &=-k t \end{aligned} \end{equation} \begin{equation} \ln [\mathrm{A}]=-k t+\ln [\mathrm{A}]_{0}(\text { integrated rate law }) \end{equation} \begin{equation} \begin{array}{l}{\text { a. Use a procedure similar to the one just shown to derive an inte- }} \\ {\text { grated rate law for a reaction } A \longrightarrow \text { products, which is one-half- }} \\ {\text { order in the concentration of } A \text { (that is, Rate }=k[A]^{1 / 2} )}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Use the result from part a to derive an expression for the half-life }} \\ {\text { of a one-half-order reaction. }}\end{array} \end{equation}

Short answer

Part a: Integrated Rate Law: \( [A] = ([A]_0^{1/2} - \frac{1}{2}kt)^2 \). Part b: Half-life: \( t_{1/2} = \frac{ [A]_{0}^{1/2} }{k} \) for a one-half-order reaction.

Step-by-step solution

Write the half-order rate law

Write the expression for the rate of the reaction which is one-half-order in the concentration of A. The rate is given by Rate = k[A]^{1/2}.

Express the rate as the change in concentration over time

Replace the expression for Rate with the change in concentration over time to create an equation that can be integrated. \(\frac{d[A]}{dt} = k[A]^{1/2}\)

Rearrange the equation to separate variables

Rearrange the differential equation to separate the variables (concentration [A] and time t). \(\frac{d[A]}{[A]^{1/2}} = k dt\)

Integrate both sides of the equation

Integrate the left side from \( [A]_0 \) to \( [A] \) and the right side from 0 to t. \(\int_{[A]_0}^{[A]}\frac{d[A]}{[A]^{1/2}} = \int_{0}^{t}k dt\)

Evaluate the integrals

Evaluate both integrals to find the integrated rate law. For the left side, use the integral \(\int x^{-1/2} dx = 2x^{1/2}\), and for the right side, the integral of a constant is the constant times the variable. \(2[A]^{1/2}\Big|_{[A]_0}^{[A]} = kt\Big|_{0}^{t}\)

Simplify the integrated rate law

Subtract the lower limit from the upper limit in the evaluated integrals and simplify the equation to solve for \( [A] \) in terms of t. \(2[A]^{1/2} - 2[A]_{0}^{1/2} = kt\)

Solve for the half-life

Using the concept of half-life, let \( [A] = \frac{1}{2}[A]_0 \) and solve for the time t, which gives the half-life (\( t_{1/2} \) of the reaction. \(2(\frac{1}{2}[A]_0)^{1/2} - 2[A]_{0}^{1/2} = k t_{1/2}\)

Finalize the half-life expression

Simplify the expression for the half-life to find a formula for \( t_{1/2} \) in terms of the rate constant k and the initial concentration \( [A]_0 \) for a one-half-order reaction. \( t_{1/2} = \frac{ [A]_{0}^{1/2} }{k} \)

Key concepts

First-Order Reaction

Understanding first-order reactions is crucial for students tackling reaction kinetics topics. A first-order reaction is defined by the rate of reaction being directly proportional to the concentration of one reactant. In mathematical terms, the rate law is expressed as Rate = k[A], where 'Rate' is the change in concentration of reactant A over time, 'k' is the rate constant, and [A] is the concentration of A. When you plot the natural logarithm of concentration against time, you get a straight line, indicative of a first-order kinetics.

In educational contexts, applying calculus to the rate law allows us to derive an integrated rate equation, providing a relationship between concentration and time that doesn't involve rate directly. The first-order integrated rate law, expressed as \( \ln [A] = -kt + \ln [A]_0 \), is pivotal when it comes to predicting the concentration of reactant at any point in time or the time required to reach a certain concentration.

Differential Equations in Chemistry

Differential equations play a pivotal role in chemistry, especially when it comes to describing how the concentration of substances changes over time in reaction kinetics. A differential equation relates some function with its derivatives, representing how that function changes. For instance, in the context of a chemical reaction, a differential rate equation such as \( \frac{d[A]}{dt} = -k[A] \) allows chemists to calculate reaction rates and concentrations over time.

The separation of variables and integration techniques are used to solve these equations, a skill that links chemistry with math. After separating the variables and integrating, you obtain an integrated rate law that describes how the concentration of reactants decreases over time in a predictable manner. It's essential for students to grasp calculus concepts to navigate through chemical kinetics effectively.

Reaction Kinetics

Reaction kinetics is the study of the speed or rate at which chemical reactions occur and the factors affecting this rate. It's central to predicting how long a chemical reaction will take to reach completion or how different conditions might affect the speed of a reaction. Reaction rates can be influenced by various factors, including the concentration of reactants, temperature, presence of a catalyst, and the physical state of the reactants.

The rate law expresses the rate of a reaction in terms of the concentration of reactants and the rate constant. The order of the reaction tells us how the rate is affected by the concentration of reactants. Understanding reaction kinetics is crucial for many practical applications, such as in the pharmaceutical industry where it informs the stability and shelf life of drugs, in environmental science to model pollution breakdown, and in food science for spoilage and preservation processes.

Half-life of a Reaction

The half-life of a reaction is an important concept that refers to the time required for the concentration of a reactant to decrease to half its initial value. It's a measure of how quickly a reactant is being consumed in a reaction. In a first-order reaction, the half-life is constant and is not dependent on the initial concentration of reactant. Mathematically, for a first-order reaction, it is expressed as \( t_{1/2} = \frac{\ln 2}{k} \), where \( t_{1/2} \) is the half-life and 'k' is the rate constant.

Understanding half-life provides valuable insight into the stability of reactants and can help predict the duration of a reaction. This concept finds relevance in a variety of fields, including radioactive decay in nuclear chemistry, medication dosing in pharmacology, and the decay of organic matter in environmental studies.