Problem 39 - Chemistry Matter and Change

Question

Calculate A reaction between A and B to form AB is first order in A and first order in \(\mathrm{B}\) . The rate constant, \(k,\) equals 0.500 \(\mathrm{mol} /(\mathrm{L} \cdot \mathrm{s}) .\) What is the rate of the reaction when \([\mathrm{A}]=2.00 \times 10^{-2} \mathrm{M}\) and \([\mathrm{B}]=1.50 \times 10^{-2} \mathrm{M} ?\)

Step-by-Step Solution

Answer

The rate of the reaction when \([\mathrm{A}] = 2.00 \times 10^{-2} \ \mathrm{M}\) and \([\mathrm{B}] = 1.50 \times 10^{-2} \ \mathrm{M}\) is \(1.5 \times 10^{-3} \ \mathrm{mol}/(\mathrm{L} \cdot \mathrm{s})\), as calculated using the rate law for a first-order reaction.

Step 1: 1. Recall the rate law for a first-order reaction

The rate law for a reaction involving two reactants A and B that is first order in A and first order in B is given by \[Rate=k[\mathrm{A}]^{a}[\mathrm{B}]^{b}\] where: - \(Rate\) is the rate of the reaction - \(k\) is the rate constant - \([\mathrm{A}]\) and \([\mathrm{B}]\) are the concentrations of A and B, respectively - \(a\) and \(b\) are the orders of reaction, with \(a=1\) and \(b=1\) in this case.

Step 2: 2. Substitute given values into the rate law equation

We have the rate constant \(k=0.500 \ \mathrm{mol}/(\mathrm{L} \cdot \mathrm{s}),\) \( [\mathrm{A}] = 2.00 \times 10^{-2} \ \mathrm{M},\) and \([\mathrm{B}] = 1.50 \times 10^{-2} \ \mathrm{M}\). We will substitute these values into the rate law equation. Since the reaction is first order in A and first order in B, we have \(a=1\) and \(b=1\), so the rate law simplifies to: \[Rate=k[\mathrm{A}][\mathrm{B}]\]

Step 3: 3. Calculate the rate of the reaction

Now, we can calculate the rate of the reaction by plugging in the values for \(k\), \([\mathrm{A}]\), and \([\mathrm{B}]\) into the simplified rate law equation: \[Rate=(0.500 \ \mathrm{mol}/(\mathrm{L} \cdot \mathrm{s})) \times (2.00 \times 10^{-2} \ \mathrm{M}) \times (1.50 \times 10^{-2} \ \mathrm{M})\] After multiplying the values, we obtain the rate of the reaction: \[Rate=1.5 \times 10^{-3} \ \mathrm{mol}/(\mathrm{L} \cdot \mathrm{s}) \] The rate of the reaction when \([\mathrm{A}] = 2.00 \times 10^{-2} \ \mathrm{M}\) and \([\mathrm{B}] = 1.50 \times 10^{-2} \ \mathrm{M}\) is \(1.5 \times 10^{-3} \ \mathrm{mol}/(\mathrm{L} \cdot \mathrm{s})\).

Key Concepts

First-Order Reaction

Understanding first-order reactions is crucial for grasping the basics of reaction kinetics. These reactions depend linearly on the concentration of one reactant. In layman's terms, if you were to double the amount of the reactant, the rate of the reaction would also double. This property makes them easily identifiable. For example, many radioactive decay processes follow first-order kinetics, where the rate at which the substance decays is proportional to its current amount.

Mathematically, a first-order reaction has a rate that is determined directly by the concentration of a single reactant raised to the first power. This can be concisely represented as \( Rate = k[Reactant] \), where \( k \) is the rate constant unique to the reaction and \( [Reactant] \) is the molar concentration of the reactant under consideration. It's important for students to recognize when a reaction fits this model as it simplifies the analysis and calculation of the reaction rate.

Rate Law Equation

The rate law equation is a mathematical representation that links the rate of a chemical reaction to the concentration of its reactants. It plays a critical role in reaction kinetics by allowing chemists to predict the speed of a chemical reaction under various conditions. The general form of the rate law is \( Rate = k[Reactant_1]^{n_1}[Reactant_2]^{n_2}...[Reactant_n]^{n_n} \), where \( k \) is the rate constant, \( [Reactant] \) represents the concentration of each reactant, and \( n \) indicates the reaction order with respect to that reactant.

The exponents, such as \( n_1 \) and \( n_2 \) in the equation, are determined experimentally and cannot be predicted from the chemical equation alone. They tell you how the rate is affected by changes in the concentration of each reactant. This makes the rate law equation a powerful tool for predicting how changes in conditions will affect reaction speed.

Reaction Kinetics

Reaction kinetics is the study of the rate at which chemical reactions occur and the factors that affect this rate. It's an essential concept in chemistry that connects the microscopic interactions of molecules to observable macroscopic properties of chemical systems. By understanding kinetics, chemists can control the speed of reactions, which is pivotal for applications ranging from manufacturing to pharmaceuticals.

Factors Influencing Reaction Rate

• Concentration: Generally, a higher concentration of reactants leads to an increased reaction rate.
• Temperature: Raising the temperature typically speeds up reactions by providing more energy to the reactant molecules.
• Catalysts: Substances that increase the reaction rate without being consumed can drastically change kinetics.
• Surface Area: In reactions involving solids, greater surface area can lead to higher reaction rates.

Chemists apply their understanding of these factors and the fundamental rate laws to optimize reactions for various needs, such as increasing the yield of a desired product or reducing the formation of byproducts.

Chemical Concentration

The concentration of a chemical in a solution is a measure of how much of that chemical is present in a given volume of solution. It is commonly expressed in units of molarity (M), which is moles of solute per liter of solution. The concept of concentration is central to the rate of reaction as it's a key variable in the rate law equation.

In the context of chemical reactions, the concentration of reactants is particularly important. As the reaction progresses, the concentrations of reactants typically decrease, leading to changes in the reaction rate unless the reaction is zero-order. Accurately measuring and controlling concentrations enables chemists to predict and manipulate reaction rates using the rate law. This concept is not only crucial for the analysis of reaction kinetics in a laboratory setting but also for understanding natural processes in biology, environmental science, and industrial applications where reactions are occurring.