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

Step-by-Step Solution

Answer

The reaction rate is \(1.50 \times 10^{-4} \, \text{mol} / (\text{L} \cdot \text{s})\).

Step 1: Understand the Rate Law

Since the reaction is first order in both reactants \(A\) and \(B\), the rate law can be written as \(\text{Rate} = k[A]^1[B]^1\). The overall reaction is second order.

Step 2: Identify Given Values

Identify the values provided: \([A] = 2.00 \times 10^{-2}\, M\), \([B] = 1.50 \times 10^{-2}\, M\), and the rate constant \(k = 0.500 \, \text{mol} / (\text{L} \cdot \text{s})\).

Step 3: Substitute Values into Rate Law

Substitute the given concentration values and the rate constant into the rate law equation: \[ \text{Rate} = (0.500 \, \text{mol} / (\text{L} \cdot \text{s}))(2.00 \times 10^{-2} \, M)(1.50 \times 10^{-2} \, M) \]

Step 4: Calculate the Reaction Rate

Perform the multiplication to find the reaction rate: \[ \text{Rate} = 0.500 \times 2.00 \times 10^{-2} \times 1.50 \times 10^{-2} \, \text{mol} / (\text{L} \cdot \text{s}) \] \[ \text{Rate} = 1.50 \times 10^{-4} \, \text{mol} / (\text{L} \cdot \text{s}) \]

Key Concepts

First-order reactions

A first-order reaction is a chemical reaction where the rate depends linearly on the concentration of only one reactant. In simple terms, if you double the concentration of this reactant, the rate of reaction also doubles. This makes it easy to predict how changes in concentration affect the reaction rate.

For example, in a reaction where substance A is the only reactant showing first-order behavior, the rate of reaction is directly proportional to the concentration of A, expressed as \[\text{Rate} = k[A]^1\]where:

• \( k \) is the rate constant, which you determine experimentally.
• \([A]\) is the concentration of reactant A.

This concept is fundamental in predicting and understanding how changes in reactant concentrations influence the rate of chemical reactions.

Rate law

The rate law is an equation that relates the rate of a chemical reaction to the concentration of its reactants. Understanding the rate law helps chemists understand how fast a reaction proceeds and how different factors may influence it.

In a given chemical reaction, the rate law is expressed in the form:\[\text{Rate} = k[A]^m[B]^n\]where:

• \( k \) is the rate constant specific to the reaction at a given temperature.
• \([A]\) and \([B]\) are the concentrations of the reactants.
• \( m \) and \( n \) are the reaction orders with respect to each reactant.

The example problem presents a reaction that is first order in both A and B, which means the exponent for both is one. This shows that the rate is proportional to the concentration of each reactant. In the given case, the overall reaction order is obtained by adding the exponents, equaling two, indicating a second-order reaction overall.

Rate constant

The rate constant, denoted by \( k \), is a crucial component in the rate law equation. It provides the necessary conversion factor that connects the concentration of reactants to the rate of the reaction. The rate constant varies for different reactions and usually depends on external conditions like temperature.

In the context of the initial problem, the rate constant is given as \(0.500 \text{ mol} / (\text{L}\cdot\text{s})\). This numerical value helps us calculate the reaction rate when substituting the concentrations of reactants into the rate law. The units of \( k \) will differ based on the order of the reaction. For our second-order reaction, it is expressed in \(\text{mol}^{-1} \cdot \text{L}\cdot\text{s}^{-1}\), to balance with the concentration units \([A]\) and \([B]\).

Understanding the role and value of the rate constant is key to predicting how fast a reaction will occur under specific conditions.

Chemical kinetics

Chemical kinetics is the branch of chemistry focused on studying the speed, or rate, of chemical reactions. It explores not just how fast reactions happen, but also what factors affect these rates. Through studying kinetics, chemists can gain insights into reaction mechanisms and factors like temperature, pressure, and catalysts that influence reaction rates.

Key concepts in chemical kinetics include understanding:

• Rate laws that help establish the link between reactant concentrations and reaction rates.
• Reaction orders which define the power dependency of reaction rates on reactant concentrations.
• Rate constants that bridge concentration with reaction speed, adjusted for external conditions.

All these concepts provide the foundation for predicting and controlling reaction speeds—a valuable insight for fields spanning industrial chemistry to pharmacology.