Problem 20 - Chemistry The Molecular Science

Question

A reaction has the experimental rate law, Rate = \(k[\mathrm{~A}]^{2}[\mathrm{~B}]\). If the concentration of \(\mathrm{A}\) is doubled and the concentration of \(\mathrm{B}\) is halved, what happens to the reaction rate?

Step-by-Step Solution

Answer

The reaction rate is doubled.

Step 1: Understanding the Rate Law

The rate law for the reaction is given by Rate = \( k[A]^2[B] \). This means the rate depends on the concentration of \( A \) squared and \( B \) linearly.

Step 2: Identify Initial Rate Expression

Initial rate can be expressed as \( ext{Rate}_1 = k[A]^2[B] \). This represents the original conditions of the reaction.

Step 3: Adjust Concentrations

The concentration of \( A \) is doubled, so \( [A] \to 2[A] \). The concentration of \( B \) is halved, so \( [B] \to \frac{1}{2}[B] \).

Step 4: Substitute Adjusted Concentrations in Rate Law

Substitute the changed concentrations into the rate law to find the new rate: \( ext{Rate}_2 = k(2[A])^2(\frac{1}{2}[B]) \).

Step 5: Simplify the Expression

Calculate the new rate: \( ext{Rate}_2 = k(4[A]^2)(\frac{1}{2}[B]) = 4[A]^2 \times \frac{1}{2}[B] = 2k[A]^2[B] \).

Step 6: Compare Old and New Rates

The original rate was \( k[A]^2[B] \) and the new rate is \( 2k[A]^2[B] \). Therefore, the rate is doubled.

Key Concepts

Rate Law Expression

The rate law expression is a mathematical relationship that describes the rate of a chemical reaction based on the concentration of reactants. It is represented in the general form, \( ext{Rate} = k[A]^m[B]^n\), where \(k\) is the rate constant, and \(m\) and \(n\) are the orders of the reaction concerning reactants \(A\) and \(B\) respectively. These orders indicate how the concentration of each reactant affects the rate. In our example, Rate = \(k[A]^2[B]\), the reaction order is 2 with respect to \(A\) and 1 with respect to \(B\). Understanding the exponents helps predict how changes in concentrations will affect the rate.

Concentration Effects on Reaction Rate

The concentration of reactants greatly influences the reaction rate, as described by the rate law. If you increase the concentration of a reactant, you generally increase the rate at which the reaction occurs. In our given problem, the concentration of \([A]\) is doubled, and \([B]\) is halved. This change directly affects the rate, which can be calculated by plugging the new concentrations into the rate law expression.

• Doubling \([A]\) implies substituting \(2[A]\) for \([A]\) in the rate law, leading to \((2[A])^2 = 4[A]^2\).
• Halving \([B]\) means replacing \([B]\) with \(\frac{1}{2}[B]\), modifying the terms that include \([B]\).

By calculating, we see that the new rate is twice the original rate, demonstrating the marked effect of concentration changes on reaction kinetics.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate of chemical reactions. It helps predict how different conditions influence the speed and mechanism of the reaction. Through understanding kinetics, we can comprehend why some reactions are instantaneous while others take years to reach completion. Rate laws, including both concentration effects and temperature dependencies, are fundamental components of this field.
In practice, chemical kinetics allows chemists to optimize reactions to be faster, more efficient, and economically feasible. By manipulating variables such as concentration, which we analyzed using our specific reaction rate law, scientists can determine the best conditions for industrial or laboratory chemical processes. This ability to control reactions is crucial in a wide range of fields, including pharmaceuticals, agriculture, and materials science.