Question

A reaction has the following experimental rate equation: 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?

Short answer

The reaction rate doubles.

Step-by-step solution

Identify the Initial Rate Equation

The initial rate equation is given as: Rate \(= k[A]^2[B]\), where \([A]\) and \([B]\) are the concentrations of A and B, respectively, and \(k\) is the rate constant.

Modify the Concentrations

The problem states that the concentration of \(A\) is doubled and \(B\) is halved. Therefore, the new concentrations are \([A]' = 2[A]\) and \([B]' = \frac{1}{2}[B]\).

Substitute New Concentrations into the Rate Equation

Insert the new concentrations into the rate equation: Rate' \(= k(2[A])^2\left(\frac{1}{2}[B]\right)\).

Simplify the Expression

Calculate the expression: \((2[A])^2 = 4[A]^2\), and then multiply by \(\frac{1}{2}[B]\): Rate' \(= k \cdot 4[A]^2 \cdot \frac{1}{2}[B] = 2k[A]^2[B]\).

Compare New Rate with Original Rate

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

Key concepts

Rate Equation

A rate equation is an expression that links the rate of a chemical reaction to the concentration of its reactants. In the given exercise, the rate equation is: Rate \(= k[A]^2[B] \). This tells us how the rate of reaction is affected by the concentrations of substances \( A \) and \( B \). The \( k \) in the equation is the rate constant, a number that helps scale the relationship between the concentrations of reactants and the rate. The exponents on \([A]\) and \([B]\) indicate how the concentration changes affect the rate. For example, in this equation, since \([A]\) is squared, any change in its concentration has a significant impact on the rate. Understanding how modifications in reactant concentrations translate into changes in the reaction rate is crucial in predicting how the reaction proceeds under varying conditions.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate at which chemical reactions occur. It helps determine how different factors like concentration, temperature, and catalysts impact reaction rates. In the provided exercise, we see a practical application of chemical kinetics. By altering the concentrations of \( A \) and \( B \), we're able to observe the effect on the reaction rate. Kinetics gives insights into the mechanism of the reaction, explaining the step-by-step process at the molecular level that leads to product formation. Kinetics is essential for both academic research and industrial applications, as it can optimize reactions to make them faster and more efficient.

Concentration

Concentration refers to the amount of a substance present in a certain volume of solution. It plays a pivotal role in reactions as it influences the rate at which they occur. The rate equation in our exercise demonstrates this by showing that the rate is proportional to \([A]^2[B] \), meaning that an increase or decrease in these concentrations will directly affect the speed of the reaction. For instance, doubling the concentration of \( A \) results in a quadrupled effect since it is squared in the rate equation, illustrating a relatively larger impact on the reaction rate compared to changes in \( B \). Understanding how concentration impacts reaction rates can help in designing better experimental conditions and achieving desired reaction outcomes more efficiently.