Problem 55 - Chemistry and Chemical Reactivity

Question

To determine the concentration dependence of the rate of the reaction $$ \mathrm{H}_{2} \mathrm{PO}_{3}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{HPO}_{3}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) $$ you might measure \(\left[\mathrm{OH}^{-}\right]\) as a function of time using a pH meter. (To do so, you would set up conditions under which \(\left[\mathrm{H}_{2} \mathrm{PO}_{3}\right]\) remains constant by using a large excess of this reactant.) How would you prove a second-order rate dependence for \(\left[\mathrm{OH}^{-}\right] ?\)

Step-by-Step Solution

Answer

Plot \( 1/\left[\mathrm{OH}^{-}\right] \) vs. time; a straight line confirms second-order.

Step 1: Understanding Rate Laws

A second-order rate law has the form \( rate = k[A]^2 \) or \( rate = k[A][B] \). Here, we need to determine if the reaction is second-order with respect to \( \mathrm{OH}^{-} \).

Step 2: Setting Up the Experiment

To determine if the reaction is second-order with respect to \( \mathrm{OH}^{-} \), the experiment should be set up so that \( \left[\mathrm{H}_{2} \mathrm{PO}_{3}^{-}\right] \) is in large excess. This ensures that its concentration remains approximately constant during the reaction, simplifying the analysis to focus on \( \mathrm{OH}^{-} \).

Step 3: Measuring the Concentration Over Time

Use a pH meter to measure the concentration of \( \mathrm{OH}^{-} \) as a function of time. Since pH is related to the concentration of \( \mathrm{OH}^{-} \), these measurements will allow you to track how the concentration changes.

Step 4: Plotting the Data

Make a plot of \( \frac{1}{\left[\mathrm{OH}^{-}\right]} \) versus time. For a reaction that is second-order with respect to \( \mathrm{OH}^{-} \), this plot should result in a straight line.

Step 5: Analyzing the Plot

If the plot from Step 4 is a straight line, this indicates a second-order reaction with respect to \( \mathrm{OH}^{-} \). The slope of this line is equal to the rate constant \( k \) for the reaction.

Key Concepts

Second-Order Reactions

Second-order reactions are fascinating as they involve the rate depending on the concentration of two reactants, or one reactant squared. Understanding this is pivotal when studying chemical kinetics. The mathematical expression for second-order reactions can be written as either: - \( rate = k[A]^2 \) or - \( rate = k[A][B] \). In these expressions, \( k \) represents the rate constant, and \([A]\) and \([B]\) are the concentrations of the reactants. Second-order reactions have unique characteristics, such as their half-life being inversely proportional to the initial concentration of the reactant(s). To confirm a second-order reaction, you would often plot \( \frac{1}{[A]} \) against time, which should yield a straight line. The slope of this line is equal to the rate constant \( k \), and its linearity is evidence that the reaction follows second-order kinetics.

Rate Laws

Rate laws are the equations that express the rate of a reaction as a function of the concentration of its reactants. They are crucial for understanding how different factors influence the speed of chemical reactions. Each reaction has its unique rate law, which is determined experimentally and can take different forms—commonly zero-order, first-order, or second-order, depending on how the rate is influenced by the concentration of one or more reactants. The general form of a rate law is: - \( rate = k[A]^m[B]^n \), where \( m \) and \( n \) are the reaction orders with respect to reactants \( A \) and \( B \), respectively. The rate constant \( k \) varies with temperature but stays constant for a given reaction under specific conditions. By studying rate laws, chemists can predict how the rate changes when concentrations are altered, helping control industrial processes, for instance.

Concentration Dependence

Concentration dependence is a key factor in determining how the rate of a reaction relates to the concentration of reactants. For second-order reactions, this dependency is particularly pronounced, as the rate is either proportional to the square of one reactant's concentration or the product of two different reactants' concentrations. To investigate this, scientists often set up experiments where one reactant's concentration is kept constant in large excess, simplifying analysis and clarifying the dependency. In practice, you'd measure the concentration changes of a key reactant over time, making it possible to determine the order of the reaction with respect to each reactant. This process typically involves plotting calibration or integrated rate laws, and various graphical methods can help elucidate the concentration dependencies clearly.

pH Measurement

pH measurement plays a crucial role in assessing the concentration of ions in solution, such as hydroxide ions \( \left(\text{OH}^-\right) \). A pH meter precisely registers how acidic or basic a solution is, linking closely to the concentration of hydrogen ions. In the case of a reaction involving \( \text{OH}^- \), the pH offers an indirect but highly useful measure where \([\text{OH}^-]\) can be calculated from the pH using the formula: \[ [\text{OH}^-] = 10^{-(14 - \text{pH})} \] Monitoring the pH over time allows you to track changes in concentration, essential for proving the order of a reaction. Such measurements are integral to kinetic studies, enabling the determination of how fast a reaction proceeds and verifying kinetic models.