Problem 34 - Chemistry The Molecular Science

Question

When phenacyl bromide and pyridine are both dissolved in methanol, they react to form phenacylpyridinium bromide. \(\mathrm{C}_{6} \mathrm{H}_{5}-\stackrel{\mathrm{O}}{\|} \mathrm{C}-\mathrm{CH}_{2} \mathrm{Br}+\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{~N} \longrightarrow\) \(\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{C}-\mathrm{CH}_{2} \mathrm{NC}_{5} \mathrm{H}_{5}^{+}+\mathrm{Br}^{-}\) When equal concentrations of reactants were mixed in methanol at \(35^{\circ} \mathrm{C}\), these data were obtained: \begin{tabular}{rc|rr} \hline Time \((\min )\) & [Reactant] \((\mathrm{mol} / \mathrm{L})\) & Time \((\mathrm{min})\) & \([\mathrm{Reactant}]\) \((\mathrm{mol} / \mathrm{L})\) \\ \hline 0 & 0.0385 & \(500 .\) & 0.0208 \\ \(100 .\) & 0.0330 & \(600 .\) & 0.0191 \\ \(200 .\) & 0.0288 & \(700 .\) & 0.0176 \\ \(300 .\) & 0.0255 & \(800 .\) & 0.0163 \\ \(400 .\) & 0.0220 & \(1000 .\) & 0.0143 \\ \hline \end{tabular} (a) Determine the rate law for this reaction. (b) Determine the overall order of this reaction. (c) Determine the rate constant for this reaction. (d) Determine the rate constant for this reaction when the concentration of each reactant is \(0.030 \mathrm{~mol} / \mathrm{L}\)

Step-by-Step Solution

Answer

(a) Rate law is first-order: Rate = k[Reactant]. (b) Overall order: 1. (c) Calculate k from the slope of ln[Reactant] vs. time plot. (d) Rate constant k is the same for [0.030 mol/L].

Step 1: Identify the Reaction Order

To determine the reaction order, plot the data to see if it fits a zero, first, or second-order reaction. For a first-order reaction, a plot of \(\ln[\text{Reactant}]\) vs. time should be linear.

Step 2: Verify First-Order Reaction

Plot \(\ln[\text{Reactant}]\) against time. Calculate \(\ln([0.0385])\), \(\ln([0.0330])\), etc., and plot these against the given times. A linear relationship confirms a first-order reaction.

Step 3: Write the Rate Law

Since the data shows a linear relationship in the first-order plot, the rate law is determined to be first-order in the reactant: \( \text{Rate} = k[\text{Reactant}] \).

Step 4: Determine the Overall Order

The overall order of the reaction is the sum of the orders with respect to each reactant. Since it's first-order in this reactant, the overall order is 1.

Step 5: Calculate the Rate Constant (k)

Use the slope of the line from the \(\ln[\text{Reactant}]\) vs. time plot to find \(k\). Slope \( = -k \), so find the slope and convert it into \(k\).

Step 6: Calculate Rate Constant for Specific Concentration

Since the reaction is first-order, \(k\) is constant and doesn't change with concentration. Thus, the rate constant for any concentration, including \(0.030 \, \text{mol/L}\), is the same as calculated in the previous step.

Key Concepts

Rate Law

The rate law of a chemical reaction tells us how the rate of the reaction depends on the concentration of the reactants. It's a fundamental aspect of reaction kinetics and helps in determining the speed at which a reaction progresses. In our original exercise, we derived that the rate law is of the form: \[ \text{Rate} = k [\text{Reactant}] \] Here, \(k\) is the rate constant, and \([\text{Reactant}]\) is the concentration of one of the reactants. This expression indicates that the reaction rate is directly proportional to the concentration of the reactant. This type of rate law is typical for reactions that follow first-order kinetics. Understanding the rate law is crucial because it allows chemists to predict how changes in concentration affect the reaction rate. It also plays a vital role in industrial and lab settings where controlling reaction speeds is necessary. The rate constant \(k\) is specific to a particular reaction at a given temperature and pressure.

Reaction Order

The reaction order is a classification of reactions based on how the rate is affected by the concentration of the reactants. For the exercise we studied, the reaction was determined to be first-order. Determining the reaction order involves analyzing how the concentration of the reactant influences the rate of reaction through experimental data. In general, there are:

• Zero-order reactions: The rate is independent of the concentration of reactants.
• First-order reactions: The rate is directly proportional to the concentration of one reactant.
• Second-order reactions: The rate is proportional to the square of the concentration of one reactant or to the product of two reactants.

Knowing the reaction order is vital for studying reaction mechanisms. It provides insight into the steps involved in a reaction. In our case, as both reactants are present in equal concentrations at the start, the reaction order was determined through plotting data, which confirmed the first-order behavior.

First-order Reaction

A first-order reaction means that the rate of the reaction depends linearly on the concentration of one reactant. This is a common type of reaction in chemical kinetics and has a simplest mathematical form. For these reactions, the rate law can be represented as already shown: \[ \text{Rate} = k [\text{Reactant}] \] A distinctive feature of first-order reactions is their logarithmic integration. If you plot the natural logarithm of the concentration of the reactant against time, you get a straight line. The slope of this line is equal to \(-k\), the negative rate constant. This makes it straightforward to determine \(k\) from experimental data. First-order reactions are often encountered in radioactive decay and some simple decomposition reactions. They exhibit an exponential decrease of the concentration of reactants over time. Their halving times, or half-lives, are constant and independent of the initial concentration, making them relatively easy to work with in practical situations.