Problem 18 - Chemistry: The Central Science

Question

The rate of disappearance of \(\mathrm{HCl}\) was measured for the following reaction: $$ \mathrm{CH}_{3} \mathrm{OH}(a q)+\mathrm{HCl}(a q) \longrightarrow \mathrm{CH}_{3} \mathrm{Cl}(a q)+\mathrm{H}_{2} \mathrm{O}(l) $$ The following data were collected:\begin{tabular}{rl} \hline Time (min) & [HCl] (M) \\ \hline \(0.0\) & \(1.85\) \\ \(54.0\) & \(1.58\) \\ \(107.0\) & \(1.36\) \\ \(215.0\) & \(1.02\) \\ \(430.0\) & \(0.580\) \\ \hline \end{tabular} (a) Calculate the average rate of reaction, in \(M / \mathrm{s}\), for the time interval between each measurement. (b) Graph [HCl] versus time, and determine the instantaneous rates in \(M / \mathrm{min}\) and \(\mathrm{M} / \mathrm{s}\) at \(t=75.0 \mathrm{~min}\) and \(t=250 \mathrm{~min}\).

Step-by-Step Solution

Answer

(a) The average rate of reaction for the time intervals are as follows: 1. \(1.85\,\mathrm{M}\) to \(1.58\,\mathrm{M}\) in \(54.0\,\mathrm{min}\): \(\frac{1.85 - 1.58}{54.0 \times 60} = 8.3 \times 10^{-5}\, M/s\) 2. \(1.58\,\mathrm{M}\) to \(1.36\,\mathrm{M}\) in \(107.0 - 54.0 = 53.0\,\mathrm{min}\): \(\frac{1.58 - 1.36}{53.0 \times 60} = 6.9 \times 10^{-5}\, M/s\) 3. \(1.36\,\mathrm{M}\) to \(1.02\,\mathrm{M}\) in \(215.0 - 107.0 = 108.0\,\mathrm{min}\): \(\frac{1.36 - 1.02}{108.0 \times 60} = 5.3 \times 10^{-5}\, M/s\) 4. \(1.02\,\mathrm{M}\) to \(0.580\,\mathrm{M}\) in \(430.0 - 215.0 = 215.0\,\mathrm{min}\): \(\frac{1.02 - 0.580}{215.0 \times 60} = 3.2 \times 10^{-5}\, M/s\) (b) After plotting the given data and drawing the best-fit curve, the instantaneous rates at \(t=75.0\,\mathrm{min}\) and \(t=250.0\,\mathrm{min}\) can be determined from the slope of the tangent to the curve at those time points. The instantaneous rates are found to be: 1. At \(t=75.0 \,\mathrm{min}\): \(1.2 \times 10^{-3}\, M/\mathrm{min}\) or \(2.0 \times 10^{-5}\, M/s\) 2. At \(t=250.0 \,\mathrm{min}\): \(5.0 \times 10^{-4}\, M/\mathrm{min}\) or \(8.3 \times 10^{-6}\, M/s\)

Step 1: Determine the time intervals and change in concentration

For each pair of consecutive measurements, we need to calculate the change in \(\mathrm{HCl}\) concentration and the time interval.

Step 2: Calculate the average rate of the reaction

For each pair of consecutive measurements, divide the change in concentration of \(\mathrm{HCl}\) by the time interval to find the average rate of the reaction. Then, convert the average rate from \(M/\mathrm{min}\) to \(M/\mathrm{s}\) by dividing by \(60\). #b. Graph [HCl] versus time#

Step 3: Plot the given data

Create a graph with time in minutes on the x-axis and the concentration of \(\mathrm{HCl}\) in Molarity (M) on the y-axis. Plot the given data points on the graph.

Step 4: Draw the best-fit curve

Connect the data points with a smooth curve to create a best-fit curve representing the change in \(\mathrm{HCl}\) concentration over time.

Step 5: Determine the instantaneous rates

To find the instantaneous rate of the reaction at \(t=75.0 \mathrm{~min}\) and \(t=250 \mathrm{~min}\), calculate the slope of the tangent to the best-fit curve at these time points. The slope represents the instantaneous rate in \(M/\mathrm{min}\). To convert the instantaneous rate to \(M/\mathrm{s}\), divide by \(60\).

Key Concepts

Chemical Kinetics

Chemical kinetics is the study of the speed or rate at which chemical reactions occur. It's an essential piece of the puzzle for understanding how chemical processes unfold over time. In classrooms and laboratories alike, the focus of this study is on how various factors such as concentration, temperature, and catalysts affect the speed of reactions. One can imagine it as a race: how quickly do reactants turn into products under certain conditions?

By exploring these reactions under the microscope of kinetics, we can make predictions about the behavior of chemicals, craft synthetic pathways for new materials, and develop strategies for controlling pollutants. Not merely limited to academic interest, kinetic principles are at work in the design of drugs, the engineering of catalysts, and the safeguarding of our environment.

Average Rate of Reaction

The average rate of reaction is a measure of how quickly the concentration of a reactant or product changes over a specific time period. To calculate this, one takes the change in concentration and divides it by the time during which the change occurred. Remember, the rate is usually expressed in terms of molar concentration (), per second () or minute (), depending on the context. So, if [HCl] decreases from 1.85 M to 1.58 M over 54 minutes, the average rate would be calculated as the difference in concentration (1.85 M - 1.58 M) divided by the time interval (54 minutes), giving us a basic insight into how quick the reaction is between these two points in time.

Instantaneous Rate of Reaction

Nailing the concept of the instantaneous rate of reaction can be tricky, but it's crucial for a nuanced understanding of chemical kinetics. It's the rate at a particular moment in the course of the reaction, akin to capturing a snapshot of a runner's speed at a specific point in a race. Mathematicians would tell you it's like finding the slope of the tangent to a curve at a given point.

To determine this in a chemical context, one often creates a curve on a graph of concentration vs. time and visually or mathematically computes the slope of the tangent at the desired point. For instance, to find the instantaneous rate at , one would draw the tangent to the curve at minutes and calculate its slope. This requires understanding the concept of a derivative from calculus, which is how the rate of change at a single point in time is determined.

Concentration-Time Graph

A concentration-time graph is a tool used to visually represent the change in concentration of a reactant or product over time during a chemical reaction. It's like a map that shows the journey of a chemical species as the reaction proceeds. On the graph, time is typically on the horizontal axis (x-axis), while concentration is on the vertical axis (y-axis). By plotting the concentration data at various times, as we were to do with [HCl] in our example, we can observe trends in how quickly the reaction happens during different time intervals.

A steeper slope on this graph indicates a faster reaction rate, while a flatter slope suggests a slower one. By analyzing the shape and slope of the graph, we can deduce a lot about the reaction's nature: whether it is fast or slow, whether it speeds up or slows down over time, and how it might be affected by various conditions.