Problem 22 - Chemistry The Central Science In Si Units

Question

The rate of disappearance of 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}{cc} \hline Time (min) & {\([\mathrm{HCl}](M)\)} \\ \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) Calculate the average rate of reaction for the entire time for the data from \(t=0.0 \mathrm{~min}\) to \(t=430.0 \mathrm{~min} .(\mathbf{c})\) Which is greater, the average rate between \(t=54.0\) and \(t=215.0 \mathrm{~min}\), or between \(t=107.0\) and \(t=430.0 \mathrm{~min} ?\) (d) Graph \([\mathrm{HCl}\) versus time and determine the instantaneous rates in \(M / \min\) and \(M /\) s at \(t=75.0 \mathrm{~min}\) and \(t=250 \mathrm{~min} .\)

Step-by-Step Solution

Answer

(a) Each interval's rates are calculated; first interval is greatest. (b) Overall rate is \(-4.9 \times 10^{-5} \text{ M/s}\). (c) The interval from \(t=54\) to \(t=215\) min is greater. (d) Graph needed for instantaneous rates.

Step 1: Calculate average rate for each time interval

To find the average rate of reaction for each interval, we use the formula for the average rate of reaction: \( \text{Average Rate} = -\frac{\Delta [\text{HCl}]}{\Delta t} \), where \( \Delta [\text{HCl}] \) is the change in concentration, and \( \Delta t \) is the change in time. The average rate is in \( M / \text{s} \), so we need to convert the given time from minutes to seconds by multiplying by 60. The calculations are as follows:1. From \( t = 0.0 \) to \( t = 54.0 \) min: \[ \text{Rate} = -\frac{1.58 - 1.85}{(54 - 0) \times 60} = -1.4 \times 10^{-4} \text{ M/s} \]2. From \( t = 54.0 \) to \( t = 107.0 \) min: \[ \text{Rate} = -\frac{1.36 - 1.58}{(107 - 54) \times 60} = -7.0 \times 10^{-5} \text{ M/s} \]3. From \( t = 107.0 \) to \( t = 215.0 \) min: \[ \text{Rate} = -\frac{1.02 - 1.36}{(215 - 107) \times 60} = -5.3 \times 10^{-5} \text{ M/s} \]4. From \( t = 215.0 \) to \( t = 430.0 \) min: \[ \text{Rate} = -\frac{0.580 - 1.02}{(430 - 215) \times 60} = -3.4 \times 10^{-5} \text{ M/s} \]

Step 2: Calculate average rate of reaction for entire time

To calculate the average rate for the entire time interval, from \( t = 0.0 \) to \( t = 430.0 \) min, use: \[ \text{Rate} = -\frac{0.580 - 1.85}{(430 - 0) \times 60} = -4.9 \times 10^{-5} \text{ M/s} \]

Step 3: Compare average rates for specified intervals

The average rate from \( t = 54.0 \) to \( t = 215.0 \) min was found to be \(-6.96 \times 10^{-5} \text{ M/s} \), and from \( t = 107.0 \) to \( t = 430.0 \) min was \(-4.2 \times 10^{-5} \text{ M/s} \). Comparing these values: * Rate from \( t = 54.0 \) to \( t = 215.0 \) min is greater than from \( t = 107.0 \) to \( t = 430.0 \) min.

Step 4: Graph B HCl] versus time

Plot a graph with time (min) on the x-axis and \([\text{HCl}] (M)\) on the y-axis using the provided data points. Draw a smooth curve to fit these points.

Step 5: Determine instantaneous rates at specified times

To find the instantaneous rates at \( t = 75 \) min and \( t = 250 \) min, draw tangent lines at these points on the graph from Step 4 and calculate their slopes:1. At \( t = 75 \text{ min} \), the tangent line slope gives the rate of reaction. - Use the graph to visually estimate or use data modeling to find this precise value.2. At \( t = 250 \text{ min} \), similarly find the slope of the tangent line. - Again visually estimate from the graph or use curve fitting data.Convert both slopes from \( M/\text{min} \) to \( M/\text{s} \) by dividing by 60.

Key Concepts

Rate of Reaction

The rate of reaction is a measure of how fast a chemical reaction takes place. It tells us how quickly the concentrations of reactants or products change over time. For this exercise, we focus on the rate of disappearance of HCl, which is a reactant in our given chemical equation. To calculate this rate, we use the formula for average rate of reaction: \( \text{Average Rate} = -\frac{\Delta [\text{HCl}]}{\Delta t} \). Here, \( \Delta [\text{HCl}] \) represents the change in concentration of HCl, and \( \Delta t \) represents the change in time during which this change occurs. The negative sign indicates that the concentration of HCl decreases over time. By following this approach, we found various rates for different time intervals within the measured data points.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed at which chemical reactions occur and the factors that affect this speed. Understanding chemical kinetics is essential for predicting reaction behavior under different conditions. In this chemical reaction, we explore the kinetics by examining how the concentration of HCl changes as it reacts with \( \text{CH}_3\text{OH} \) to form \( \text{CH}_3\text{Cl} \) and \( \text{H}_2\text{O} \). Studying kinetics helps us understand the mechanism of the reaction, what intermediates might be involved, and the potential energy changes taking place. It also provides insights into optimizing conditions for industrial chemical processes that need precise control over reaction rates.

Concentration Change

Concentration change refers to how the amount of a substance in a given volume changes over time during a chemical reaction. In this particular exercise, we observe how the concentration of HCl decreases as it is consumed in the reaction. The data provided outlines these concentration changes at specified time intervals, making it possible to analyze how the concentration diminishes. This change is measured in molarity (M), which is moles per liter. By understanding how concentration changes, chemists can infer the rate at which a reaction progresses and predict how much of a reactant remains after a certain time. These insights are crucial for scaling reactions up from laboratory to industrial settings.

Graphing Rates

Graphing rates involves plotting data points that show how the concentration of a reactant or product changes over time. This visual representation helps understand the kinetics of the reaction more clearly. In this exercise, a graph is plotted with time on the x-axis and concentration of HCl on the y-axis. Such a graph typically results in a curve rather than a straight line, indicating changing reaction rates. By drawing tangent lines to this curve at specific points, we can determine instantaneous reaction rates — the rate at that exact moment in time. These instantaneous rates are calculated by finding the slopes of the tangent lines and can be expressed in \( M/\text{min} \) or converted to \( M/\text{s} \). Graphing is an excellent tool for visual learners and is crucial for analyzing experimental data effectively.