Problem 54 - Chemistry The Molecular Nature of Matter

Question

The following data were collected for the decomposition of acetaldehyde. \(\mathrm{CH}_{3} \mathrm{CHO},\) (used in the manufacture of a variety of chemicals including perfumes, dyes, and plastics), into methane and carbon monoxide. The data were collected at \(535^{\circ} \mathrm{C}\). $$\mathrm{CH}_{3} \mathrm{CHO} \longrightarrow \mathrm{CH}_{4}+\mathrm{CO}$$ $$\begin{array}{cc}{\left[\mathrm{CH}_{3} \mathrm{CHO}\right]\left(\mathrm{mol}\mathrm{L}^{-1}\right)} & \text {Time (s) } \\ 0.200 & 0 \\\0.153 & 0.20 \times 10^{2} \\\0.124 & 0.40 \times 10^{2} \\\0.104 & 0.60 \times 10^{2} \\\0.090 & 0.80 \times 10^{2} \\\0.079 & 1.00 \times 10^{2} \\\0.070 & 1.20 \times 10^{2} \\\0.063 & 1.40 \times 10^{2} \\\0.058 & 1.60 \times 10^{2} \\\0.053 & 1.80 \times 10^{2} \\\0049 & 2.00 \times 10^{2} \\\\\hline\end{array}$$ Make a graph of concentration versus time and determine, using the tangent to the curve, the instantaneous rate of reaction of \(\mathrm{CH}_{3} \mathrm{CHO}\) after 60 seconds and after 120 seconds.

Step-by-Step Solution

Answer

To determine the instantaneous rates of reaction for CH3CHO after 60 seconds and 120 seconds, graph the concentration vs. time, draw tangent lines at the respective times, and calculate the slopes of these tangents.

Step 1: Plot the Data on a Graph

Using the given data, you will plot the concentration of CH3CHO on the y-axis against time on the x-axis to create a concentration vs. time graph. Each point corresponds to a pair of values from the table, where the time is converted to seconds (given in scientific notation in the table) and the concentration is in mol L^-1.

Step 2: Draw Tangents at 60 and 120 seconds

Once the graph is plotted, you need to draw a tangent line at the point corresponding to 60 seconds (0.60 x 10^2 seconds) and another tangent at the point corresponding to 120 seconds (1.20 x 10^2 seconds). A tangent line represents the slope of the curve at that precise point, which corresponds to the instantaneous rate of reaction.

Step 3: Calculate the Slope of Each Tangent

The slope of the tangent line will give you the instantaneous rate at the specific time. To calculate the slope, choose two points on the tangent line, and use the formula slope = (change in y) / (change in x), which in this context is rate = −(change in concentration) / (change in time). You will do this for both tangent lines drawn in Step 2.

Step 4: Determine Instantaneous Rates of Reaction

After calculating the slopes, those values represent the instantaneous rates of reaction for CH3CHO after 60 seconds and after 120 seconds. Note that the rate will be negative, indicating a decrease in concentration over time, but usually, rates are reported as positive values.

Key Concepts

Chemical Kinetics

Chemical kinetics, broadly speaking, is the study of rates of chemical processes. It involves an in-depth analysis of how the speed of a reaction is influenced by various factors such as concentration, temperature, and the presence of a catalyst. A fundamental aspect of chemical kinetics is the understanding of reaction mechanisms, which are the step-by-step sequences of elementary reactions that lead to the overall chemical change.

In the context of our example with acetaldehyde, the decomposition into methane and carbon monoxide at a certain temperature, chemical kinetics becomes relevant not just for academic study but also in industrial applications, such as the synthesis of chemicals used in perfumes, dyes, and plastics. By understanding the reaction mechanism and kinetics, chemists can optimize processes, ensuring maximum yield and cost efficiency.

Kinetic studies not only provide insight into the speed of reactions but also help in deducing the steps involved in complex reactions. These studies are vital in areas such as the design of chemical reactors, environmental modeling, and even in pharmacology where the rate of drug metabolism is of interest.

Reaction Rate Calculation

Reaction rate calculation is an essential tool in chemical kinetics used to determine the speed of a chemical reaction. The rate is typically expressed as the change in concentration of a reactant or product per unit time. For a reaction like the decomposition of acetaldehyde, the rate would be the decrease in concentration of acetaldehyde over time. Calculating this requires discerning how rapidly the concentration changes as the reaction proceeds.

To find the rate, you can use the formula: \[\text{rate} = -\frac{\Delta[\text{reactant}]}{\Delta t}\] where \(\Delta[\text{reactant}]\) is the change in concentration of the reactant over the time interval \(\Delta t\). The negative sign indicates that the concentration of the reactant decreases over time.

For instantaneous rates, as with our example at specific times of 60 and 120 seconds, we don't look at overall change but rather the slope of the tangent to the curve at that specific time. This tells us how fast the reaction is proceeding exactly at that moment, different from average rates which are over larger time spans. These calculations are critical for chemists to understand and control reactions for research and industrial purposes.

Concentration vs Time Graph

A concentration versus time graph is a visual representation that depicts how the concentration of a particular species in a reaction changes over time. Plotting this graph is a common way to illustrate the kinetics of a chemical reaction and provides valuable insights into the reaction's dynamics.

In the decomposition of acetaldehyde example, plotting the concentration of CH3CHO on the y-axis and time on the x-axis creates a graph that can be used to visually analyze the reaction rate. By drawing a tangent to the curve at a specific time point, you can determine the instantaneous rate of reaction at that moment.

Drawing Tangents for Instantaneous Rates

When a straight line is drawn tangent to the curve at a particular time point, it reflects the slope of the curve at that instant. This slope is numerically equal to the instantaneous rate of the reaction, showing how quickly the reaction is proceeding at that precise moment. By calculating the slope, or the steepness, of these tangent lines, we can determine the rate at which the concentration is changing right at 60 seconds and 120 seconds for our acetaldehyde example.

Understanding how to read and analyze such graphs is crucial for students and researchers alike, as it is a fundamental skill in the study and application of chemical kinetics.