Problem 21 - Chemistry: The Central Science in SI Units, Global Edition

Question

The isomerization of methyl isonitrile \(\left(\mathrm{CH}_{3} \mathrm{NC}\right)\) to acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) was studied in the gas phase at \(215^{\circ} \mathrm{C}\), and the following data were obtained: $$ \begin{array}{rc} \hline \text { Time (s) } & {\left[\mathrm{CH}_{3} \mathrm{NC}\right](M)} \\ \hline 0 & 0.0165 \\ 2000 & 0.0110 \\ 5000 & 0.00591 \\ 8000 & 0.00314 \\ 12,000 & 0.00137 \\ 15,000 & 0.00074 \\ \hline \end{array} $$ (a) Calculate the average rate of reaction, in \(M / s\), for the time interval between each measurement. (b) Calculate the average rate of reaction over the entire time of the data from \(t=0\) to \(t=15,000 \mathrm{~s} .(\mathbf{c})\) Which is greater, the average rate between \(t=2000\) and \(t=12,000 \mathrm{~s}\), or between \(t=8000\) and \(t=15,000 \mathrm{~s} ?(\mathbf{d})\) Graph \(\left[\mathrm{CH}_{3} \mathrm{NC}\right]\) versus time and determine the instantaneous rates in \(M / \mathrm{s}\) at \(t=5000 \mathrm{~s}\) and \(t=8000 \mathrm{~s}\).

Step-by-Step Solution

Answer

In summary, the average rates of reaction for various time intervals are: - Between \(t=0\) and \(t=2000\) s: \(2.75 \times 10^{-6} \frac{M}{s}\) - Between \(t=2000\) and \(t=5000\) s: \(1.70 \times 10^{-6} \frac{M}{s}\) - Between \(t=5000\) and \(t=8000\) s: \(9.23 \times 10^{-7} \frac{M}{s}\) - Between \(t=8000\) and \(t=12000\) s: \(4.43 \times 10^{-7} \frac{M}{s}\) - Between \(t=12000\) and \(t=15000\) s: \(2.10 \times 10^{-7} \frac{M}{s}\) The overall average rate of reaction is \(1.05 \times 10^{-6} \frac{M}{s}\). The average rate between \(t=2000\) and \(t=12000\) s is greater than the average rate between \(t=8000\) and \(t=15000\) s. To find the instantaneous rates of reaction at \(t=5000\) s and \(t=8000\) s, plot the concentration data as a function of time and calculate the slope of the tangent line at these points.

Step 1: Calculate the average rate of reaction

The average rate of reaction is given by the formula: \[ \text{Average rate} = \frac{\Delta [\text{concentration}]}{\Delta \text{time}} \] We will use this formula to calculate the average rate of reaction between each measurement. 1. Between \(t=0\) and \(t=2000\) s: \[ \text{Average rate} = \frac{0.0165 - 0.0110}{2000 - 0} = \frac{0.0055}{2000} = 2.75 \times 10^{-6} \frac{M}{s} \] 2. Between \(t=2000\) and \(t=5000\) s: \[ \text{Average rate} = \frac{0.0110 - 0.00591}{5000 - 2000} = \frac{0.00509}{3000} = 1.70 \times 10^{-6} \frac{M}{s} \] 3. Between \(t=5000\) and \(t=8000\) s: \[ \text{Average rate} = \frac{0.00591 - 0.00314}{8000 - 5000} = \frac{0.00277}{3000} = 9.23 \times 10^{-7} \frac{M}{s} \] 4. Between \(t=8000\) and \(t=12000\) s: \[ \text{Average rate} = \frac{0.00314 - 0.00137}{12000 - 8000} = \frac{0.00177}{4000} = 4.43 \times 10^{-7} \frac{M}/{s} \] 5. Between \(t=12000\) and \(t=15000\) s: \[ \text{Average rate} = \frac{0.00137 - 0.00074}{15000 - 12000} = \frac{0.00063}{3000} = 2.10 \times 10^{-7} \frac{M}{s} \] #b) Calculate the average rate of reaction over the entire time of the data#

Step 2: Calculate the overall average rate of reaction

To find the average rate of reaction for the entire time, we will use the formula for the average rate between \(t=0\) and \(t=15000\) s: \[ \text{Average rate} = \frac{0.0165 - 0.00074}{15000 - 0} = \frac{0.01576}{15000} = 1.05 \times 10^{-6} \frac{M}{s} \] #c) Compare the average rate between t=2000 and t=12000 s, and between t=8000 and t=15000 s#

Step 3: Compare the average rates

We have previously calculated the average rates between these time intervals: - Between \(t=2000\) and \(t=12000\) s, the average rate is equal to the sum of the average rates between \(t=2000\), \(t=5000\), \(t=8000\) and \(t=12000\) which is: \(2.75 \times 10^{-6} + 1.70 \times 10^{-6} + 9.23 \times 10^{-7} + 4.43 \times 10^{-7} = 5.66 \times 10^{-6} \frac{M}{s}\) - Between \(t=8000\) and \(t=15000\) s, the average rate is equal to the sum of the average rates between \(t=8000\), \(t=12000\) and \(t=15000\) which is: \(9.23 \times 10^{-7} + 4.43 \times 10^{-7} + 2.10 \times 10^{-7} = 1.58 \times 10^{-6} \frac{M}{s}\) The average rate between \(t=2000\) and \(t=12000\) s is greater than the average rate between \(t=8000\) and \(t=15000\) s. #d) Graph CH3NC vs. time and determine the instantaneous rates at t=5000 s and t=8000 s#

Step 4: Determine the instantaneous rates

We'll graph the concentration of CH3NC as a function of time. Using a graph, we can calculate the slope of the tangent line to the curve at specific time points (like \(t=5000\) and \(t=8000\) s) to find the instantaneous rate. Unfortunately, we cannot draw the graph in this format, so we suggest using any graph plotting tool or software to create a graph with the data provided and then determining the tangent's slopes at \(t=5000\) s and \(t=8000\) s.

Key Concepts

Isomerization

Isomerization is a fascinating chemical process that involves the transformation of a molecule into another molecule with the same atoms, but in a different arrangement. In the context of this exercise, isomerization is demonstrated through the conversion of methyl isonitrile ( CH_3NC ) to acetonitrile ( CH_3CN ) in the gas phase. This rearrangement involves breaking and forming bonds, and it highlights an essential aspect of chemical kinetics, where the molecular structure is altered without changing the chemical formula.

• It's a vital process in synthetic chemistry.
• Plays a role in many biological pathways.
• It can affect physical properties like boiling point and solubility.

Understanding isomerization helps us grasp how molecular changes can impact reactions and processes.

Average Reaction Rate

The average reaction rate gives us an overview of how fast a reaction is occurring over a specific time interval. It is calculated by measuring how much the concentration of a reactant or product changes over a given time. In this exercise, we use the formula:\[\text{Average rate} = \frac{\Delta [\text{concentration}]}{\Delta \text{time}}\]This allows us to see the reaction's speed between each measurement.

• Between \(t=0\) to \(t=2000\) s, we saw a rate of \(2.75 \times 10^{-6} M/s\).
• The rate decreases over time, indicating the reaction slows as it proceeds.
• This approach is great for comparing reaction rates over different time spans.

Using average rates can help scientists determine the efficiency of chemical processes.

Instantaneous Reaction Rate

The instantaneous reaction rate offers a snapshot of how quickly a reaction is happening at a specific moment. Unlike the average rate, which covers a time span, the instantaneous rate is the slope of the tangent line at a particular point on a concentration vs. time graph.

• Provides precise information on reaction speed at specific times.
• Useful for understanding dynamic changes in reaction mechanisms.
• Calculated by finding the derivative of the concentration concerning time.

You can visualize this by plotting the concentration vs. time data and finding the tangent's slope at the point of interest, such as \(t=5000\) or \(t=8000\) seconds.

Concentration vs. Time Graph

A concentration vs. time graph is a powerful tool in chemical kinetics, illustrating how the concentration of a reactant or product changes over time. By plotting the concentration of CH_3NC against time from the data given, we can analyze both average and instantaneous rates.

• Helps in visualizing the progress of a reaction.
• Allows the determination of reaction order.
• Essential for understanding the reaction’s kinetics.

This graph shows the downward trend in concentration as the reaction proceeds. The steeper the curve, the faster the reaction at that point. Software tools and graphing calculators can be used to create these graphs, enhancing analysis and understanding of chemical kinetics.