Problem 19 - Chemistry The Central Science In Si Units

Question

Consider the following hypothetical aqueous reaction: \(A(a q) \rightarrow B(a q)\). A flask is charged with 0.065 mol of \(A\) in total volume of \(100.0 \mathrm{~mL}\). The following data are collected: \begin{tabular}{lccccc} \hline Time \((\min )\) & 0 & 10 & 20 & 30 & 40 \\ \hline Moles of A & 0.065 & 0.051 & 0.042 & 0.036 & 0.031 \\ \hline \end{tabular} (a) Calculate the number of moles of \(\mathrm{B}\) at each time in the table, assuming that there are no molecules of \(\mathrm{B}\) at time zero and that A cleanly converts to B with no intermediates. (b) Calculate the average rate of disappearance of A for each 10 -min interval in units of \(\mathrm{M} / \mathrm{s}\). \((\mathbf{c})\) Between \(t=0 \mathrm{~min}\) and \(t=30 \mathrm{~min},\) what is the average rate of appearance of B in units of \(M /\) s? Assume that the volume of the solution is constant.

Step-by-Step Solution

Answer

(a) Moles of B: 0, 0.014, 0.023, 0.029, 0.034. (b) Avg rate of A: 0.000233, 0.000150, 0.000100, 0.000083 M/s. (c) Avg rate of B: 0.000161 M/s.

Step 1: Calculate Moles of B

Since A completely converts to B, the number of moles of B at any given time is equal to the initial moles of A minus the moles of A remaining at that time. - **At 0 min**: Moles of B = 0.065 - 0.065 = 0.000 mol - **At 10 min**: Moles of B = 0.065 - 0.051 = 0.014 mol - **At 20 min**: Moles of B = 0.065 - 0.042 = 0.023 mol - **At 30 min**: Moles of B = 0.065 - 0.036 = 0.029 mol - **At 40 min**: Moles of B = 0.065 - 0.031 = 0.034 mol

Step 2: Calculate Concentration of A

Calculate the concentration of A by dividing the moles of A by the volume in liters (0.1 L). - **At 0 min**: [A] = 0.065 / 0.1 = 0.650 M - **At 10 min**: [A] = 0.051 / 0.1 = 0.510 M - **At 20 min**: [A] = 0.042 / 0.1 = 0.420 M - **At 30 min**: [A] = 0.036 / 0.1 = 0.360 M - **At 40 min**: [A] = 0.031 / 0.1 = 0.310 M

Step 3: Calculate Average Rate of Disappearance of A

Use the formula for average rate of disappearance \(-\Delta [A]/\Delta t\) for each 10-min interval. Convert 10 minutes to seconds (600 seconds).- **0 to 10 min**: rate = (0.650 - 0.510) / 600 = 0.000233 M/s- **10 to 20 min**: rate = (0.510 - 0.420) / 600 = 0.000150 M/s- **20 to 30 min**: rate = (0.420 - 0.360) / 600 = 0.000100 M/s- **30 to 40 min**: rate = (0.360 - 0.310) / 600 = 0.000083 M/s

Step 4: Calculate Average Rate of Appearance of B

Between 0 and 30 min, calculate the change in concentration of B (\[\Delta [B]\]) and divide by the time in seconds.- **[B] at 0 min**: 0.000 M- **[B] at 30 min**: 0.029 mol / 0.1 L = 0.290 M- Rate of appearance = (0.290 - 0.000) / 1800 s = 0.000161 M/s

Key Concepts

Average Rate of Reaction

In chemical kinetics, the average rate of reaction is a measure of how quickly reactants turn into products over a specific time period. To illustrate this concept, consider the decomposition of substance A to form substance B in a closed flask, as given in the exercise.

The average rate of reaction is typically expressed in terms of the concentration change of reactants and/or products over time. For example, if we are looking at the disappearance of A, we calculate this as the change in concentration of A divided by the time interval. Conversely, the appearance of B is measured by its concentration change over the time interval as well.

• The formula for the average rate of disappearance of A is: \[-\frac{\Delta [A]}{\Delta t}\]where \(-\Delta [A]\) is the change in concentration of A and \(\Delta t\) is the change in time.
• For B, the formula is: \[\frac{\Delta [B]}{\Delta t}\]where \(\Delta [B]\) represents the change in concentration of B.

In practice, the rates can be different at different time intervals due to factors such as temperature, concentration, and the presence of catalysts.

Calculating changes over various intervals provides insights into the dynamic nature of the reaction, helping understand how the conversion between reactants and products proceeds over time.

Concentration Calculations

Concentration calculations are fundamental in determining how much of a substance is present in a given volume of solution. This is vital in kinetics as it allows for an understanding of the reaction's progress over time.

In the context of the given reaction, determining the concentration of A at any point in time involves using the formula: \[[A] = \frac{\text{moles of A}}{\text{volume of solution in liters}}\]This formula helps in finding the molarity, which is expressed as moles per liter (M).

• To find the concentration of A initially (0 min):\[[A] = \frac{0.065}{0.1} = 0.650 \, \text{M}\]
• At subsequent intervals, concentrations are recalculated based on the remaining moles of A.

When these concentrations are plotted over time, they illustrate how A decreases, informing us about the progression and completion of the reaction.

By understanding concentration calculations, students can quantitatively analyze not just how much reactant is left, but also infer about the speed of a reaction.

Chemical Reaction Rates

Chemical reaction rates quantify the speed at which reactants are converted into products. This is an essential part of reaction kinetics and significantly influences how we manipulate reactions in real-world applications.

The rate of a reaction can vary and is determined by several factors:

• Concentration: A higher concentration of reactants often leads to increased collision frequency, usually speeding up the reaction.
• Temperature: Raising the temperature generally increases the reaction rate by providing more energy, allowing molecules to collide more effectively.
• Pressure: In reactions involving gases, increased pressure can lead to reaction acceleration due to a similar collision frequency increase.
• Catalysts: These substances lower the activation energy required for a reaction, increasing the rate without being consumed in the process.

For our example reaction, we can calculate the rate of appearance of B during a specific period, such as from 0 to 30 minutes. Using the concentration data provided:\[\text{Rate of appearance of B} = \frac{\Delta [B]}{\Delta t} = \frac{0.290 \, \text{M} - 0.000 \, \text{M}}{1800 \, \text{s}} = 0.000161 \, \text{M/s}\]Such calculations help predict how changes in conditions could influence reaction speed and inform practical decisions in chemical manufacturing and research. Understanding how these rates are determined and adjusted is crucial for controlling reactions efficiently.