Question

A flask is charged with \(0.100 \mathrm{~mol}\) of \(\mathrm{A}\) and allowed to react to form B according to the hypothetical gas-phase reaction \(\mathrm{A}(\mathrm{g}) \longrightarrow \mathrm{B}(\mathrm{g})\). The following data are collected: \begin{tabular}{lccccc} \hline Time (s) & 0 & 40 & 80 & 120 & 160 \\ \hline Moles of A & \(0.100\) & \(0.067\) & \(0.045\) & \(0.030\) & \(0.020\) \\ \hline \end{tabular} (a) Calculate the number of moles of \(\mathrm{B}\) at each time in the table. (b) Calculate the average rate of disappearance of \(\mathrm{A}\) for each 40 -s interval, in units of \(\mathrm{mol} / \mathrm{s}\). (c) What additional information would be needed to calculate the rate in units of concentration per time?

Short answer

The moles of B at each time interval are 0, 0.033, 0.055, 0.070, and 0.080. The average rates of disappearance of A for each 40-second interval are \(8.25 \times 10^{-4}, 5.50 \times 10^{-4}, 3.75 \times 10^{-4},\) and \(2.50 \times 10^{-4} \mathrm{~mol/s}\). To calculate the rate in units of concentration per time, we need the volume of the reaction mixture to convert moles into concentration (Molarity, M).

Step-by-step solution

(a) Calculate the moles of B at each time interval

As A is converted into B in a 1:1 molar ratio, the moles of B formed will be the same as the moles of A disappeared. Moles of A disappeared = Initial moles of A - Moles of A at a given time So, we can calculate the moles of B at each time interval as: 1. At time 0 s: Moles of B = Initial moles of A - Moles of A at 0 s = \(0.100 - 0.100 = 0\) 2. At time 40 s: Moles of B = Initial moles of A - Moles of A at 40 s = \(0.100 - 0.067 = 0.033\) 3. At time 80 s: Moles of B = Initial moles of A - Moles of A at 80 s = \(0.100 - 0.045 = 0.055\) 4. At time 120 s: Moles of B = Initial moles of A - Moles of A at 120 s = \(0.100 - 0.030 = 0.070\) 5. At time 160 s: Moles of B = Initial moles of A - Moles of A at 160 s = \(0.100 - 0.020 = 0.080\) So, the moles of B at each time interval are 0, 0.033, 0.055, 0.070, and 0.080.

(b) Calculate the average rate of disappearance of A for each 40-second interval

To calculate the average rate of disappearance of A, we will use the formula: Average rate = \(\frac{\text{Change in moles of A}}{\text{Change in time}}\) For each 40-second interval, the average rates of disappearance are: 1. From 0 s to 40 s: Average rate = \(\frac{0.100 - 0.067}{40 - 0} = \frac{0.033}{40} = 8.25 \times 10^{-4} \mathrm{~mol/s}\) 2. From 40 s to 80 s: Average rate = \(\frac{0.067 - 0.045}{80 - 40} = \frac{0.022}{40} = 5.50 \times 10^{-4} \mathrm{~mol/s}\) 3. From 80 s to 120 s: Average rate = \(\frac{0.045 - 0.030}{120 - 80} = \frac{0.015}{40} = 3.75 \times 10^{-4} \mathrm{~mol/s}\) 4. From 120 s to 160 s: Average rate = \(\frac{0.030 - 0.020}{160 - 120} = \frac{0.010}{40} = 2.50 \times 10^{-4} \mathrm{~mol/s}\) Thus, the average rates of disappearance of A are \(8.25 \times 10^{-4}, 5.50 \times 10^{-4}, 3.75 \times 10^{-4},\) and \(2.50 \times 10^{-4} \mathrm{~mol/s}\).

(c) Additional information to calculate the rate in units of concentration per time

To calculate the rate in units of concentration per time (M/s), we need to know the volume of the reaction mixture. The volume will allow us to convert moles into concentration (Molarity, M) and then calculate the rate based on the change in concentration over time.

Key concepts

Reaction Rate

The reaction rate tells us how fast a reaction occurs. It measures how quickly the reactants are converted into products. In the context of the given exercise, we're analyzing the rate at which A turns into B. This is done by computing the average rate of disappearance of A. The rate of reaction can change over time, and it is crucial to note that the average rate provides a general overview over a specific time interval, rather than a specific moment.

To find this rate, we determine how much of A has been used up at various time intervals, dividing this change by the time period, as shown in \[\text{Average rate} = \frac{\text{Change in moles of A}}{\text{Change in time}}.\]This helps us see the pattern of the reaction, often starting fast, then slowing down over time.

Molarity

Molarity is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters. In chemical reactions, knowing the molarity is crucial when we need to calculate reaction rates in terms of concentration changes over time.

If we have the volume of the reaction vessel, we can convert the moles of A and B into their respective molarities using the formula:\[\text{Molarity} (M) = \frac{\text{moles of solute}}{\text{liters of solution}}.\]This conversion is necessary to express reaction rates in units like \(M / s\). Molarity gives a clear picture of how concentrated, or diluted, products and reactants are at any stage of the reaction.

Gas-Phase Reaction

Gas-phase reactions involve reactants and products in the gaseous state, just like the reaction between A and B given in the exercise. Such reactions are unique because of their dependency on factors like temperature and pressure, besides concentration.

In comparison to liquid phase reactions, gas-phase reactions often have higher energies and faster rates due to the greater freedom of movement of gas molecules. The reaction rate itself can depend on how often these molecules collide and with what energy, known as the collision theory of gases.

• Gas particles are in constant, random motion.
• Collisions need sufficient energy to break bonds and form new ones.
• Temperature changes can greatly affect particle speed and collision frequency.

Understanding the nature of gas-phase reactions is essential, as it influences how we predict and control reaction outcomes in real-world applications.