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 \(40-\mathrm{s}\) 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

(a) The number of moles of B at each time are as follows: Time 0 s: 0 moles Time 40 s: 0.033 moles Time 80 s: 0.055 moles Time 120 s: 0.070 moles Time 160 s: 0.080 moles (b) The average rate of disappearance of A for each 40-second interval: Interval 0 - 40 s: 0.000825 mol/s Interval 40 - 80 s: 0.000550 mol/s Interval 80 - 120 s: 0.000375 mol/s Interval 120 - 160 s: 0.000250 mol/s (c) To calculate the rate in units of concentration per time, the additional information needed is the volume of the system containing the reacting gases.

Step-by-step solution

(a) Number of Moles of B at Each Time

To find the number of moles of B at each time, we can use the initial number of moles of A and the moles of A remaining at each time point to calculate the moles of B formed. Since there is a ratio of 1:1 between A and B in the reaction, the number of moles of B at a given time is equal to the difference between the initial moles of A and the remaining moles of A at that time. At each time, using the formula: Moles of B = Initial moles of A - Moles of A remaining Time 0 s: Moles of B = \(0.100 - 0.100 = 0\) Time 40 s: Moles of B = \(0.100 - 0.067 = 0.033\) Time 80 s: Moles of B = \(0.100 - 0.045 = 0.055\) Time 120 s: Moles of B = \(0.100 - 0.030 = 0.070\) Time 160 s: Moles of B = \(0.100 - 0.020 = 0.080\)

(b) Average Rate of Disappearance of A

The average rate of disappearance of A can be determined by finding the difference in the number of moles of A over each 40-second interval and dividing it by the time difference (40 seconds). At each interval, using the formula: Average rate of disappearance of A = (Moles of A at initial time - Moles of A at final time) / Time interval Interval 0 - 40 s: Average rate = \((0.100 - 0.067) / 40 = 0.000825\,\mathrm{mol/s}\) Interval 40 - 80 s: Average rate = \((0.067 - 0.045) / 40 = 0.000550\,\mathrm{mol/s}\) Interval 80 - 120 s: Average rate = \((0.045 - 0.030) / 40 = 0.000375\,\mathrm{mol/s}\) Interval 120 - 160 s: Average rate = \((0.030 - 0.020) / 40 = 0.000250\,\mathrm{mol/s}\)

(c) Additional Information Needed

In order to calculate the rate in units of concentration per time, additional information that is needed includes the volume of the system containing the reacting gases. This would allow us to convert the number of moles of A into concentration (in units like M or mol/L) at each time point and then calculate the rate of disappearance in terms of concentration per time.

Key concepts

Reaction Rates

Have you ever wondered just how fast a chemical reaction happens? That speed is defined by a concept called the reaction rate. In essence, the reaction rate tells you how quickly a reactant is consumed or a product is formed in a chemical reaction.

For example, in our exercise, the reaction of gas-phase \(\mathrm{A}\longrightarrow\mathrm{B}\) demonstrates this concept. By measuring the moles of \(\mathrm{A}\) over time, we can determine how fast \(\mathrm{A}\) is being converted into \(\mathrm{B}\). This data allows us to calculate the average rate of disappearance and gain insights into what influences the reaction's speed.

In practice, you'll often find that reaction rates are affected by several factors, such as temperature, concentration, and presence of a catalyst. In laboratory settings, controlling these variables helps us understand the reaction kinetics and predict the reaction's progress.

Gas-Phase Reactions

Gas-phase reactions occur when reactants in the gaseous state undergo chemical changes to form products. These types of reactions are particularly intriguing because they often occur faster than those in liquid or solid phases. This is because gas molecules move freely and collide more frequently, a key factor in facilitating reactions.

Our given reaction of \(\mathrm{A}\rightarrow\mathrm{B}\) is a simple gas-phase reaction, indicating its reactants and products are both gases. Understanding this provides insights into the reaction mechanics since gases expand to fill their containers entirely. This leads to uniform distribution, helping us accurately measure reaction rates without the complexity introduced by mixing seen in liquids and solids.

Studying gas-phase reactions is crucial in fields like atmospheric science and industrial chemistry because they help explain phenomena like pollution and the manufacturing of chemicals.

Moles Calculation

In chemistry, the concept of the "mole" is fundamental. The mole is a unit that measures the amount of substance, where one mole of any substance contains the same number of entities (like atoms or molecules), which is Avogadro's number: \(6.022 \times 10^{23}\) entities.

The calculation of moles is pivotal for understanding chemical reactions since it assists in stoichiometry, which deals with the quantitative relationships between reactants and products in a chemical reaction. In our exercise, by knowing the initial and remaining moles of \(\mathrm{A}\), we can calculate the number of moles of \(\mathrm{B}\) formed during the reaction.

• At 0 seconds, moles of \(\mathrm{B}\) are \(0\)
• At 40 seconds, moles of \(\mathrm{B}\) increase to \(0.033\)

The 1:1 ratio between \(\mathrm{A}\) and \(\mathrm{B}\) calls for a straightforward subtraction of moles, reinforcing the simplicity yet importance of mole calculations in understanding reaction progress.

Concentration Units

When dealing with chemical reactions, especially those involving gases, understanding concentration is crucial. Concentration is defined as the amount of a substance within a specified volume and is typically expressed in units like molarity (M), moles per liter (mol/L).

In the context of our exercise, while we are given data in terms of moles, converting these into concentration units would require knowledge of the system's volume. This conversion allows for the calculation of reaction rates in terms of concentration change over time, which is often more meaningful than moles because it accounts for how tightly packed the molecules are within a space.

Once we know the volume, we can calculate concentration at each time interval using:

• Concentration (M) = Number of moles / Volume (L)

This approach highlights why having complete data, like the system's volume, is essential for calculating rates of reaction in concentration units. Such calculations give clearer insight into the reaction dynamics and kinetics.