Problem 39 - General Chemistry: Principles and Modern Applications

Question

For the reaction \(A \longrightarrow\) products, the data tabulated below are obtained. (a) Determine the initial rate of reaction (that is, \(-\Delta[\mathrm{A}] / \Delta t)\) in each of the two experiments. (b) Determine the order of the reaction. $$\begin{array}{ll} \hline \text { First Experiment } & \\ \hline[\mathrm{A}]=1.512 \mathrm{M} & t=0 \mathrm{min} \\ \begin{array}{l} | \mathrm{A}\rfloor=1.490 \mathrm{M} \\ {[\mathrm{A}]=1.469 \mathrm{M}} \end{array} & \begin{array}{l} t=1.0 \mathrm{min} \\ t=2.0 \mathrm{min} \end{array} \\ \hline & \\ \hline \text { Second Experiment } & \\ \hline[\mathrm{A}]=3.024 \mathrm{M} & t=0 \mathrm{min} \\ {[\mathrm{A}]=2.935 \mathrm{M}} & t=1.0 \mathrm{min} \\ {[\mathrm{A}]=2.852 \mathrm{M}} & t=2.0 \mathrm{min} \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The initial rates for the reactions are \(0.022\, M/min\) and \(0.086\, M/min\) respectively. The reaction is of second order.

Step 1: Determine the rate of reaction for the first experiment

The rate of reaction is calculated by the formula \(\Delta[\mathrm{A}]/ \Delta t\). Using the provided data for the first experiment, the rate of reaction is calculated: \[ -((1.469\, M - 1.512\, M)/(2.0\, min - 0\, min)) = 0.022\, M/min\]

Step 2: Determine the rate of reaction for the second experiment

Using the same formula, and the provided data for the second experiment, calculate the rate of reaction: \[ -((2.852\, M - 3.024\, M)/(2.0\, min - 0\, min)) = 0.086\, M/min\].

Step 3: Determine the order of the reaction

The order of a reaction is determined by observing how the rate changes when the concentration of the reactant is changed. From the rates, it can be seen that the reaction rate of Experiment 2 is almost 4 times that of Experiment 1 while the concentration of reactant A is twice as much. Therefore, it can be concluded that the reaction is of second order because when the concentration doubles, the rate quadruples.

Key Concepts

Rate of Reaction

The rate of reaction is an essential concept in reaction kinetics. It tells us how quickly a reaction proceeds. You may represent it as the change in concentration of a reactant or product per unit time. Essentially, it's all about how fast or slow substances are transformed in a chemical reaction.

To calculate the rate of reaction, use the formula:

• \( Rate = -\frac{\Delta[A]}{\Delta t} \)

Here, \([A]\) denotes the concentration of the reactant, and \(\Delta t\) is the change in time. The negative sign indicates a decrease in the concentration of reactants.

For example, in our exercise, the rate of reaction for the first experiment was determined using initial and final concentrations over a given time. By substituting values into the formula, we calculated a reaction rate of \(0.022\, M/min\) for the first experiment. Repeating this process for the second experiment yielded a rate of \(0.086\, M/min\).

Understanding the rate of reaction helps predict how fast a reaction will occur under different conditions, which is crucial in both industrial applications and laboratory experiments.

Reaction Order

Reaction order provides insight into how the concentration of reactants influences the rate of a reaction. In essence, it shows the relationship between the rate and the concentrations of reactants.

The order of a reaction is determined by the exponents in the rate law, which applies only in specific scenarios calculated from experimental data. These orders can be zero, first, second, or even fractional. Each exponent reveals the reactants' impact on the reaction rate:

• Zero-order: Rate is independent of the concentration of reactants.
• First-order: Rate is directly proportional to one reactant's concentration.
• Second-order: Rate is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants.

In the exercise, we discovered that when the concentration doubled from Experiment 1 to Experiment 2, the rate quadrupled. This observation is characteristic of a second-order reaction, suggesting that the concentration of the reactant has a squared influence on the rate.

Chemical Kinetics Data Analysis

Chemical kinetics data analysis is pivotal for interpreting the results of reaction experiments, and it involves using data collected during experiments to draw conclusions about reaction behaviors. This process is key when you're analyzing how a set of conditions, such as temperature or concentration, impacts the rate.

To perform such an analysis, chemists collect data like concentrations at various times and then calculate reaction rates. This involves plotting concentration vs. time and examining the resulting graph to understand trends. From these trends, they can deduce the order and rate constant of the reaction.

In our specific problem, we collected concentration data over time for two different experiments. By comparing changes in concentrations, we calculated the rates of reaction. Analyzing the differences between these rates gave us insights about the reaction order. Such analysis is not only crucial for understanding a specific reaction but also aids in predicting how similar reactions might behave outrightly and under similar conditions. This predictive capability can optimize reaction conditions in industrial processes and experiments.