Problem 33 - Chemistry in 30 days

Question

Give the following data for the reaction: \(X+Y \rightarrow Z\) \begin{tabular}{|c|c|c|} \hline\([X]\) & {\([Y]\)} & Rate \(\times 10^{-1} \mathrm{~ms}^{-1}\) \\ \hline \(1.0 \mathrm{M}\) & \(1.0 \mathrm{M}\) & \(0.25\) \\ \hline \(2.0 \mathrm{M}\) & \(1.0 \mathrm{M}\) & \(0.50\) \\ \hline \(1.0 \mathrm{M}\) & \(2.0 \mathrm{M}\) & \(0.25\) \\ \hline \(1.0 \mathrm{M}\) & \(3.0 \mathrm{M}\) & \(0.25\) \\ \hline \end{tabular} Which one is the rate law equation? (a) Rate \(=k[X][Y]\) (b) Rate \(=k[X]^{0}[Y]^{1}\) (c) Rate \(=k[X][Y]^{\circ}\) (d) Rate \(=k[X][Y]^{2}\).

Step-by-Step Solution

Answer

The rate law equation is (c) Rate \(=k[X][Y]^{\circ}\).

Step 1: Examine Experiment Data

First, look at the changes in concentration of the reactants and how these changes affect the rate of reaction. **Compare experiments 1 and 2:** Concentration of \([X]\) doubles from 1.0 M to 2.0 M while \([Y]\) remains constant at 1.0 M. The rate goes from 0.25 to 0.50, indicating that the rate is directly proportional to \([X]\). This suggests a first-order dependency on \([X]\).

Step 2: Determine Influence of Y

Next, consider experiments where the concentration of \([Y]\) changes. **Compare experiments 1 and 3:** Concentration of \([Y]\) doubles from 1.0 M to 2.0 M while \([X]\) remains constant at 1.0 M. The rate remains 0.25. Also, compare experiments 1 and 4, where \([Y]\) triples, and the rate still remains 0.25. This indicates the rate is independent of \([Y]\). This suggests a zero-order dependency on \([Y]\).

Step 3: Derive the Rate Law Equation

Based on the observations from Step 1 and Step 2: The rate has a first-order dependency on \([X]\) and zero-order dependency on \([Y]\). Therefore, the rate law is **Rate** \(= k[X]^{1}[Y]^{0}\). This means option (c) **Rate** \(=k[X][Y]^{\circ}\) is the correct rate law equation.

Key Concepts

Reaction Order

Understanding reaction order is crucial in determining how different reactants influence the rate of a chemical reaction. The reaction order indicates the power to which the concentration of a reactant is raised in the rate law equation. Simply put, it tells us how the rate is affected by the concentration of each reactant involved. In the given exercise, by observing how changes in the concentration of reactants affect the reaction rate, scientists can deduce the overall order of the reaction.

• First-order reactions — the rate changes linearly with a change in concentration.
• Zero-order reactions — the rate does not change with the concentration of one of the reactants.

This concept helps to identify which reactants have significant effects on how fast or slow a reaction occurs.

Concentration Dependency

Concentration dependency describes how alterations in the concentration of reactants affect the rate of chemical reactions. In chemical kinetics, concentration is one of the pivotal factors that influences how quickly a reaction proceeds. When examining concentration dependency, researchers look at how altering the concentration impacts the speed of the overall reaction.
In our example, it is seen that the concentration of reactant \([X]\) doubles the rate when its concentration is doubled, which suggests a direct relationship, highlighting its first-order nature in the reaction. On the other hand, varying the concentration of another reactant, \([Y]\), yields no change in rate. Hence, \([Y]\) shows no dependency, indicating a zero-order reaction concerning this reactant.
Recognizing how concentrations impact reaction rates aids in determining the specific order with respect to each reactant.

Chemical Kinetics

Chemical kinetics is the field of study that dives into the rates of chemical reactions and the factors affecting them. It allows scientists to understand and predict how different conditions impact the speed at which reactions occur. By analyzing reaction rates under various scenarios, one can determine the rate expression and reaction order.
In the provided problem, an understanding of chemical kinetics was instrumental in deducing the rate law. By interpreting experimental data, the relationship between the reactants and reaction rate was established. This insight into the reaction mechanism leads to identifying the specific role and order of each reactant.
The study of kinetics not only helps in writing the correct rate laws but also aids in optimizing conditions to maximize the reaction rate for industrial and laboratory processes.

Zero-Order Reaction

Zero-order reactions are unique because their rate is directly independent of the concentration of any reactants. This means that even if the concentration of a reactant changes, it has no impact on the rate of the reaction. For such reactions, the rate is constant and equals the rate constant \(k\).
In the given problem, reactant \([Y]\) is identified as having zero-order dependency. Increasing its concentration doesn’t result in any change in the rate of the reaction. This aspect of zero-order reactions is clear from the constant rate observed in trials involving changes to \([Y]\).
Highlighting zero-order characteristics is crucial in processes where maintaining a stable reaction rate, regardless of concentration fluctuations, is vital. This makes zero-order reactions particularly advantageous in specific industrial applications where consistency is key.