Question

You perform a series of experiments for the reaction \(\mathrm{A} \longrightarrow \mathrm{B}+\mathrm{C}\) and \(\mathrm{find}\) that the rate law has the form rate \(=k[\mathrm{~A}]^{x} .\) Determine the value of \(x\) in each of the following cases: (a) There is no rate change when [A] is tripled. (b) The rate increases by a factor of 9 when \([\mathrm{A}]\) is tripled. (c) When [A] is doubled, the rate increases by a factor of 8 . [Section \(14.3]\)

Short answer

For the given reaction with rate law equation rate \(=k[\mathrm{A}]^x\), the values of \(x\) for each case are as follows: (a) \(x = 0\) when the rate of reaction does not change when the concentration of A is tripled. (b) \(x = 2\) when the rate of reaction increases by a factor of 9 when the concentration of A is tripled. (c) \(x = 3\) when the rate of reaction increases by a factor of 8 when the concentration of A is doubled.

Step-by-step solution

Case (a): No change in rate when [A] is tripled

When the concentration of A is tripled, the rate remains unchanged: rate_before = rate_after \[k [\mathrm{A}]^x = k [3\mathrm{A}]^x\] Divide both sides by \(k[\mathrm{A}]^x\): \[1 = 3^x\] Take the logarithm of both sides: \[\log (1) = x \log (3)\] As log(1) equals 0, then x=0: \[x = 0\] In case (a), the value of \(x\) is 0.

Case (b): Rate increases by a factor of 9 when [A] is tripled

In this case, the following relationship exists: rate_after = 9 * rate_before \[k [3\mathrm{A}]^x = 9k [\mathrm{A}]^x\] Divide both sides by \(k[\mathrm{A}]^x\): \[3^x = 9\] As \(3^2 = 9\), then x=2: \[x = 2\] In case (b), the value of \(x\) is 2.

Case (c): Rate increases by a factor of 8 when [A] is doubled

In this case, we have the following relationship: rate_after = 8 * rate_before \[k [2\mathrm{A}]^x = 8k [\mathrm{A}]^x\] Divide both sides by \(k[\mathrm{A}]^x\): \[2^x = 8\] As \(2^3 = 8\), then x=3: \[x = 3\] In case (c), the value of \(x\) is 3. This concludes the analysis of the given reactions. For each case, we determined the value of \(x\) in the rate law equation based on the reaction's rate behavior when changing the concentration of reactant A.

Key concepts

Reaction Order

Reaction order is a central part of understanding the dynamics of how chemical reactions proceed. It signifies the power to which the concentration of a reactant is raised in the rate law equation. This concept helps us predict how changes in the concentration of a reactant will affect the overall rate of the reaction.

For the exercise you are dealing with, you need to determine the reaction order with respect to reactant \( [A] \). Identifying this order gives insights into how the reaction's rate responds to varying concentrations of \( [A] \). By examining each case, we find that the reaction order changes depending on how the rate changes:

• In case (a), a tripling of \( [A] \) has no effect, so the order is zero.
• In case (b), a tripling of \( [A] \) causes the rate to increase nine times, indicating a second order reaction.
• In case (c), doubling \( [A] \) leads to an eight-fold increase in rate, suggesting a third order reaction.

Each of these cases illustrates different powers of the concentration \( x \), supporting the detailed understanding of the reaction's dynamics.

Concentration Dependence

The dependence of a reaction rate on the concentration of its reactants is pivotal for predicting and controlling chemical processes. When you alter the concentration of a reactant, it often causes a corresponding change in the reaction rate.

In this exercise, concentration variance plays a major role. Recognizing how each case responds to changes in concentration helps in determining the reaction order:

• With no change in rate in response to tripling \( [A] \), the reaction is zero-order for case (a).
• In case (b), if tripling \( [A] \) multiplies the rate by nine, it shows a second-degree dependence.
• For case (c), doubling \( [A] \) results in an eight-fold increase, underlining a third-degree relationship.

Comprehending concentration dependence allows chemists to accurately devise how to speed up or slow down reactions by adjusting concentrations accordingly.

Chemical Kinetics

Chemical kinetics is a fascinating field that studies the rate of chemical reactions. It provides the framework to understand the speed at which reactants convert into products.

In your exercise, chemical kinetics principles are applied to deduce how the concentration of reactant \( [A] \) affects the reaction rate. By calculating the order of the reaction, you harness kinetic data to forecast reaction behavior under different scenarios:

• Understanding zero-order kinetics helps explain why some reactions remain unaffected by changes in reactant concentrations.
• Identifying second-order kinetics clarifies how rates dramatically increase with concentration changes, particularly in case (b).
• Grasping third-order kinetics reveals the more complex scenarios as shown in case (c).

These kinetics concepts facilitate precise predictions of reaction progress and the necessary conditions to drive desired outcomes efficiently.

Rate Law Expression

The rate law expression is a vital tool in quantifying the speed of a chemical reaction. It establishes a quantitative relationship between reaction rate and the concentrations of the reactants.

The general form is given by \( ext{rate} = k[ ext{Reactant}]^x \), where \( k \) is the rate constant, and \( x \) represents the reaction order with respect to a reactant. Through this setup, one can evaluate:

• A zero-order reaction where the rate is constant irrespective of concentration, as seen in case (a).
• A second-order reaction where the rate heavily depends on concentration changes, evident in case (b).
• A third-order reaction where even small changes cause significant effects, presented in case (c).

The expression helps chemists form a complete picture of the reaction's pace, enabling better control and optimization of industrial and laboratory processes.