Question

For a reaction \(\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}+\mathrm{D}\) if the concentration of \(\mathrm{A}\) is doubled without altering the concentration of \(B\), the rate gets doubled. If the concentration of is increased by nine times without altering the concentration of \(\mathrm{A}\), the rate gets tripled. The order of the reaction is (a) 2 (b) 1 (c) \(3 / 2\) (d) \(4 / 3\)

Short answer

The order of the reaction is \( \frac{3}{2} \), so the answer is (c).

Step-by-step solution

Write the Rate Law Equation

For the reaction \( \mathrm{A} + \mathrm{B} \rightarrow \mathrm{C} + \mathrm{D} \), the rate law can be expressed as:\[ \text{Rate} = k [A]^x [B]^y\]where \( x \) and \( y \) are the orders of reaction with respect to \( A \) and \( B \), respectively, and \( k \) is the rate constant.

Analyze the First Condition

The problem states that if the concentration of \( A \) is doubled (\( [A] \longrightarrow 2[A] \)) without changing \( [B] \), the rate doubles. Substitute these conditions into the rate equation:\[k (2[A])^x [B]^y = 2 imes (k [A]^x [B]^y)\]This simplifies to:\[2^x = 2\]Therefore, \( x = 1 \).

Analyze the Second Condition

It is given that increasing \( [B] \) by nine times (\( [B] \longrightarrow 9[B] \)) without altering \( [A] \) leads to the rate tripling. Apply this to the rate equation:\[k [A]^x (9[B])^y = 3 imes (k [A]^x [B]^y)\]This simplifies to:\[9^y = 3\]Solving for \( y \), we find \( y = \frac{1}{2} \).

Calculate the Overall Order of Reaction

The overall order of the reaction is the sum of the individual orders with respect to \( A \) and \( B \):\[\text{Overall Order} = x + y = 1 + \frac{1}{2} = \frac{3}{2}\]

Select the Correct Answer

From the calculated overall order of \( \frac{3}{2} \), the correct choice is option (c) \( \frac{3}{2} \).

Key concepts

Rate Law

Understanding the rate law is crucial for grasping how reactions proceed over time. The rate law provides insight into how the concentration of reactants influences the reaction rate. For a reaction such as \( \mathrm{A} + \mathrm{B} \rightarrow \mathrm{C} + \mathrm{D} \), the rate law is given as follows:\[ \text{Rate} = k [A]^x [B]^y \]In this equation, \( k \) represents the rate constant, while \( x \) and \( y \) are the orders of the reaction with respect to the concentrations of \( \mathrm{A} \) and \( \mathrm{B} \), respectively. The exponents \( x \) and \( y \) indicate how the rate is affected when either \([A]\) or \([B]\) are changed.

• If \( x = 1 \), as in our problem, doubling \([A]\) doubles the rate, indicating a direct relationship.
• Similarly, if \( x = 2 \), doubling \([A]\) would quadruple the rate.
• The same logic applies to \( y \) with respect to \( \mathrm{B} \).

The rate law must often be determined experimentally, as it conveys the dependency of the reaction rate on reactant concentrations.

Order of Reaction Determination

Determining the order of a reaction is essential for understanding its kinetics. It involves figuring out the values of \( x \) and \( y \) in the rate law equation from experimental data, which in turn reveals the relationship between reactant concentrations and reaction rate.To determine the reaction order:

• Start by looking at changes in concentration and their impact on the rate. If doubling the concentration of a reactant doubles the rate, that reactant is first order with respect to the reaction.
• If increasing a concentration by three times increases the rate ninefold, it's second order with respect to that reactant.
• For the given exercise, doubling \([A]\) doubles the rate, so \( A \) is first order (\( x = 1 \)).
• Tripling the rate by increasing \([B]\) ninefold indicates a half order for \( B \) (\( y = \frac{1}{2} \)).

Understanding these patterns aids in predicting how changes in concentration influence the rate of reaction.

Rate Equation Analysis

Analyzing the rate equation helps synthesize all the kinetic information of a chemical reaction. From the rate law, you can predict how alterations in reactants' concentrations affect the overall reaction rate.For the reaction \( \text{A} + \text{B} \rightarrow \text{C} + \text{D} \), we use data from concentration changes:\[ \text{Rate} = k [A]^1 [B]^{1/2} \]The overall order is the sum of the individual orders (\( x + y \)):

• From our exercise, with \( x = 1 \) and \( y = \frac{1}{2} \), the reaction is overall order \( \frac{3}{2} \).

By assessing the rate equation, you can:

• Understand how fast or slow a reaction proceeds under varying conditions.
• Identify the influence each reactant has on the rate.
• Enhance your ability to control reaction conditions in practical applications or industrial processes.

Engaging deeply with rate equation analysis equips you with the tools to predict and manipulate chemical reaction behaviors effectively.