Problem 114 - A Complete Resource Book in Chemistry for JEE Main

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 \(\mathrm{B}\), 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) \(4 / 3\)

Step-by-Step Solution

Answer

The order of the reaction is \( \frac{3}{2} \).

Step 1: Understand the rate law

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

Step 2: Analyze the effect of concentration change on rate for A

When the concentration of \(A\) is doubled, the rate doubles. Thus, we can say that:\[\frac{k(2[A])^x[B]^y}{k[A]^x[B]^y} = 2 \]Simplifying this gives \(2^x = 2\), which implies \(x = 1\).

Step 3: Analyze the effect of concentration change on rate for B

If the concentration of \(B\) is increased by nine times and the rate triples, then:\[\frac{k[A]^x(9[B])^y}{k[A]^x[B]^y} = 3 \]Simplifying this gives \(9^y = 3\), therefore \(y = \frac{1}{2}\).

Step 4: Determine the overall order of the reaction

The overall order of the reaction is the sum of the individual orders, thus:\[ ext{Order} = x + y = 1 + \frac{1}{2} = \frac{3}{2}\]

Key Concepts

Rate Law

In chemical kinetics, the rate law provides a mathematical expression that relates the rate of a reaction to the concentration of the reactants. For a reaction like \[ A + B \rightarrow C + D \]we can express the rate law as:\[ \text{Rate} = k[A]^x[B]^y \]

• \(k\): The rate constant, a unique value for every reaction that depends on factors like temperature and presence of catalysts.
• \([A]^x\) and \([B]^y\): These express how the concentration of each reactant affects the reaction rate. The powers, \(x\) and \(y\), are the orders with respect to their respective reactants.

The rate law helps predict how changes in concentration affect the speed of a reaction. Understanding this allows chemists to control reactions in industrial processes and laboratories.

Reaction Order

Reaction order is a key concept in determining how a reaction responds to changes in reactant concentrations. The order with respect to a specific reactant is represented by the exponent in the rate law equation, like \(x\) for \([A]^x\). The overall reaction order is simply the sum of the individual orders from all reactants, such as \(x + y\) for a reaction involving two reactants.
It's crucial to distinguish between the two:

• Individual Reaction Order: Pertains to one reactant. Exponent values indicate how rate changes with concentration variances.
• Overall Reaction Order: The total sum of all exponents in the rate law. It provides insight into the reaction's overall sensitivity to concentration changes.

In our exercise, the overall reaction order is determined to be \(\frac{3}{2}\), showing that the reaction rate depends on multiple reactants in a non-uniform way.

Concentration Effects

Concentration effects illustrate how altering reactant concentrations impacts the speed of a reaction. They are directly tied to the orders determined in the rate law.
If doubling a reactant concentration doubles the reaction rate, the order with respect to that reactant equals one, as seen in our case with reactant \(A\). Similarly, if increasing a concentration ninefold results in rate tripling, as with reactant \(B\), it suggests a fractional order, like \(\frac{1}{2}\).
Understanding concentration effects is vital for:

• Predicting how fast products form under different conditions.
• Optimizing reaction conditions in practical applications.
• Designing safer, more efficient chemical processes.

These effects highlight the importance of precision in chemical experimentation and industrial application.

Rate Equation Analysis

Rate equation analysis involves dissecting the rate law equation to understand the dynamics of a chemical reaction fully. It combines mathematics and chemistry to explore reaction kinetics deeply. This analysis helps us:

• Identify relationship patterns between reactants and reaction rates.
• Calculate individual and overall reaction orders accurately.
• Develop strategies to alter reaction conditions favorably.

The analysis in our exercise began with writing the rate expression:\[\text{Rate} = k[A]^x[B]^y\]By examining concentration changes and their impact on rates, we determined:

• \(x = 1\) for reactant \(A\).
• \(y = \frac{1}{2}\) for reactant \(B\).
• Overall order is \(\frac{3}{2}\), revealing the nuanced nature of the reaction.

Through careful calculation and consideration, rate equation analysis enables chemists to understand and manipulate chemical reactions more effectively.