Problem 88 - A Complete Resource Book in Chemistry for JEE Main

Question

The experimental rate law for a reaction \(2 \mathrm{~A}+3 \mathrm{~B} \longrightarrow\) Product, is \(\mathrm{V} \alpha \mathrm{C}_{\mathrm{A}} \mathrm{C}_{\mathrm{B}}^{-2} .\) If the concentration of both \(\mathrm{A}\) and are doubled the rate of reaction increases by a factor of (a) \(\sqrt{2}\) (b) 2 (c) \(2 . \sqrt{2}\) (d) 4

Step-by-Step Solution

Answer

There seems to be a discrepancy; doubling reduces the rate by half, unlike the choices.

Step 1: Understand the given rate law

The experimental rate law is given by \( V \propto C_A C_B^{-2} \). This means that the rate \( V \) is directly proportional to the concentration of \( A \) and inversely proportional to the square of the concentration of \( B \).

Step 2: Determine the initial rate factor

Initially, the rate is proportional to \( C_A C_B^{-2} \). If we denote the initial concentrations as \( C_{A_0} \) and \( C_{B_0} \), then the initial rate \( V_0 \) is proportional to \( C_{A_0} C_{B_0}^{-2} \).

Step 3: Calculate the new rate factor

When the concentration of both \( A \) and \( B \) is doubled, the new concentrations become \( 2C_{A_0} \) and \( 2C_{B_0} \). The new rate \( V' \) is proportional to \( (2C_{A_0})(2C_{B_0})^{-2} \).

Step 4: Simplify the new rate expression

The new rate can be simplified as follows: \[ V' \propto 2C_{A_0} \cdot (2^2C_{B_0}^2)^{-1} = 2C_{A_0} \cdot \left(\frac{1}{4C_{B_0}^2}\right) = \frac{2C_{A_0}}{4C_{B_0}^2} = \frac{1}{2} \cdot \frac{C_{A_0}}{C_{B_0}^2} \]This simplifies to \( \frac{1}{2} \times V_0 \), which indicates the rate is halved.

Step 5: Compare the new rate to the initial rate

The result \( \frac{1}{2} imes V_0 \) implies the new rate is half the original rate when concentrations of \( A \) and \( B \) are doubled. Hence, the factor increase is \( \frac{1}{2} \), which suggests the choices given in the problem may contain an error.

Key Concepts

Rate Law

In chemical reactions, a rate law is an equation that gives insight into how the rate of a reaction depends on the concentration of reactants. The rate law for a reaction dictates that the rate is proportional to the product of the concentration of the reactants, each raised to a specific power. These powers are not necessarily the stoichiometric coefficients from the balanced chemical equation. Instead, they are determined experimentally.

For the given reaction, the rate law is expressed as \( V \propto C_A C_B^{-2} \). This implies that the reaction rate \( V \) increases with an increase in the concentration of \( A \) but decreases with an increase in the concentration of \( B \). This unique setup reveals how each reactant influences the rate distinctly, making the concentration of \( A \) and \( B \) critical in calculating \( V \). Understanding the rate law is pivotal for predicting how changes in conditions affect reaction speed.

Concentration Effect

Changes in reactant concentrations can significantly affect the rate of a reaction, a concept known as the concentration effect. According to our rate law, \( V \propto C_A C_B^{-2} \), the reaction rate is directly proportional to the concentration of \( A \) and inversely proportional to the square of the concentration of \( B \).

In the scenario where the concentrations of both reactants \( A \) and \( B \) are doubled, the reaction rate is influenced as follows:

• The concentration of \( A \) doubled contributes to doubling the factor \( C_A \).
• Doubling \( B \)'s concentration impacts the rate by a factor of \( \left(2C_B\right)^{-2} = \frac{1}{4C_B^2} \).

This interaction leads to an overall halving of the reaction rate, due to their combined effects following the rate law.

Proportionality in Rate Laws

Proportionality in rate laws is a fundamental aspect of understanding reaction kinetics. It specifies that the rate of a reaction is not solely dependent on the concentration magnitude, but on how these concentrations relate to one another per the rate law parameters.

In this exercise, the direct proportionality concerning \( C_A \) and the inverse squared relation to \( C_B \) demonstrates this intricacy. Simply put:

• If \( A \)'s concentration doubles, the rate doubles as well.
• If \( B \)'s concentration doubles, the rate reduces by a factor of four, due to the inverse square relationship \( C_B^{-2} \).

The conflicting effects of doubling both concentrations illustrate the nuanced, non-linear nature of many chemical reaction kinetics. This highlights the importance of understanding proportionality within rate laws to predict outcomes accurately in reaction scenarios.