Problem 169 - A Complete Resource Book in Chemistry for JEE Main

Question

If a is the initial concentration of reactant and \((\mathrm{a}-\mathrm{x})\) is the remaining concentration after time 't' in a first order reaction of rate constant \(k_{1}\), then which of the following relations is /are correct? (a) \(k_{1}=\frac{2.303}{t} \log \left(\frac{\mathrm{a}}{\mathrm{a}-x}\right)\)

Step-by-Step Solution

Answer

(a) is correct; the formula matches the derived first-order reaction relation.

Step 1: Understand the First Order Reaction Formula

A first order reaction can be described using the formula: \[ k_1 = \frac{1}{t} \ln \left( \frac{[A]_0}{[A]} \right) \] where \( [A]_0 \) is the initial concentration of the reactant, and \( [A] \) is the remaining concentration at time \( t \). In this problem, \([A]_0 = a\) and \([A] = a-x\).

Step 2: Relate the Natural Logarithm to the Common Logarithm

The given formula uses the logarithm base 10 (common logarithm). To relate it to the natural logarithm, use the conversion: \[ \ln(x) = 2.303 \cdot \log(x) \].

Step 3: Apply the Conversion to the First Order Reaction Formula

Substitute the conversion from the natural logarithm to the common logarithm into the first order reaction formula:\[ k_1 = \frac{1}{t} \ln \left( \frac{a}{a-x} \right) = \frac{1}{t} \cdot 2.303 \cdot \log \left( \frac{a}{a-x} \right) \] Simplifying gives:\[ k_1 = \frac{2.303}{t} \log \left( \frac{a}{a-x} \right) \].

Step 4: Evaluate the Options

Option (a) compares directly to our derived formula. It states \( k_1 = \frac{2.303}{t} \log \left( \frac{a}{a-x} \right) \), which matches our result exactly. Therefore, this relation is correct.

Key Concepts

Rate Constant

In the realm of chemical kinetics, the rate constant is a crucial concept. It symbolizes how quickly a reaction proceeds. For a first order reaction, the rate constant, denoted as \( k_1 \), is directly related to the concentration of the reactant. Unlike reactions of other orders, first order reactions depend linearly on the concentration of a single reactant.
The rate constant has units that depend on the order of the reaction. For a first order reaction, these units are \( s^{-1} \).
This dependency means that the rate of reaction can be determined by focusing on the initial concentration \( [A]_0 \) and the concentration at time \( t \), \( [A] \).

• The rate constant provides insight into the reaction's speed; a larger \( k_1 \) value indicates a faster reaction.
• It remains constant only under constant temperature and pressure conditions.

This intrinsic property of the reaction, \( k_1 \), allows chemists to predict how swiftly a reaction will move to completion.

Logarithm Conversion

Logarithms are powerful mathematical tools used to transform multiplicative relationships into additive ones. In chemical kinetics, this conversion is valuable for simplifying calculations. The common logarithm (base 10) and the natural logarithm (base \( e \)) are often used.
The natural logarithm is usually represented by \( \ln \), while the common logarithm is represented by \( \log \).
To convert between these two, we use the handy conversion factor:\[ \ln(x) = 2.303 \cdot \log(x) \]

• This equation allows us to transition between natural and common logarithms smoothly.
• The factor 2.303 is derived from the natural logarithm of 10, as \( \ln(10) \approx 2.303 \).

By employing logarithm conversion, chemists can easily work with equations in the form most convenient for their calculations.

Reaction Kinetics

Reaction kinetics is the study of reaction rates and how they change under different conditions. It explores how molecular collisions, temperature, concentration, and catalysts affect the speed of chemical reactions. For a first order reaction, the focus is on understanding how a single reactant's concentration impacts the reaction rate.
One of the foundational aspects in reaction kinetics is the rate law, which describes how the rate depends on the concentration of the reactants.
In first order reactions, the rate of the reaction can be described with the formula:\[ k_1 = \frac{1}{t} \ln \left( \frac{[A]_0}{[A]} \right) \]

• The equation shows how reaction kinetics helps determine \( k_1 \) using experimentally measured concentrations over time.
• Understanding first order reaction kinetics is pivotal for fields like pharmacokinetics, environmental science, and industrial chemistry.

Overall, reaction kinetics provide essential insights into the fundamental governmental principles of chemical reaction behavior.