Problem 15 - A Complete Resource Book in Chemistry for JEE Main

Question

The unit of second-order reaction rate constant is (a) \(\mathrm{L}^{-1}, \mathrm{~mol}^{-1} \mathrm{~d} \mathrm{~s}^{-1}\) (b) \(\mathrm{L}^{2} \mathrm{~mol}^{-2} \mathrm{~s}^{-1}\) (c) \(\mathrm{L} \cdot \mathrm{mol}^{-1} \mathrm{~s}^{-1}\) (d) \(\mathrm{s}^{-1}\)

Step-by-Step Solution

Answer

Option (c): \(\mathrm{L} \cdot \mathrm{mol}^{-1} \mathrm{~s}^{-1}\).

Step 1: Understand Second-Order Reaction

For a second-order reaction, the rate is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. It's generally given as \( \text{Rate} = k[A]^2 \) or \( \text{Rate} = k[A][B] \).

Step 2: Define the Rate Units

The rate of reaction has units of \( \text{mol} \cdot \text{L}^{-1} \cdot \text{s}^{-1} \). This is because rate is defined as the change in concentration over time.

Step 3: Determine Units for Rate Constant

For a reaction of the form \( \text{Rate} = k[A]^2 \), rearrange to \( k = \frac{\text{Rate}}{[A]^2} \). Since \( [A] \) has units of \( \text{mol} \cdot \text{L}^{-1} \), then \( [A]^2 \) has units of \( \text{mol}^2 \cdot \text{L}^{-2} \).

Step 4: Calculate Units for \(k\)

Divide the rate units \( \text{mol} \cdot \text{L}^{-1} \cdot \text{s}^{-1} \) by \( \text{mol}^2 \cdot \text{L}^{-2} \) to find the units of \( k \):\[ k = \frac{\text{mol} \cdot \text{L}^{-1} \cdot \text{s}^{-1}}{\text{mol}^2 \cdot \text{L}^{-2}} = \text{L} \cdot \text{mol}^{-1} \cdot \text{s}^{-1} \]

Step 5: Match Result with Options

Compare this result \( \text{L} \cdot \text{mol}^{-1} \cdot \text{s}^{-1} \) with the given options. This matches option (c).

Key Concepts

Reaction Rate

A reaction rate measures how quickly a chemical reaction occurs. It indicates the change in concentration of reactants or products over a certain period. The factors influencing reaction rates typically include concentration, temperature, surface area, and catalysts.

For example, in a simple reaction such as - A + B → Products The rate of reaction can be expressed as the decrease in concentration of reactant A per unit time. This can be denoted as −d[A]/dt. It is important to monitor this to either speed up or slow down a reaction based on the desired outcomes.

When studying reaction rates, you look at initial rates or the rate over a specific time interval. It is indispensable for predicting how a reaction proceeds over time and for controlling industrial chemical processes.

Units of Rate Constant

The units of a rate constant are essential in understanding the type of reaction taking place. Rate constants allow chemists to quantify the speed of a reaction under given conditions. Depending on the order of the reaction, the units of the rate constant will vary.

For a second-order reaction, the rate constant ( **k**) has units expressed in terms of - L - mol ^{-1} - s ^{-1} This is derived from the formula for the reaction rate, which involves concentration terms raised to certain powers. In the case of a second-order reaction, the units can be determined using the formula - ( text{mol} ^{-1} L imes s ^{-1} ) for one reactant squared or a combination of two reactants. The units ensure dimensional consistency across the rate equation.
Understanding and correctly using these units is critical in experimental settings because they direct how measurements should be recorded and analyzed.

Second-Order Reaction

Second-order reactions are characterized by a reaction rate that is proportional to either the product of the concentrations of two reactants or the square of the concentration of a single reactant. This can be mathematically expressed as:
- Rate = k[A]^2 or - Rate = k[A][B] Where - [A] and [B] are the concentrations of the reactants.
In second-order reactions, doubling the concentration of a reactant has a squared effect on the rate. For example, if [A] doubles, the rate increases by a factor of four ( 2^2 = 4 ).
These reactions are important in chemical kinetics and show how concentrations of substances in a reaction mix evolve over time. They are often found in biological and environmental systems where two components must come together to react and where precision in predicting the reaction rate is critical in scientific research.