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

Question

The unit of second-order reaction rate constant is (a) \(\mathrm{L}^{-1} \cdot \mathrm{mol}^{-1} \mathrm{ds}^{-1}\) (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): \( \mathrm{L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1} \) is the correct unit for the second-order reaction rate constant.

Step 1: Understanding Second-order Reactions

In a second-order reaction, the rate of reaction is proportional to either the square of the concentration of one reactant or to the product of the concentrations of two different reactants. The general form of the rate equation for a second-order reaction is \( r = k [A]^2 \) or \( r = k [A][B] \).

Step 2: Identifying the Units for Rate

The rate of reaction (\( r \)) is typically expressed in terms of concentration change per unit time, which has units of \( \mathrm{mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{s}^{-1} \).

Step 3: Determining Units for Second-order Rate Constant

Since the rate of reaction for a second-order process is given by \( r = k [A]^2 \) or \( r = k [A][B] \), the units of \([A]\) or \([B]\) are \(\mathrm{mol} \cdot \mathrm{L}^{-1}\). Substituting into the rate equation \( \mathrm{mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{s}^{-1} = k (\mathrm{mol} \cdot \mathrm{L}^{-1})^2 \), we can solve for the units of \( k \), which are \( \mathrm{L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1} \).

Step 4: Matching Units to Given Options

Now, compare the derived units for the second-order rate constant \( \mathrm{L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1} \) with the provided options. Options (a), (b), and (c) are given different units, but option (c) matches our derived units exactly.

Key Concepts

Reaction Rate Constant

The reaction rate constant, often represented as \( k \), is a crucial component when studying chemical kinetics. This constant provides insight into how fast a reaction occurs. It is unique for every reaction and can help determine how changes in concentration affect the reaction speed. In the context of a second-order reaction, the rate constant can be found in the equation \( r = k [A]^2 \) or \( r = k [A][B] \), where \( r \) stands for the rate of reaction and \([A]\), \([B]\) for the concentrations of reactants involved.

The numerical value of \( k \) can be determined experimentally, often using the method of initial rates, and will vary based on factors like temperature and the presence of a catalyst. It is helpful for predicting how long it will take for reactants to turn into products under different conditions.

The rate constant is not to be confused with the reaction rate itself; while the reaction rate quantifies the speed of the reaction at a given time, the rate constant is a fixed value provided these specified conditions.

Rate Equation

A rate equation is a mathematical expression that describes the link between the concentrations of reactants and the rate of reaction. For a second-order reaction, this equation can either take the form \( r = k [A]^2 \) or \( r = k [A][B] \).

These formulae highlight how the reaction rate is dependent on the concentrations of reactants. In simple terms:

• \( k \) is the reaction rate constant.
• \([A]\) and \([B]\) are the initial concentrations of the reactants.
• \([A]^2\) or \([A][B]\) indicates that doubling the concentration has a squared effect on the reaction rate.

For instance, if \( [A] \) doubles, then \( [A]^2 \) quadruples, significantly increasing the reaction rate—a telling feature of second-order kinetics.

The rate equation aids chemists in predicting how changes in the system can influence the overall rate, making it a vital tool for both research and practical applications in chemistry.

Units of Measurement

Understanding the units of measurement in chemical kinetics is essential to interpreting the results correctly. For second-order reactions, the units of the rate constant \( k \) are crucial. Since the rate of reaction \( r \) in a second-order reaction is expressed as \( \mathrm{mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{s}^{-1} \), and concentration is expressed as \( \mathrm{mol} \cdot \mathrm{L}^{-1} \), we can derive the units for \( k \).

Consider the rate equation \( \mathrm{mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{s}^{-1} = k (\mathrm{mol} \cdot \mathrm{L}^{-1})^2 \). By rearranging the formula to solve for \( k \), we find that \( k \) must have the units \( \mathrm{L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1} \). The role of units can often clarify whether calculations have been set up and processed correctly.

Properly labeling units enables students and professionals alike to seamlessly compare and interpret different chemical processes across varying systems and experimental conditions.