Problem 239 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction \(\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{k}\left[\mathrm{H}^{+}\right]^{\mathrm{n}} .\) If \(\mathrm{pH}\) of the reaction medium changes from two to one, rate becomes 100 times that of the value at \(\mathrm{pH}=2\). The order of reaction is

Step-by-Step Solution

Answer

The order of reaction is 2.

Step 1: Understanding the Reaction Rate Equation

The given equation for the reaction rate is \( \frac{\mathrm{dx}}{\mathrm{dt}} = \mathrm{k} [\mathrm{H}^{+}]^n \), where \( k \) is the rate constant, \( [\mathrm{H}^{+}] \) is the concentration of hydrogen ions, and \( n \) is the order of the reaction with respect to \([\mathrm{H}^{+}]\).

Step 2: Calculate Initial Hydrogen Ion Concentration

At \( \text{pH} = 2 \), the hydrogen ion concentration, \([\mathrm{H}^{+}]\), can be calculated using the formula \([\mathrm{H}^{+}] = 10^{-\text{pH}}\). Thus, \([\mathrm{H}^{+}] = 10^{-2} = 0.01 \text{M}\).

Step 3: Calculate Hydrogen Ion Concentration at New pH

For \( \text{pH} = 1 \), the hydrogen ion concentration is \([\mathrm{H}^{+}] = 10^{-1} = 0.1 \text{M}\).

Step 4: Relate Rate Changes to pH Changes

Let the initial rate at \( \text{pH} = 2 \) be \( R_1 = k (0.01)^n \) and the rate at \( \text{pH} = 1 \) be \( R_2 = k (0.1)^n \). According to the problem, \( R_2 = 100 \times R_1 \).

Step 5: Set Up the Equation for Rate Change

We have \( k (0.1)^n = 100 \times k (0.01)^n \). Simplifying, we get \( (0.1)^n = 100 \times (0.01)^n \).

Step 6: Simplify and Solve for n

Divide both sides by \( (0.01)^n \) to get \( (0.1)^n (0.01)^{-n} = 100 \), which simplifies to \( 10^n = 100 \). Therefore, \( n = 2 \).

Step 7: Conclusion

The reaction order with respect to \([\mathrm{H}^{+}]\) is \( n = 2 \).

Key Concepts

Rate Equation

In the study of chemical kinetics, a rate equation is a mathematical expression that describes how the rate of a chemical reaction depends on the concentration of reactants. It often takes the form \( \frac{\mathrm{dx}}{\mathrm{dt}} = \mathrm{k} [A]^n \), where \( \mathrm{k} \) is the rate constant, \([A]\) is the concentration of a reactant, and \( n \) is the reaction order.

• \( \frac{\mathrm{dx}}{\mathrm{dt}} \): This symbolizes the rate of reaction, or how quickly the reactant is converted into a product over time.
• \( \mathrm{k} \): The rate constant is a proportional factor specific to a given reaction at a certain temperature.
• \( n \): The order of reaction, denoting how the concentration of a reactant influences the rate of reaction.

Understanding these components is crucial to predicting and controlling reaction speeds, which has practical applications in industries ranging from pharmaceuticals to food processing.

pH and Hydrogen Ion Concentration

The concept of pH is fundamental in chemistry as it measures the acidity or basicity of a solution. It is a logarithmic scale:

• \( \text{pH} = -\log_{10} [\mathrm{H}^+] \), where \([\mathrm{H}^+] \) is the hydrogen ion concentration.
• A lower pH indicates a higher concentration of hydrogen ions and hence a more acidic solution.
• A change by one pH unit alters the hydrogen ion concentration by a factor of 10.

For instance, changing from pH 2 to pH 1 increases \([\mathrm{H}^+]\) from 0.01 M to 0.1 M, demonstrating a tenfold increase in acidity.

Reaction Rate Dependence on pH

The rate of a reaction can be highly sensitive to changes in pH, particularly for reactions involving hydrogen ions. In the provided exercise, the reaction rate increases dramatically, becoming 100 times faster when the pH changes from 2 to 1. This rate increase can be attributed to:

• Increased concentration of hydrogen ions, \( [\mathrm{H}^+] \), boosting the rate of reaction.
• Greater accessibility of reactants to participate effectively in the reaction.

The dependency of the reaction rate on pH is crucial in biochemical processes, such as enzyme catalysis, where even slight pH changes can significantly impact rate and efficiency.

Order of Reaction Determination

The order of reaction is a key concept in chemical kinetics. It is determined by how the rate of reaction changes with the concentration of reactants:

• Experimentally, by observing how rate varies when concentrations change.
• In our scenario, if changing the pH from 2 to 1 boosts the rate by a factor of 100, we deduce this represents two powers of 10. Thus, the order \( n \) must be 2.
• This implies the rate depends quadratically on \([\mathrm{H}^+]\).

Knowing the reaction order is valuable for predicting how a change in conditions will affect the rate, assisting in optimizing reactions in industrial processes.