Problem 102 - A Complete Resource Book in Chemistry for JEE Main

Question

For a certain reaction, the activation energy is zero. What is the value of rate constant at \(300 \mathrm{~K}\), if \(\mathrm{K}=1.6\) \(\times 10^{6} \mathrm{~s}^{-1}\) at \(280 \mathrm{~K} ?\) (a) \(1.6 \times 10^{6} \mathrm{~s}^{-1}\) (b) zero (c) \(4.8 \times 10^{8} \mathrm{~s}^{-1}\) (d) \(3.2 \times 10^{12} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

(a) \(1.6 \times 10^6 \mathrm{~s}^{-1}\)

Step 1: Understanding Activation Energy and Its Effect

When the activation energy for a reaction is zero, it means the rate constant does not depend on temperature according to the Arrhenius equation. The Arrhenius equation is given by \(k = A e^{-E_a/(RT)}\), where \(E_a\) is the activation energy. If \(E_a = 0\), the equation simplifies to \(k = A\), indicating the rate constant \(k\) remains constant regardless of temperature.

Step 2: Determining Rate Constant at 300 K

Given the activation energy is zero, the rate constant \(k\) remains constant over different temperatures. This means that the rate constant at \(300 \mathrm{~K}\) will be the same as the rate constant at \(280 \mathrm{~K}\), which is given as \(1.6 \times 10^6 \mathrm{~s}^{-1}\).

Step 3: Conclusion and Answer Selection

Based on our understanding that the rate constant is not affected by temperature when the activation energy is zero, we conclude that the rate constant at \(300 \mathrm{~K}\) is \(1.6 \times 10^6 \mathrm{~s}^{-1}\). Therefore, the correct answer is option (a).

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula in chemistry.
It describes how the rate constant ( k ) of a reaction is influenced by temperature and activation energy. The equation is given as: \( k = A e^{-E_a/(RT)} \) where:

• \( A \) is the pre-exponential factor, representing the frequency of collisions with the correct orientation for reaction.
• \( E_a \) is the activation energy, the minimum energy required for a reaction to occur.
• \( R \) is the universal gas constant.
• \( T \) is the absolute temperature in Kelvin.

In scenarios where the activation energy \( E_a \) is zero, the Arrhenius equation simplifies significantly. Instead of a complex dependency on temperature, we derive that the rate constant \( k \) simply equals the pre-exponential factor \( A \). This outcome means the rate constant remains constant across different temperatures.
Understanding this equation helps us appreciate why certain reactions are unaffected by temperature changes, as evidenced by the problem's solution.

Rate Constant

The rate constant, symbolized as \( k \), is a crucial component in the rate of a chemical reaction.
It quantifies how quickly a reaction occurs without external impact factors like concentration or pressure affecting its value. Generally expressed in units like \( \,s^{-1} \) or \( \,mol^{-1}L \,s^{-1} \), it provides insight into a reaction's speed.

The value of the rate constant is determined by the nature of the reaction and conditions like temperature. Typically, it increases with temperature if activation energy is involved. However, if the activation energy \( E_a \) is zero, the rate constant remains unchanged regardless of temperature alterations.

In the given exercise, when the activation energy is zero, the rate constant at 300 K stays at \( 1.6 \times 10^{6} \,s^{-1} \), identical to its value at 280 K.
This stability highlights the inherent property of certain reactions and makes understanding \( k \) essential for predicting reaction behavior under various conditions.

Temperature Dependence of Reactions

Temperature is a powerful factor affecting chemical reactions. It influences both the speed and the forward reach of reactions. Most reactions quicken as temperature rises because the additional heat energy helps a greater number of molecules overcome the activation energy barrier. This increased molecular movement enhances the likelihood of successful collisions between reactant molecules.

However, in cases where the activation energy is zero, as described in the problem, this dependency is nullified. In such scenarios, rate constants stay constant across different temperatures. This intriguing exception simplifies analysis significantly because it removes temperature as a variable in reaction speed.

Understanding how temperature influences reaction rates, or doesn't, when \( E_a \) is zero, is key for comprehending broader chemical kinetics. It illustrates how the locked behavior of constant rate constants can simplify conditions where reactions occur, enabling easier control and predictability in industrial or experimental settings.