Question

In the following question two statements Assertion (A) and Reason (R) are given Mark. a. If \(\mathrm{A}\) and \(\mathrm{R}\) both are correct and \(\mathrm{R}\) is the correct explanation of \(\mathrm{A}\); b. If \(A\) and \(R\) both are correct but \(R\) is not the correct explanation of \(\mathrm{A}\); c. \(\mathrm{A}\) is true but \(\mathrm{R}\) is false; d. \(\mathrm{A}\) is false but \(\mathrm{R}\) is true, e. \(\mathrm{A}\) and \(\mathrm{R}\) both are false. (A): The rate constant increases exponentially with the increase in temperature. ( \(\mathbf{R}\) ): With the rise in temperature, the average kinetic energy of the molecules increases.

Short answer

Both A and R are correct, and R explains A; answer is (a).

Step-by-step solution

Understanding Assertion (A)

The assertion states that the rate constant in a chemical reaction increases exponentially as temperature increases. This is described by the Arrhenius equation: \( k = A \exp\left(-\frac{E_a}{RT}\right) \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. As temperature \( T \) increases, the exponent becomes less negative, leading to an increase in \( k \). Therefore, Assertion (A) is true.

Understanding Reason (R)

The reason states that with a temperature increase, the average kinetic energy of molecules increases. This is based on the kinetic molecular theory, which posits that temperature is a measure of the average kinetic energy of particles in a substance. As temperature rises, the average kinetic energy increases. Therefore, Reason (R) is true.

Evaluate if R explains A

To determine if the reason is the correct explanation of the assertion, we consider how the increase in kinetic energy affects the rate constant. As kinetic energy increases, molecules move faster and collide more frequently and with more energy, which can lead to more effective collisions and a higher reaction rate. Thus, this supports the exponential increase in the rate constant described in (A). Hence, Reason (R) properly explains Assertion (A).

Key concepts

Rate Constant

The rate constant, often symbolized as \( k \), is a fundamental factor in determining the speed of a chemical reaction. In chemistry, the Arrhenius equation describes how the rate constant varies with temperature:\[k = A \exp\left(-\frac{E_a}{RT}\right)\]

• \( A \) is the pre-exponential factor, representing the frequency of collisions with the correct orientation.
• \( E_a \) is the activation energy, which is the minimum energy that reacting species need for a successful collision.
• \( R \) is the gas constant, a fundamental constant in chemistry.
• \( T \) is the temperature in Kelvin.

As the temperature \( T \) increases, the exponent \(-\frac{E_a}{RT}\) becomes less negative, resulting in a larger value for \( k \). This implies that an increase in temperature usually leads to an increase in the rate constant \( k \). Thus, reactions tend to occur faster at higher temperatures.

Kinetic Molecular Theory

The kinetic molecular theory provides a framework for understanding how temperature influences molecular motion and reaction rates. It essentially relates temperature with the average kinetic energy of molecules:

• Temperature as Average Kinetic Energy: Temperature is a direct measure of the average kinetic energy of the particles in a substance.
• Molecular Movement: As the temperature increases, molecules within a substance move more rapidly.
• Impact on Reaction Rates: Higher molecular speeds result in more frequent and energetic collisions during a reaction.

With elevated temperature, more molecules possess the energy exceeding the activation energy \( E_a \). Consequently, a greater number of effective collisions occur, thereby enhancing the reaction rate.

Average Kinetic Energy

Average kinetic energy is a crucial concept, particularly when understanding how temperature affects the speed of a reaction. In terms of its mathematical expression, the average kinetic energy \( \overline{E_k} \) is given by:\[\overline{E_k} = \frac{3}{2} k_B T\]where \( k_B \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

• Energy Distribution: At a given temperature, not all molecules have identical energies. The distribution is often described by the Maxwell-Boltzmann distribution.
• Influence on Chemical Reactions: Only molecules with sufficient kinetic energy to overcome the activation energy barrier contribute to the reaction.

Therefore, as average kinetic energy rises with temperature, the proportion of molecules with energy exceeding \( E_a \) increases, leading to a higher reaction rate. This effect explains why reactions generally proceed faster at elevated temperatures.