Question

The rate constant, the activation energy and Arrhenius parameter of a chemical reaction at \(300 \mathrm{~K}\) are \(\mathrm{K}, \mathrm{Ea}\) and \(\mathrm{A}\) respectively. The value of rate constant at \(\mathrm{T} \rightarrow\) \(\infty\) is (a) \(\mathrm{A}\) (b) \(\mathrm{Ea}\) (c) \(\mathrm{Ea} \times \mathrm{A}\) (d) \(\mathrm{A}-\mathrm{Ea}\)

Short answer

The rate constant at \( T \to \infty \) is \( A \).

Step-by-step solution

Understanding Arrhenius Equation

The Arrhenius equation describes how the rate constant ( \(k\) ) of a reaction depends on temperature and is given by \( k = A e^{-\frac{E_a}{RT}} \), where \( R \) is the gas constant, \( E_a \) is the activation energy, \( T \) is the temperature, and \( A \) is the pre-exponential factor, also known as the Arrhenius constant.

Evaluating the Equation at High Temperatures

As the temperature \( T \) approaches infinity, the fraction \( \frac{E_a}{RT} \) approaches zero since \( E_a \) is constant and \( RT \) becomes very large. This makes the exponent \(-\frac{E_a}{RT}\) tend towards zero.

Simplifying the Exponent

When \( \frac{E_a}{RT} \) approaches zero, the expression \( e^{-\frac{E_a}{RT}} \) approaches \( e^0 = 1 \), because the exponential of zero is 1.

Final Calculation of Rate Constant

Substitute the simplified exponential back into the Arrhenius equation: \( k = A \cdot e^{0} = A \cdot 1 = A \). Hence, at \( T \to \infty \), \( k = A \). Therefore, the rate constant approaches \( A \) at infinite temperature.

Key concepts

Reaction Rate Constant

A reaction rate constant, often denoted by the symbol \( k \), provides a measure of the speed of a chemical reaction at a given temperature. It quantifies how quickly reactants are converted into products. The magnitude of \( k \) is influenced by several factors, including temperature and the intrinsic properties of the reacting substances.

• High \( k \) values indicate fast reactions, whereas low \( k \) values suggest slower processes.
• \( k \) is a crucial parameter in the Arrhenius equation, connecting with temperature and activation energy.

The rate constant is a reflection of the underlying kinetic phenomena within a reaction system. Understanding \( k \) helps chemists design and control chemical reactions by choosing conditions that optimize the rate.

Activation Energy

Activation energy, represented as \( E_a \), is the minimum energy required for a chemical reaction to proceed. It acts as an energy barrier that reactants must overcome to transform into products.

• A low \( E_a \) means that less energy is needed for the reaction to occur, typically leading to faster reactions.
• Conversely, a high \( E_a \) suggests that more energy input is necessary to initiate the reaction, often slowing it down.

This concept is integral in kinetic studies because it explains why certain reactions only occur under specific temperature conditions. Activation energy also provides insight into the stability of molecules and the mechanism of the reaction, making it a fundamental concept in understanding chemical kinetics.

Temperature Dependence of Reactions

The temperature dependence of reactions is a critical aspect of chemical kinetics, which describes how changes in temperature impact the rate of a chemical reaction. The Arrhenius equation clarifies this relationship, showing that even small changes in temperature can significantly affect the rate constant \( k \).

• As temperature increases, the kinetic energy of molecules increases, leading to a higher probability of successful collisions between reactant molecules.
• The equation \( k = A e^{-\frac{E_a}{RT}} \) implies that at higher temperatures, the exponential factor approaches 1, making \( k \approx A \), thus enhancing the reaction rate.

By understanding temperature dependence, scientists can predict how reaction rates will change with varying thermal conditions, which is crucial in fields like environmental chemistry, industrial processes, and even biological systems.