Problem 26 - A Complete Resource Book in Chemistry for JEE Main

Question

The equation for the rate constant is \(k=\) Ae \(^{-E a R T} . A\) chemical reaction will proceed more rapidly if there is a decrease in (a) \(k\) (b) \(\mathrm{A}\) (c) \(\mathrm{E}\) (d) \(\mathrm{T}\)

Step-by-Step Solution

Answer

The reaction will proceed more rapidly if there is a decrease in activation energy \(E\) (option c).

Step 1: Analyze the Rate Constant Equation

The given equation is: \(k = A e^{-\frac{E_a}{RT}}\). This is known as the Arrhenius equation. Here, \(A\) is the pre-exponential factor or frequency factor, \(E_a\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the temperature expressed in Kelvin. The rate constant \(k\) indicates how fast a reaction proceeds.

Step 2: Understand the Effect of Activation Energy

Activation energy \(E_a\) represents the minimum energy required for a reaction to occur. A higher \(E_a\) results in a smaller \(k\), meaning the reaction would proceed more slowly. Therefore, decreasing \(E_a\) generally increases \(k\), speeding up the reaction.

Step 3: Consider Temperature's Role

Temperature \(T\) is in the denominator of the exponential part of the Arrhenius equation. An increase in \(T\) leads to a larger exponent \(-\frac{E_a}{RT}\) being less negative, thus increasing \(k\). Conversely, a decrease in \(T\) would make \(k\) smaller, slowing down the reaction.

Step 4: Evaluate Pre-exponential Factor Influence

The pre-exponential factor \(A\) does not affect the activation energy directly but is related to the frequency of molecular collisions. Reducing \(A\) would typically decrease \(k\), slowing the reaction.

Step 5: Determine the Parameter That Increases Reaction Rate

From the analysis, decreasing \(E_a\) will increase \(k\) and speed up the reaction. Decreasing \(T\) or \(A\) would generally reduce \(k\), slowing the reaction. Thus, decreasing \(E_a\) is the only means to make the reaction proceed more rapidly.

Key Concepts

Rate Constant

In the realm of chemical kinetics, the rate constant, denoted by \(k\), is a pivotal player in determining the speed of a chemical reaction. This constant is derived from the well-known Arrhenius equation: \[ k = A e^{-\frac{E_a}{RT}} \] Here, \(A\) is the pre-exponential factor, \(E_a\) the activation energy, \(R\) the universal gas constant, and \(T\) the temperature in Kelvin. The rate constant tells us how fast or slow a reaction proceeds under specific conditions. - A higher \(k\) value means a faster reaction rate.- Conversely, a lower \(k\) implies a slower rate.The beauty of \(k\) lies in its sensitivity to both the activation energy and temperature, acting as a beacon for understanding reaction speed.
Thus, exploring the parameters that affect \(k\), like temperature and activation energy, becomes crucial.

Activation Energy

Activation energy, \(E_a\), represents the minimum amount of energy that reactant molecules must possess for a successful transformation to products. It is essentially the energy barrier that must be overcome for a reaction to occur.- A high \(E_a\) means the reaction requires more energy to proceed.- A low \(E_a\) makes it easier for the reaction to occur, increasing its rate.In the Arrhenius equation, \(E_a\) appears in the exponential term as \(e^{-\frac{E_a}{RT}}\).
The larger the \(E_a\), the smaller the value of this term becomes, leading to a lower rate constant \(k\).
Lowering \(E_a\) is akin to lowering the hurdles in a race, making it easier for reactants to transform into products, thereby speeding up the reaction.
This concept illustrates why catalysts are so crucial—they lower the activation energy, allowing reactions to proceed more rapidly.

Temperature Effect

Temperature plays a fascinating role in chemical reaction rates. It is found in the Arrhenius equation as part of the denominator in the exponential expression: \(e^{-\frac{E_a}{RT}}\). An increase in temperature enhances the kinetic energy of molecules, leading to more frequent and energetic collisions. As a result:- The \\(\frac{E_a}{RT}\) \/ exponent becomes less negative.- Consequently, the exponential term grows larger.- This increase results in a higher rate constant, \(k\), thus accelerating the reaction.Conversely, lowering the temperature decreases molecular movement, resulting in less frequent collisions and a smaller \(k\), slowing down the reaction. Therefore, temperature is a critical factor in controlling how fast or slow a reaction takes place.

Pre-exponential Factor

The pre-exponential factor, \(A\), also known as the frequency factor, is often overlooked but is quite essential. It accounts for the frequency of collisions and the proper orientation required for molecules to successfully react.While it doesn't alter the activation energy, it plays a significant role in determining \(k\):- A higher \(A\) increases the likelihood of effective collisions, thus raising \(k\).
- A lower \(A\), meanwhile, decreases the chance of productive interactions, resulting in a lower rate constant.\(A\) provides a quantitative measure of the inherent collision frequency and correctness of orientation, independent of energy barriers.