Problem 54 - A Complete Resource Book in Chemistry for JEE Main

Question

A reaction rate is given by \(k=1.5 \times 10^{15} \exp (-25000 / \mathrm{RT}) \mathrm{s}^{-1}\) it means that (a) half-life period of the reaction will be smaller at high temperature (b) log vs \(\mathrm{T}\) will give a straight line (c) half-life of the reaction will be smaller at lower temperature (d) \(\log\) vs \(1 / \mathrm{T}\) will give a straight line having a slope of \(-25000\)

Step-by-Step Solution

Answer

Correct options: (a) and (d).

Step 1: Understanding the Arrhenius Equation

The given equation \( k = 1.5 \times 10^{15} \exp\left(-\frac{25000}{RT}\right) \ s^{-1} \) is an expression using the Arrhenius equation format \( k = A \exp\left(-\frac{E_a}{RT}\right) \). Here, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy (25000 in this case), \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Analyzing the Arrhenius Plot

From the Arrhenius equation \( \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \), we know that a plot of \( \ln k \) versus \( \frac{1}{T} \) will yield a straight line with a slope of \(-\frac{E_a}{R} \). In this problem, \( E_a = 25000 \), indicating a straight line with slope \(-25000/R\). This makes option (d) correct.

Step 3: Evaluating Temperature Effects on Reaction Rate

According to the Arrhenius equation, as the temperature \( T \) increases, the value of \( k \) increases, which implies that the reaction rate increases and the half-life decreases at higher temperatures. Thus, option (a) is correct, and option (c) is incorrect.

Step 4: Checking Options Involving Direct Temperature Plots

Reviewing options (b) and (d), a direct plot of \( \log \) vs \( T \) doesn't match the Arrhenius formula, which suggests incorrect assumptions about the relationship. Option (d) is indeed correct as explained before.

Key Concepts

Activation Energy

Activation energy is a crucial concept in chemical kinetics. It's the minimum energy that reacting molecules must have for a reaction to occur. Think of activation energy like a hill that reactants need to climb before a reaction can happen. If they don't have enough energy, they can't get over the hill. In the \[ k = 1.5 \times 10^{15} \exp\left(-\frac{25000}{RT}\right) \, s^{-1} \]equation from the exercise, the number 25000 represents the activation energy, expressed in Joules per mole. This number helps determine how sensitive a reaction is to changes in temperature.

• Higher activation energy means slower reactions, as it requires more energy to get going.
• Lower activation energy results in faster reactions, given the same conditions.

Understanding activation energy is key to predicting how various factors, like temperature, influence reaction rates.

Reaction Rate

The reaction rate tells us how quickly a reaction proceeds. It's influenced by several factors, including the activation energy and temperature. In the equation \( k = 1.5 \times 10^{15} \exp\left(-\frac{25000}{RT}\right) \, s^{-1} \), the variable \( k \) is the rate constant. This constant changes with temperature, but gives us a fixed rate of reaction for specific conditions.

• Higher \( k \) implies a faster reaction.
• Lower \( k \) suggests a slower reaction.

The Arrhenius equation links the rate constant to temperature, offering insights into how likely molecules will successfully collide and react. When the activation energy is high, fewer molecules have the needed energy to cross the activation threshold, slowing down the reaction.

Temperature Effect on Reaction Rate

Temperature plays a pivotal role in the speed of chemical reactions. According to the Arrhenius equation, the reaction rate increases exponentially with an increase in temperature. This can be explained by the fact that higher temperatures provide energy to molecules, increasing their movement and the probability of successful collisions.

• As temperature rises, molecules gain kinetic energy.
• Increased energy means a higher number of effective collisions.
• This results in a higher reaction rate as more molecules can surpass the activation energy.

Thus, as deduced from the original exercise, higher temperatures lead to a decrease in the reaction's half-life, as the reaction completes faster due to increased rates.

Half-life of Reaction

The half-life of a reaction is the time required for half of a substance to be consumed in a reaction. Unlike in radioactive decay, the half-life in chemical reactions can vary with the condition, especially with temperature. From the Arrhenius perspective, this occurs because the reaction rate, \( k \), is not constant but varies with temperature. Higher temperatures increase the reaction rate \( k \), leading to a shorter half-life.

• At high temperatures, reactions complete quicker, reducing their half-life.
• At low temperatures, reactions proceed slower, extending their half-life.

Thus, understanding the dependence of half-life on temperature is vital for managing reactions in chemical processes, ensuring they proceed at desired rates under specific conditions.