Problem 61 - Chemistry: The Central Science

Question

(a) A certain first-order reaction has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\). What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if \(E_{a}=75.5 \mathrm{~kJ} / \mathrm{mol}\) ? (b) Another first-order reaction also has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\) What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if \(E_{a}=125 \mathrm{~kJ} / \mathrm{mol}\) ? (c) What assumptions do you need to make in order to calculate answers for parts (a) and (b)?

Step-by-Step Solution

Answer

Using the Arrhenius equation, we find the rate constants at 60ºC for both reactions: (a) For the first reaction, with an activation energy of \(E_a = 75.5 \times 10^3 J\text{mol}^{-1}\), we calculate the rate constant at 60ºC as \(k_2 = 2.07 \times 10^{-1} \mathrm{s^{-1}}\). (b) For the second reaction, with an activation energy of \(E_a = 125 \times 10^3 J\text{mol}^{-1}\), we calculate the rate constant at 60ºC as \(k_2 = 6.81 \times 10^{-3} \mathrm{s^{-1}}\). Assumptions made in this calculation include: (1) the reactions follow the first-order kinetics, (2) the activation energy and pre-exponential factor are constant over the temperature range considered, and (3) the temperature change does not affect the reaction mechanism, and the rate-determining step remains the same.

Step 1: Get the k value at the initial temperature

Use the Arrhenius equation: \(k_1 = k_0 e^{-\frac{E_a}{R T_1}}\) Now, we need to find k0 (the pre-exponential factor) using the given values.

Step 2: Solve for k0

Rearrange the equation to solve for k0: k0_=\(\frac{k_1}{e^{-\frac{E_a}{R T_1}}}\) Using the given values: k0_= \(\frac{2.75 \times 10^{-2} \mathrm{s^{-1}}}{e^{\frac{-75.5 \times 10^3 J \text{mol}^{-1}}{(8.314 J \text{mol}^{-1}K^{-1})(293.15K)}}\) Calculate k0.

Step 3: Calculate k at the new temperature

Use the Arrhenius equation and plug in the calculated k0 and the new temperature T2: \(k_2=k_0 e^{-\frac{E_a}{R T_2}}\) Calculate k2, which is the rate constant at 60ºC for reaction (a). (b) We are given the same initial k and temperature values, but now: Ea_= \(125 \times 10^3 J \text{mol}^{-1}\) Repeat steps 1-3 using this new Ea value to find the rate constant at 60ºC for reaction (b). (c) To answer the assumptions made for parts (a) and (b):

Step 4: Assumption 1

The reaction follows first-order kinetics, as mentioned in the problem.

Step 5: Assumption 2

The activation energy (Ea) and pre-exponential factor (k0) are constant over the temperature range considered.

Step 6: Assumption 3

The temperature change does not affect the reaction mechanism, and the rate-determining step remains the same.

Key Concepts

First-Order Reaction

A first-order reaction is a type of chemical reaction where the rate depends linearly on the concentration of a single reactant. This means the rate of the reaction can be expressed as:

• \( ext{Rate} = k[A] \)

In this equation, \( A \) represents the concentration of the reactant, and \( k \) is the rate constant. Such reactions are characterized by a constant half-life that doesn't depend on the initial concentration of the reactants.
This makes them predictable and easy to study in laboratory settings.
A typical example of a first-order reaction is radioactive decay.

• Easy prediction with fixed half-lives
• Simplifies mathematical treatment due to linear relationships

Rate Constant

The rate constant, denoted as \( k \), plays a crucial role in determining the speed of a chemical reaction. It is a proportionality constant in the rate law equations. In the context of a first-order reaction, its unit is inverse time, such as \( ext{s}^{-1} \).
The value of the rate constant indicates how fast or slow a reaction proceeds. Higher values of \( k \) correspond to faster reactions and vice versa.
The rate constant is temperature dependent and can be calculated using the Arrhenius equation:

• \( k = A e^{-rac{E_a}{RT}} \)

Where:

• \( A \) is the pre-exponential factor
• \( E_a \) is the activation energy
• \( R \) is the gas constant
• \( T \) is the temperature in Kelvin

Activation Energy

Activation energy, often symbolized as \( E_a \), is the minimum energy that reacting molecules need to undergo a reaction. It is a threshold energy barrier that must be overcome for reactants to transform into products.
In practical terms, activation energy affects the rate of reaction; the lower the energy, the faster the reaction can proceed.
Typically measured in kilojoules per mole (kJ/mol), activation energy can be determined through experimental observations of reaction rates at different temperatures.
It's a critical factor in the Arrhenius equation, influencing how \( k \) changes with temperature.
Many reactions require the input of activation energy to proceed, even if the overall process is exothermic.

Pre-Exponential Factor

The pre-exponential factor, often denoted as \( A \), is a term within the Arrhenius equation that embodies the frequency of collisions and the probability that the collisions have the correct orientation to lead to a reaction.
It is also known as the frequency factor or the Arrhenius constant.
Usually, \( A \) has similar units as the rate constant and is temperature-independent over the range considered in many reactions.
Although it might seem like a constant with little significance, \( A \) greatly influences the rate of reaction by dictating the general number of effective collisions per unit time.

• Represents the likelihood of successful collisions
• A higher \( A \) indicates more frequent and oriented collisions leading towards a reaction