Question

You study the effect of temperature on the rate of two reactions and graph the natural logarithm of the rate constant for each reaction as a function of \(1 / T .\) How do the two graphs compare (a) if the activation energy of the second reaction is higher than the activation energy of the first reaction but the two reactions have the same frequency factor, and (b) if the frequency factor of the second reaction is higher than the frequency factor of the first reaction but the two reactions have the same activation energy? [Section 14.5]

Short answer

In conclusion, when comparing the graphs of the natural logarithm of the rate constant as a function of the inverse of the temperature: a) If the activation energy of the second reaction is higher than that of the first reaction, but the frequency factors are the same, the graph for the second reaction will have a steeper negative slope and the same y-intercept as the first reaction. b) If the frequency factor of the second reaction is higher than that of the first reaction, but the activation energies are the same, the graph for the second reaction will have the same slope but a higher y-intercept than the first reaction.

Step-by-step solution

Understand the Arrhenius equation

The Arrhenius equation is given by: \(k = A \cdot e^{-\frac{E_a}{RT}}\) Where \(k\) is the rate constant, \(A\) is the frequency factor (also called the pre-exponential factor), \(E_a\) is the activation energy, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. The natural logarithm of the Arrhenius equation gives: \(\ln{k} = \ln{A} - \frac{E_a}{RT}\) By plotting \(\ln{k}\) versus \(\frac{1}{T}\), we can find the relationship between the rate constant and temperature for each reaction.

Compare the graphs for the first scenario

Scenario (a) has the same frequency factor, \(A\), and a higher activation energy, \(E_{a2}>E_{a1}\), for the second reaction. For the first reaction: \(\ln{k_1} = \ln{A} - \frac{E_{a1}}{RT}\) For the second reaction: \(\ln{k_2} = \ln{A} - \frac{E_{a2}}{RT}\) When comparing these two equations, we can see that the slopes of the graphs will differ, as the activation energies are different. Since \(E_{a2}>E_{a1}\), the graph for the second reaction will have a steeper negative slope than the graph for the first reaction. The two graphs will have the same y-intercept, as the frequency factors are equal.

Compare the graphs for the second scenario

Scenario (b) has the same activation energy, \(E_a\), and a higher frequency factor, \(A_2>A_1\), for the second reaction. For the first reaction: \(\ln{k_1} = \ln{A_1} - \frac{E_a}{RT}\) For the second reaction: \(\ln{k_2} = \ln{A_2} - \frac{E_a}{RT}\) When comparing these two equations, the slopes of the graphs will be equal as the activation energies are the same. The y-intercepts of the graphs will differ, with the second reaction having a higher y-intercept than the first reaction. This is because \(\ln{A_2} > \ln{A_1}\), as \(A_2 > A_1\). In conclusion, when comparing the graphs of the natural logarithm of the rate constant as a function of the inverse of the temperature: a) If the activation energy of the second reaction is higher than that of the first reaction, but the frequency factors are the same, the graph for the second reaction will have a steeper negative slope and the same y-intercept as the first reaction. b) If the frequency factor of the second reaction is higher than that of the first reaction, but the activation energies are the same, the graph for the second reaction will have the same slope but a higher y-intercept than the first reaction.

Key concepts

Activation Energy

The concept of activation energy is central to the understanding of chemical kinetics, which deals with the rates of chemical reactions. In simple terms, activation energy, often denoted as \(E_a\), is the minimum amount of energy that reacting molecules must possess in order to undergo a chemical transformation. This energy is required to break certain bonds within the reactant molecules so that new bonds can form to create the products. Imagine a hill where reactants must climb up to the top (the highest point being the activation energy) before they can roll down to form products.

The activation energy essentially acts as a gatekeeper; only those particles with sufficient energy to reach or surpass this threshold can successfully react. It's important to note that activation energy is not the total energy needed for a reaction but rather the necessary energy to initiate the reaction. The value of \(E_a\) can be influenced by various factors, such as the presence of a catalyst, which is a substance that lowers the activation energy, thereby increasing the rate of reaction. In our textbook exercise, when we compare two reactions with different activation energies but the same frequency factor, we see that a higher \(E_a\) results in a steeper negative slope on a \(\ln{k}\) versus \(\frac{1}{T}\) plot, illustrating that the reaction rate decreases more rapidly with temperature for the reaction with higher activation energy.

Frequency Factor

The frequency factor, denoted as \(A\), is another key element of the Arrhenius equation. It represents the number of times that reactant particles collide with each other with the correct orientation per unit time. Also referred to as the pre-exponential factor, it is essentially a measure of the likelihood that collisions will be effective in leading to a reaction.

This factor includes not only the frequency of collisions but also the orientation and steric factors; reactants must be aligned properly to react. When the frequency factor is high, there is a higher chance that reactant molecules will collide with the proper orientation, leading to more frequent reaction events. Our exercise demonstrates that if the frequency factor of one reaction is higher than another, but both have the same activation energy, the graph will show the same slope but different y-intercepts. A higher frequency factor translates to a higher rate constant \(k\), resulting in a higher y-intercept on the graph. This situation describes a reaction that, inherently, has a greater tendency to occur at a given temperature compared to one with a lower frequency factor.

Temperature and Reaction Rate

Temperature plays a crucial role in chemical kinetics, particularly in determining the reaction rate. Increasing the temperature typically increases the average kinetic energy of the molecules involved in the reaction. When molecules move faster, they collide more frequently, and with greater energy, thus more of these collisions have enough energy to overcome the activation energy barrier.

According to the Arrhenius equation, as temperature \(T\) increases, the fraction \(e^{-\frac{E_a}{RT}}\) increases, leading to an increased rate constant \(k\) and, consequently, a faster reaction rate. This relationship is why we may observe that reactions tend to proceed faster as the temperature rises. By plotting the natural logarithm of the rate constant against the inverse of temperature, we can visually assess how temperature affects the reaction rate. A steeper slope indicates a more pronounced effect of temperature changes on the reaction rate, which is tied to the magnitude of the activation energy of the reaction.

Natural Logarithm of Rate Constant

In kinetics studies, the natural logarithm of the rate constant (\(\ln{k}\)) is often plotted against the inverse of the temperature (\(\frac{1}{T}\)) to provide a clearer picture of the reaction's temperature dependence. This graph is instrumental in determining the activation energy and frequency factor from experimental data. The slope of this line corresponds to \(-\frac{E_a}{R}\), and the y-intercept corresponds to \(\ln{A}\).

The linear relationship, defined by the Arrhenius plot, allows chemists to predict how a reaction will behave under different temperature conditions with a considerable degree of accuracy. By using this plot, it can be seen that a higher activation energy leads to a steeper slope, and a larger frequency factor translates to a higher intercept. This method of analysis facilitates the comparison of different reactions and the determination of their kinetic parameters, which is precisely what the exercise seeks to clarify for students. Understanding the interpretation of the Arrhenius plot is fundamental for students tackling reaction kinetics, as it provides a window into the molecular world and how temperature influences the pace of chemical transformations.