Question

a. Using the free energy profile for a simple one-step reaction, show that at equilibrium \(K=k_{\mathrm{f}} / k_{\mathrm{r}}\), where \(k_{\mathrm{f}}\) and \(k_{\mathrm{r}}\) are the rate constants for the forward and reverse reactions. Hint: Use the relationship \(\Delta G^{\circ}=-R T \ln (K)\) and represent \(k_{\mathrm{f}}\) and \(k_{\mathrm{r}}\) using the Arrhenius equation \(\left(k=A e^{-E_{2} / R T}\right)\). b. Why is the following statement false? "A catalyst can increase the rate of a forward reaction but not the rate of the reverse reaction."

Short answer

In summary, we have shown that at equilibrium for a simple one-step reaction, the equilibrium constant K is equal to the ratio of the forward and reverse rate constants, \(K=\frac{k_\mathrm{f}}{k_\mathrm{r}}\), using the relationship between Gibbs free energy change and the Arrhenius equation. Furthermore, we have debunked the false statement, explaining that a catalyst increases the rate of both the forward and reverse reactions by providing an alternative reaction pathway with a lower activation energy.

Step-by-step solution

a. Proving the relationship at equilibrium

To prove that at equilibrium, \(K=\frac{k_\mathrm{f}}{k_\mathrm{r}}\), we need to use the relationship between the Gibbs free energy change under standard conditions (\(\Delta G^\circ\)) and the equilibrium constant (K), which is \(\Delta G^\circ=-R T \ln (K)\). We also need to represent the rate constants of the forward and reverse reactions using the Arrhenius equation, which is \(k=A e^{-E_\mathrm{a}/R T}\).

Write down the Arrhenius equations for forward and reverse reactions

For the forward reaction with a rate constant \(k_\mathrm{f}\), the Arrhenius equation can be written as: \(k_\mathrm{f}=A_\mathrm{f} e^{-E_\mathrm{a_\mathrm{f}}/R T}\) Similarly, for the reverse reaction with a rate constant \(k_\mathrm{r}\), the Arrhenius equation will be: \(k_\mathrm{r}=A_\mathrm{r} e^{-E_\mathrm{a_\mathrm{r}}/R T}\)

Calculate the standard Gibbs free energy change for the reaction

The standard Gibbs free energy change for the reaction (\(\Delta G^\circ\)) can be calculated as the difference between the activation energy for the forward reaction (\(E_\mathrm{a_\mathrm{f}}\)) and the activation energy for the reverse reaction (\(E_\mathrm{a_\mathrm{r}}\)) as follows: \(\Delta G^\circ = E_\mathrm{a_\mathrm{r}} - E_\mathrm{a_\mathrm{f}}\)

Substitute the Arrhenius equations into the given relationship

Now we need to substitute the Arrhenius equations for \(k_\mathrm{f}\) and \(k_\mathrm{r}\) into the relationship between \(\Delta G^\circ\) and the equilibrium constant (\(K\)): \(-R T \ln (K) = E_\mathrm{a_\mathrm{r}} - E_\mathrm{a_\mathrm{f}}\)

Solve for K

Next, use the relations for the forward and reverse rate constants from the Arrhenius equation to solve for K: \(\ln(K) = -\frac{E_\mathrm{a_\mathrm{r}} - E_\mathrm{a_\mathrm{f}}}{R T}\) Expanding this further, we get: \(K = e^{\frac{E_\mathrm{a_\mathrm{f}} - E_\mathrm{a_\mathrm{r}}}{R T}}\) Now substitute the Arrhenius equations and divide the equations: \(K = \frac{k_\mathrm{f}}{k_\mathrm{r}}= \frac{A_\mathrm{f} e^{-E_\mathrm{a_\mathrm{f}}/R T}}{A_\mathrm{r} e^{-E_\mathrm{a_\mathrm{r}}/R T}}\) Thus, we have shown that at equilibrium, \(K=\frac{k_\mathrm{f}}{k_\mathrm{r}}\).

b. Debunking the false statement

The statement "A catalyst can increase the rate of a forward reaction but not the rate of the reverse reaction" is false. A catalyst works by lowering the activation energy of both the forward and reverse reactions in a chemical reaction. This results in an increased rate for both the forward and reverse reactions without affecting the equilibrium constant. The catalyst achieves this by providing an alternative reaction pathway with a lower activation energy. Therefore, the statement is false because a catalyst does indeed increase the rate of both the forward and reverse reactions.

Key concepts

Gibbs Free Energy

Gibbs free energy, denoted as \(\Delta G\), is a thermodynamic quantity that predicts the direction of chemical reactions and whether they occur spontaneously. At equilibrium, the change in Gibbs free energy equals zero, indicating no net change in the composition of the system over time. This concept ties in closely with the equilibrium constant, \(K\), through the relationship \(\Delta G^\circ=-R T \ln(K)\), where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin. In essence, knowing the Gibbs free energy change provides insight into the favorability of a reaction and the position of equilibrium.

For example, a negative \(\Delta G\) suggests the reaction proceeds spontaneously in the forward direction, leading to the formation of products. Conversely, a positive \(\Delta G\) indicates the reaction favors the reverse direction, or the formation of reactants. In the context of a chemical reaction at equilibrium, understanding \(\Delta G\) offers a quantitative measure of how reactants and products are balanced.

Arrhenius Equation

The Arrhenius equation provides a mathematical relationship between the rate constant \(k\) of a reaction and the temperature \(T\). It is expressed as \(k=A e^{-E_a/R T}\), where \(A\) is the pre-exponential factor (or frequency factor), \(E_a\) is the activation energy of the reaction, and \(R\) is the universal gas constant. This important equation in chemical kinetics quantifies how the rate of a reaction increases with temperature, primarily due to an increase in the number of molecules that have enough energy to overcome the energy barrier, or activation energy, of the reaction.

This equation is a crucial tool for chemists because it allows them to predict how changes in temperature will affect the speed of chemical reactions. For instance, as temperature increases, \(k\) generally increases, meaning the reaction speeds up. This concept is inherently related to chemical kinetics and catalysis, as both involve understanding and manipulating reaction rates.

Equilibrium Constant

The equilibrium constant, \(K\), is a dimensionless value that characterizes the composition of reactants and products at equilibrium in a chemical reaction. It is calculated from the concentrations (or pressures) of the reactants and products. For a generic reaction \(aA + bB \leftrightarrow cC + dD\), the equilibrium constant \(K\) is expressed as \(K=[C]^c[D]^d/[A]^a[B]^b\), where the square brackets denote concentrations. At equilibrium, the rates of the forward and reverse reactions are equal, and thus, \(K\) remains constant at a given temperature.

The value of \(K\) indicates the extent to which a reaction will proceed. A large \(K\) suggests a greater concentration of products relative to reactants, while a small \(K\) indicates the opposite. The equilibrium constant is also intimately connected to Gibbs free energy, as mentioned previously, and provides crucial information for predicting the position of equilibrium in a chemical process.

Activation Energy

Activation energy, \(E_a\), is the minimum amount of energy that reacting molecules must possess for a reaction to occur. This energy acts as a barrier that reactants must overcome to be transformed into products. The concept of activation energy is central to the Arrhenius equation, as it determines the temperature dependency of reaction rates. Lower activation energy results in a higher rate constant \(k\), assuming other factors remain constant.

If a reaction has a high activation energy, it will proceed slowly since fewer molecules possess the energy necessary to overcome the energy barrier. Understanding activation energy aids in the design of catalysis that lowers this barrier, hence speeding up the reaction without changing the equilibrium position, as catalysis doesn't alter the energy of reactants or products, but rather provides an alternative pathway with a lower activation energy for the reaction to proceed.

Rate Constants

Rate constants, \(k\), are numerical values used in chemical kinetics to quantify the speed of a chemical reaction. They reflect how quickly reactants are converted into products. The rate of a reaction can usually be described by an equation that involves the rate constant and the concentrations of reactants. For example, in a simple first-order reaction \(A \rightarrow B\), the rate \(v\) is given by \(v=k[A]\), where \(k\) is the first-order rate constant.

Rate constants are temperature-dependent and can be calculated using the Arrhenius equation. They may vary dramatically with temperature, and this relationship allows chemists to understand and control reaction rates. Additionally, for a reaction at equilibrium, the ratio of the rate constants for the forward (\(k_f\)) and reverse (\(k_r\)) reactions provide the equilibrium constant \(K=\frac{k_f}{k_r}\), indicating a deep interconnection between chemical kinetics and equilibrium chemistry.

Chemical Kinetics

Chemical kinetics is the study of rates of chemical processes and the factors that affect these rates. It involves understanding how various variables such as concentration, temperature, and the presence of catalysts influence the speed at which reactants are converted into products. The laws of chemical kinetics are governed by rate expressions and the Arrhenius equation, which relate to the concepts of rate constants and activation energy.

One of the primary goals in the study of chemical kinetics is to understand the mechanism of a reaction, meaning the step-by-step sequence of elementary steps that make up the overall process. Kinetics can also differentiate between reactions that occur in a single step versus those that involve multiple steps, which may include intermediates and transition states. These insights are fundamental in designing chemical processes, selecting catalysts, and in the development of new materials and pharmaceuticals.

Catalysis

Catalysis refers to the process by which a substance, the catalyst, speeds up a chemical reaction without being consumed in the reaction itself. Catalysts function by lowering the activation energy for a reaction, enabling a larger proportion of the reactant molecules to participate in the reaction at a given temperature. They commonly provide an alternative pathway for the reaction to follow, which requires less energy for the transformation of reactants into products.

Catalysts are pivotal in both industrial chemistry and biological systems, as they allow reactions to proceed quickly and efficiently at temperatures that would otherwise be unfeasible. For both the forward and reverse reactions, a catalyst increases the rate similarly, thus maintaining the equilibrium constant. This characteristic of catalysts illustrates their role in enhancing reaction rates while preserving the underlying thermodynamic potentials of chemical reactions, a concept that is sometimes misunderstood but crucial for the development of sustainable and efficient chemical processes.