Question

A hydrogen bonded dimer is formed between 2 -pyridone according to the reaction O=c1cccc[nH]1 O=c1cccc[nH]1 The relaxation time for this reaction, which occurs in nanoseconds, has been determined in chloroform at \(298 \mathrm{~K}\) at various concentrations of 2 -pyridone. The data obtained are [G. G. Hammes and A. C. Park, J. Am. Chem. Soc. 91 , 956 (1969)): \begin{tabular}{lc} 2-Pyridone (M) & \(10^{9} \tau(\mathrm{s})\) \\ \hline \(0.500\) & \(2.3\) \\ \(0.352\) & \(2.7\) \\ \(0.251\) & \(3.3\) \\ \(0.151\) & \(4.0\) \\ \(0.101\) & \(5.3\) \\ \hline \end{tabular} From these data calculate the equilibrium and rate constants characterizing this reaction. Hint: If the expression for the relaxation time is squared, the concentration dependence can be expressed as a simple function of the total concentration of 2-pyridone.

Short answer

Calculate \(k_f\) and \(k_r\) using linear plots of \([A]^2\), then \(K = k_f/k_r\).

Step-by-step solution

Understanding Relaxation Time Expression and Squaring

The reaction for the formation of a hydrogen-bonded dimer can be expressed as \( 2A \rightleftharpoons A_2 \), where \(A\) is the 2-pyridone and \(A_2\) is the dimer. The relaxation time \(\tau\) is related to the forward and reverse reaction rates; however, the hint suggests squaring the expression for easier manipulation. Squaring this expression reveals its dependence on the concentration of \(A\). Let \(\tau = \frac{1}{k_f[A]+k_r}\), where \(k_f\) and \(k_r\) are the forward and reverse rate constants respectively.

Establishing the Concentration Dependence

By squaring \(\frac{1}{\tau}\), our goal is to establish a linear relationship that will facilitate the calculation of equilibrium and rate constants. Expand this to express it in terms of \([A]\): \( \tau^2 = \frac{1}{(k_f [A] + k_r)^2} \), and hence \( 1/\tau^2 = (k_f [A] + k_r)^2 \).

Analyze the Linear Relationship Graphically

With the provided data, calculate \(1/\tau^2\) for each concentration \([A]\), which gives us the proportional relationship between \(1/\tau^2\) and \([A]^2\) that can be plotted graphically. This plot should yield a linear graph with a slope of \(k_f^2\) and intercept \(k_r^2\). Use the data to calculate these values and from which \(k_f\) and \(k_r\) can be derived.

Calculation of Equilibrium Constant

The equilibrium constant \(K\) is given by \(K = \frac{k_f}{k_r}\). Once \(k_f\) and \(k_r\) are determined from the slope and intercept of the graph, compute \(K\).

Derive Rate Constants from Graphical Analysis

Assume slope viable \( k_f \approx \) and intercept viable \( k_r \approx \) obtained from the linear plot. If \(k_f = 1.2 \times 10^9 \text{ M}^{-1}\text{ s}^{-1}\) and \(k_r = 2.3 \times 10^9 \text{ s}^{-1}\), then \(K = k_f/k_r \approx 0.52\). Substitute these values to validate from actual plotted data.

Key concepts

Hydrogen Bonding in Chemical Compounds

Chemical compounds often interact through hydrogen bonds, which are relatively weak but crucial interactions where a hydrogen atom is attracted to an electronegative atom like oxygen or nitrogen. In this exercise, we explore how two molecules of 2-pyridone interact via hydrogen bonding to form a dimer, a larger molecular complex.

• Nature of Hydrogen Bonds: These bonds are a special type of dipole-dipole attraction. They are not as strong as covalent bonds but are significant because they influence molecular structure and properties.
• Role in Chemical Reactions: Hydrogen bonding can significantly affect the speed and equilibrium of reactions. In this case, it contributes to forming a stable dimer from two 2-pyridone molecules.
• Factors Affecting Hydrogen Bonding: Temperature, pressure, and the presence of other chemicals can affect the strength and formation of hydrogen bonds. Here, the experiment is conducted at a set temperature of 298 K to determine these effects in a controlled manner.

To understand the extent of these interactions, we measure the relaxation time, which reflects how quickly the reaction attains equilibrium and returns to a steady-state after a disturbance. By calculating the timeline for this reaction, we gain insights into the kinetics and thermodynamics of hydrogen bonding in these systems.

Understanding the Equilibrium Constant

The equilibrium constant (\(K\)) is a fundamental concept that describes the ratio of concentrations of products to reactants at equilibrium. It provides insight into the favorability of a reaction under a set condition.

• Expression of Equilibrium Constant: For the dimerization of 2-pyridone (\(2A \rightleftharpoons A_2\)), the equilibrium constant is expressed as \(K = \frac{[A_2]}{[A]^2}\). It shows how much product is formed compared to the reactants.
• Significance of the Value: A large \(K\) indicates more products are formed, or the reaction lies to the right, favoring the dimer. A smaller \(K\) means the reactants are predominant.
• Calculation: In this exercise, \(K\) was derived using the relationship between the rate constants. The equilibrium constant reveals valuable information about reaction spontaneity and the conditions required to achieve equilibrium.

Understanding \(K\) is crucial, especially in systems where hydrogen bonding plays a significant role. It helps predict the conditions under which a reaction will reach equilibrium, guiding us in practical applications such as drug design and synthetic chemistry.

Rate Constants in Chemical Kinetics

Rate constants, denoted as \(k_f\) (forward rate constant) and \(k_r\) (reverse rate constant), play a crucial role in understanding the speed of chemical reactions.

• Definition and Importance: The forward rate constant \(k_f\) tells us how fast the reactants (in this case, two 2-pyridone molecules) convert into products (the dimer). The reverse rate constant \(k_r\) indicates the speed of the reverse transformation from product to reactants.
• Calculation from Data: By measuring relaxation times across different concentrations, we can determine these constants through graphical analysis, yielding a linear relationship where the slope gives \(k_f\) and the intercept gives \(k_r\). The provided example shows \(k_f = 1.2 \times 10^9\) M^-1s^-1 and \(k_r = 2.3 \times 10^9\) s^-1.
• Impact on Equilibrium: These constants are directly tied to the equilibrium constant \(K\). They determine how quickly equilibrium is reached and the position of equilibrium. The ratio \( \frac{k_f}{k_r} \) gives the equilibrium constant.

In summary, understanding rate constants helps chemists control reaction conditions to optimize yields and efficiency in chemical processes. They provide a window into the microscopic interactions between molecules, fundamental to the field of chemical kinetics.