Question

Imidazole is a common molecular species in biological chemistry. For example, it constitutes the side chain of the amino acid histidine. Imidazole can be protonated in solution as follows: The rate constant for the protonation reaction is \(5.5 \times 10^{10} \mathrm{M}^{-1} \mathrm{s}^{-1} .\) Assuming that the reaction is diffusion controlled, estimate the diffusion coefficient of imidazole when \(D\left(\mathrm{H}^{+}\right)=9.31 \times 10^{-5} \mathrm{cm}^{2} \mathrm{s}^{-1}, r\left(\mathrm{H}^{+}\right) \sim 1.0 \AA\) and \(r\) (imidazole) \(=6.0\) A. Use this information to predict the rate of deprotonation of imidazole by \(\mathrm{OH}^{-}\left(D=5.30 \times 10^{-5} \mathrm{cm}^{2} \mathrm{s}^{-1}\right.\) and \(r=\sim 1.5 \AA)\)

Short answer

The diffusion coefficient of imidazole when protonated is \(4.13 \times 10^{-5}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}\), and the rate constant for deprotonation by OH- is \(3.32 \times 10^{10}\,\mathrm{M}^{-1}\,\mathrm{s}^{-1}\).

Step-by-step solution

Identify the given values and the variables to be calculated

From the given exercise, we have the following values: - Rate constant for protonation reaction (k) = \(5.5 \times 10^{10}\,\mathrm{M}^{-1}\,\mathrm{s}^{-1}\) - Diffusion coefficient of H+ (\(D(\mathrm{H}^{+})\)) = \(9.31 \times 10^{-5}\, \mathrm{cm}^{2}\,\mathrm{s}^{-1}\) - Radius of H+ (r(H+)) = \(1.0\,\AA\) - Radius of imidazole (r_imidazole) = \(6.0\,\AA\) - Diffusion coefficient of OH- (D(OH-)) = \(5.30 \times 10^{-5}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}\) - Radius of OH- (r(OH-)) = \(1.5\,\AA\) We need to calculate: 1. The diffusion coefficient of imidazole when protonated (D_imidazole). 2. The rate constant of deprotonation by OH- (\((k_{dep})\)).

Calculate the diffusion coefficient of imidazole when protonated (D_imidazole)

We can use the Smoluchowski equation to calculate the diffusion coefficient of imidazole when protonated: \(k = \dfrac{4 \pi (D(\mathrm{H}^{+}) + D_{imidazole})(r(\mathrm{H}^{+}) + r_{imidazole})N_{A}}{10^8}\) where \(N_{A}\) is Avogadro's number. We will rearrange the equation to solve for D_imidazole: \(D_{imidazole} = \dfrac{10^8 k - 4 \pi D(\mathrm{H}^{+})(r(\mathrm{H}^{+}) + r_{imidazole})N_{A}}{4\pi ( r(\mathrm{H}^{+}) + r_{imidazole})N_{A}}\) Now, insert the given values into the equation: \(D_{imidazole} = \dfrac{10^8 \times 5.5 \times 10^{10} - 4 \pi \times 9.31 \times 10^{-5}(1.0 + 6.0)N_{A}}{4\pi ( 1.0 + 6.0)N_{A}}\) Calculate the D_imidazole: \(D_{imidazole} = 4.13 \times 10^{-5}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}\)

Calculate the rate constant of deprotonation by OH- (\((k_{dep})\))

We can use the Smoluchowski equation again to calculate the rate constant of deprotonation by OH-: \(k_{dep} = \dfrac{4 \pi (D(\mathrm{OH}^{-}) + D_{imidazole})(r(\mathrm{OH}^{-}) + r_{imidazole})N_{A}}{10^8}\) Insert the calculated D_imidazole and given values into the equation: \(k_{dep} = \dfrac{4 \pi (5.30 \times 10^{-5} + 4.13 \times 10^{-5})(1.5 + 6.0)N_{A}}{10^8}\) Calculate the \(k_{dep}\): \(k_{dep} = 3.32 \times 10^{10}\,\mathrm{M}^{-1}\,\mathrm{s}^{-1}\) The diffusion coefficient of imidazole when protonated is \(4.13 \times 10^{-5}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}\), and the rate constant for deprotonation by OH- is \(3.32 \times 10^{10}\,\mathrm{M}^{-1}\,\mathrm{s}^{-1}\).

Key concepts

Smoluchowski Equation

The Smoluchowski equation is essential in understanding how diffusion-controlled reactions occur. It helps calculate the rate at which molecules will encounter each other in a solution, which is especially useful in reactions that are extremely fast.
The equation considers:

• The diffusion coefficients of the reacting species
• The radii of the reacting species
• The Avogadro number to convert microscopic events to macroscopic observable rates

The formula used in this context is:\[k = \frac{4 \pi (D_A + D_B)(r_A + r_B)N_A}{10^8}\]Where:

• \(D_A\) and \(D_B\) are the diffusion coefficients of the reactants
• \(r_A\) and \(r_B\) are the radii of the reactants
• \(N_A\) is Avogadro's number

This powerful equation allows us to predict how often molecules collide, providing insight into the rate of diffusion-controlled reactions.

Protonation Reaction

In chemistry, a protonation reaction involves the addition of a proton (\(H^+\)) to a molecule, resulting in the formation of a new species. This process significantly alters the chemical properties of the molecule. Imidazole, found in biological systems, can undergo protonation, affecting its role and function.
The protonation reaction for imidazole shows that the molecule accepts a proton:- Changes the charge and the reactivity of the molecule- Plays an important part in biological functions like enzyme activity and molecular signaling- Generally a fast process, often diffusion-controlled in nature
Understanding protonation is vital in predicting how molecules like imidazole behave in biological systems, often dictating their biological roles.

Deprotonation Reaction

Deprotonation is essentially the reverse process of protonation, where a molecule loses a proton. In the context of the earlier problem, deprotonation of imidazole by hydroxide (\(OH^-\)) is considered.
Just like protonation, this process affects the charge and properties:- Results in a different ion or neutral species- Affects how the molecule interacts in a solution- Is important for maintaining pH and other chemical equilibria in cells
Deprotonation reactions are also often fast and diffusion-controlled, helping to maintain the dynamic balance between different chemical species in a biological setting. This reaction is analyzed using similar techniques and equations, like the Smoluchowski equation, to estimate kinetics and reaction rates.

Rate Constant

The rate constant is a critical factor in chemical kinetics, representing the speed at which a reaction occurs. For reactions controlled by diffusion, like the protonation of imidazole, the rate constant helps determine how quickly reactants convert to products.
Factors affecting the rate constant include:

• Temperature, where higher temperatures generally increase rates
• Nature and state of reactants; ions in solution often react quickly
• The solvent and its properties, such as viscosity and dielectric constant

The rate constant is used in equations like the Smoluchowski equation to find diffusion coefficients. It provides vital information for predicting reaction dynamics in biological and chemical systems. Understanding and calculating rate constants is key in fields like pharmacology, biochemistry, and environmental science, where fast reactions can significantly impact system behavior.