Question

The amino acid glycine \(\left(\mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}\right)\) can participate in the following equilibria in water: $$ \begin{aligned} \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \\ \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COO}^{-}+\mathrm{H}_{3} \mathrm{O}^{+} \quad K_{a}=4.3 \times 10^{-3} \\ \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \\ +\mathrm{H}_{3} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}+\mathrm{OH}^{-} \quad K_{b}=6.0 \times 10^{-5} \end{aligned} $$ (a) Use the values of \(K_{a}\) and \(K_{b}\) to estimate the equilibrium constant for the intramolecular proton transfer to form a zwitterion: $$ \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH} \rightleftharpoons{ }^{+} \mathrm{H}_{3} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COO}^{-} $$ What assumptions did you need to make? (b) What is the pH of a \(0.050 \mathrm{M}\) aqueous solution of glycine? (c) What would be the predominant form of glycine in a solution with \(\mathrm{pH} 13 ?\) With \(\mathrm{mH}^{1} ?\)

Short answer

The equilibrium constant for the zwitterion formation, \(K_{zwitterion}\), is calculated to be \(\frac{4.3 \times 10^{-3}}{6.0 \times 10^{-5}} \approx 71.67\). To find the pH of a 0.050 M aqueous glycine solution, we first determine the [H+] using the given \(K_a\) and then calculate the pH, which is found to be approximately 2.34. The predominant form of glycine at pH 13 is the deprotonated form, \(\mathrm{H}_{2}\mathrm{~N}\mathrm{-CH}_{2}\mathrm{-COO}^{-}\), and the predominant form at pH 1 is the protonated form, \(\mathrm{H}_{3}\mathrm{~N}\mathrm{-CH}_{2}\mathrm{-COOH}\).

Step-by-step solution

Determine the equilibrium constant for zwitterion formation

Using the values of \(K_{a}\) and \(K_{b}\) from the exercise, we need to calculate the equilibrium constant for the intramolecular proton transfer to form a zwitterion. This can be done using the relationship between \(K_{w}\), \(K_{a}\), and \(K_{b}\): \(K_w = K_a \times K_b\), where \(K_w\) is the ion product of water, and its value is \(1.0 \times 10^{-14}\) at 25°C. Calculate the equilibrium constant for zwitterion formation: \[K_{zwitterion} = \frac{K_a}{K_b}\]

Calculate the value of the equilibrium constant for the zwitterion formation

Using the relationship established in step 1, plug in the values of \(K_{a}\) and \(K_{b}\) given in the exercise: \[K_{zwitterion} = \frac{4.3 \times 10^{-3}}{6.0 \times 10^{-5}}\] Solve for \(K_{zwitterion}\) to get the equilibrium constant for the zwitterion formation.

Determine the pH of a 0.050 M aqueous glycine solution

Given the equilibrium constant \(K_{a}\) of the glycine, we can calculate the pH of a 0.050 M aqueous solution of glycine by first determining the [H+] (hydrogen ion concentration) using the equation: \[K_{a} = \frac{[\mathrm{H}_{3}\mathrm{O}^{+}][\mathrm{H}_{2}\mathrm{N}\mathrm{-CH}_{2}\mathrm{-COO}^{-}]}{[\mathrm{H}_{2}\mathrm{N}\mathrm{-CH}_{2}\mathrm{-COOH}]}\] Then, find the pH using the formula: \[\mathrm{pH} = -\log [\mathrm{H}^{+}]\]

Calculate the pH of a 0.050 M aqueous glycine solution

Substitute the given values into the aforementioned equations and solve for pH.

Determine the predominant form of glycine at pH 13 and pH 1.

To determine the predominant form of glycine at pH 13 and pH 1, compare the given pH values to the pKa and pKb values that can be calculated from the given \(K_{a}\) and \(K_{b}\) values for glycine: \[pK_a = -\log{K_{a}}\] \[pK_b = -\log{K_{b}}\] If the pH of the solution is significantly greater than the pKa value, the major species will be the deprotonated form. If the pH is significantly lower than the pKa value, the major species will be the protonated form. Compare the pH values to the pKa and pKb values and determine the predominant forms.

Key concepts

Zwitterion Formation

Zwitterions are intriguing molecules that have both positive and negative charges coexisting within the same structure. Specifically, when we look at amino acids like glycine, zwitterions form under certain conditions in water due to a process called intramolecular proton transfer. This means a hydrogen ion (H⁺), or proton, is transferred from the carboxylic acid group (-COOH) to the amino group (-NH₂), resulting in a zwitterion having a positively charged ammonium ion (-NH₃⁺) and a negatively charged carboxylate ion (-COO^-).

Acid Dissociation Constant (Ka)

Diving into the concept of the acid dissociation constant, denoted by Ka, sheds light on the strength of an acid in a solution. Glycine, like any other acid, tends to donate a proton to the surrounding water when dissolved, creating hydronium ions (H₃O⁺). The Ka value quantifies the extent of this proton donation, and a larger Ka signifies a stronger acid. It is crucial to understand that the acid dissociation constant is pivotal in predicting the behavior of glycine in solution, especially when it comes to how readily it will give up that proton to become a zwitterion.

In our exercise, calculating the pH of a glycine solution involves understanding how the Ka value influences the concentration of hydrogen ions (H⁺) in the solution, which is then used to calculate the pH.

Base Dissociation Constant (Kb)

In contrast to Ka, we have the base dissociation constant, Kb, illuminating the other side of glycine's character. Here, we're concerned with glycine's ability to accept a proton, functioning as a base. The Kb gives us an insight into the affinity of glycine's amino group to bond with a hydrogen ion from water, thereby forming hydroxide ions (OH^-).

While Kb is typically smaller than Ka for amino acids like glycine, implying they are more acidic than basic, it is still a crucial factor in comprehending the overall equilibrium in a glycine water solution. The balance between these tendencies to donate or accept protons underpins the establishment of equilibrium in the solution and plays a central role in determining the prevailing chemical species at a given pH.

pH Calculation

Moving on to the practical application of these concepts in our exercise involves a crucial step—calculating the pH of a solution. The pH calculation is a measure of the acidity or alkalinity of a solution on a logarithmic scale. Specifically, we calculate pH by taking the negative logarithm of the hydrogen ion concentration, indicated as [-log(H⁺)]. For a 0.050 M glycine solution, we must set up an equilibrium expression using the Ka value, solve for the concentration of H⁺, then apply the pH formula to find the solution’s acidity. This process reveals much about the glycinemolecule’s behavior under various conditions and is essential for understanding biochemistry and molecular biology.

Glycine Solution Properties

Lastly, the properties of a glycine solution are largely dictated by the equilibrium between its different forms, including the zwitterion. The pH of the solution can significantly affect which form predominates. For example, at a highly alkaline pH (such as pH 13), the glycine will mostly be in its negatively charged form since the amino group has accepted a proton. Conversely, in a highly acidic solution (pH 1), glycine will be predominantly in its positively charged form, as the carboxyl group would have donated its proton.

The exercise demonstrates this by requiring us to consider what the dominant species of glycine would be at pH values of 13 and 1, which can be assessed by comparing the pH to glycine's pKa and pKb. This comparison determines whether the amino acid exists primarily in its neutral, zwitterionic, protonated, or deprotonated state, highlighting the versatility and dynamic nature of amino acids in biological systems.