Question

In glycine, the carboxylic acid group has \(K_{\mathrm{a}}=4.3 \times 10^{-3}\) and the amino group has \(K_{b}=6.0 \times 10^{-5} .\) Use these equilibrium constant values to calculate the cquilibrium constants for the following. a. \(^{+} \mathrm{H}_{3} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}+\mathrm{H}_{3} \mathrm{O}^{+}\) b. \(\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2} \mathrm{H}+\mathrm{OH}^{-}\) c. \(\mathbf{H}_{3} \mathrm{NCH}_{2} \mathrm{CO}_{2} \mathrm{H} \rightleftharpoons 2 \mathrm{H}^{+}+\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}\)

Short answer

For the given reactions involving glycine, the equilibrium constants are: a. \(K_{a1}=4.3 \times 10^{-3}\) b. \(K_{b1}=6.0 \times 10^{-5}\) c. \(K_{c}=1.3953 \times 10^{-2}\)

Step-by-step solution

a. Calculate the equilibrium constant for the first reaction

First, let's define the equilibrium constants we already know: - Ka is the acid dissociation constant for the carboxylic acid group of glycine - Kb is the base dissociation constant for the amino group of glycine For the first reaction, we can treat it as a simple acid dissociation reaction: \(^{+} \mathrm{H}_{3} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}+\mathrm{H}_{3} \mathrm{O}^{+}\) In this reaction, the amino group of glycine is losing a proton. Therefore, Ka for this reaction would be equal to the given Ka value. So, \(K_{a1} = K_{a} = 4.3 \times 10^{-3}\).

b. Calculate the equilibrium constant for the second reaction

For the second reaction, it can be treated as a simple base dissociation reaction: \(\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2} \mathrm{H}+\mathrm{OH}^{-}\) In this reaction, the carboxylic acid group of glycine is gaining a proton. Therefore, Kb for this reaction would be equal to the given Kb value. So, \(K_{b1} = K_{b} = 6.0 \times 10^{-5}\).

c. Calculate the equilibrium constant for the third reaction

The third reaction is as follows: \(\mathrm{H}_{3} \mathrm{NCH}_{2} \mathrm{CO}_{2} \mathrm{H} \rightleftharpoons 2 \mathrm{H}^{+}+\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-}\) This reaction can be considered as a combination of two reactions: the first given reaction in the reverse direction, and the second given reaction in the forward direction. This means that the equilibrium constant for this reaction can be calculated by multiplying the inverse of the equilibrium constant for the first reaction (Ka) with the equilibrium constant for the second reaction (Kb). Hence, for the third reaction, the equilibrium constant can be calculated as follows: \(K_{c} = \frac{1}{K_{a1}} \times K_{b1} = \frac{1}{4.3 \times 10^{-3}} \times 6.0 \times 10^{-5}\) Calculation: \(K_{c} = 1.3953 \times 10^{-2}\)

Key concepts

Base Dissociation Constant

The base dissociation constant, often denoted as \(K_b\), is a fundamental concept in understanding how bases interact with water to form their respective conjugates. It essentially measures the strength of a base in terms of its tendency to accept a proton. In the case of glycine, when discussing the amino group, the \(K_b\) value gives us insight into how readily the amino group can accept a proton from water to form its conjugate acid.

Consider the second reaction of the exercise: \( \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-} + \mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2} \mathrm{H}+\mathrm{OH}^{-} \). This reaction typifies a base dissociation in which the carboxylate acts as a base, picking up a proton to become the unionized carboxylic acid.

The provided \(K_b\) value for glycine, \(6.0 \times 10^{-5}\), indicates the equilibrium balance between the reactants and products. A lower \(K_b\) suggests a weaker base, meaning the forward reaction (base acquiring a proton) is not very favorable. Understanding \(K_b\) is crucial because it helps predict the pH behavior of solutions containing glycine in different environments.

Acid Dissociation Constant

The acid dissociation constant, \(K_a\), is crucial in understanding how acids dissociate in water, shedding light on the tendency of an acid to donate its protons. For glycine, the reaction \(^+\mathrm{H}_{3} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-} + \mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-} + \mathrm{H}_{3} \mathrm{O}^{+} \) describes the acid behavior.

This is characterized by the carboxylic acid group's ability to donate a proton. The strength of this acid, as indicated by its \(K_a\) value of \(4.3 \times 10^{-3}\), outlines the extent to which it can dissociate in an aqueous solution. A higher \(K_a\) value signifies a stronger acid, correlating with a greater degree of ionization of the acid in the solution.

Understanding the \(K_a\) for glycine's functional group helps in forming predictions about its reactivity and role in various chemical processes. This knowledge is pivotal for engaging with more complex equilibrium reactions or predicting the buffering capacity of solutions containing glycine.

Glycine Chemistry

Glycine is the simplest amino acid and serves as an excellent model for exploring acid-base chemistry in biomolecules. With both an amino group and a carboxylic acid group, glycine exhibits dual behavior, acting both as an acid and as a base.

This dual character is exemplified in the reactions provided in the original exercise. Glycine's carboxylic acid group can donate a proton, behaving as an acid, while its amino group can accept a proton, acting like a base. The calculated equilibrium constants in these reactions provide valuable insights into the behavior of glycine under different conditions.

Glycine's zwitterionic form \( \mathrm{H}_{3} \mathrm{NCH}_{2} \mathrm{CO}_{2}^{-} \), in which it contains both a positive and a negative charge, plays a crucial role in its chemistry. This form is especially relevant in biological contexts where it often stabilizes or participates in various biochemical pathways.

The intricate behavior of glycine in solution involves understanding its acid and base dissociation constants, which inform how it balances its acidic and basic properties. This balancing act is essential when glycine is involved in buffer solutions, cellular processes, or chemical reactions in the laboratory. Grasping the nuances of glycine chemistry can also help predict its reactivity and interactions in biological systems.