Problem 114 - Chemistry: The Central Science

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: \(\mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\) \(\mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COO}^{-}+\mathrm{H}_{3} \mathrm{O}^{+} \quad K_{\mathrm{a}}=4.3 \times 10^{-3}\) \(\mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\) \({ }^{+} \mathrm{H}_{3} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH}+\mathrm{OH} \quad K_{\mathrm{b}}=6.0 \times 10^{-5}\) (a) Use the values of \(K_{a}\) and \(K_{b}\) to estimate the equilibrium constant for the intramolecular proton transfer to form a zwitterion: $$ \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH} \rightleftharpoons{ }^{+} \mathrm{H}_{3} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COO}^{-} $$

Step-by-Step Solution

Answer

The equilibrium constant for the intramolecular proton transfer to form a zwitterion from glycine is calculated by combining the given equilibria and dividing the equilibrium constants: \(K = \frac{K_{a}}{K_{b}} = \frac{4.3 \times 10^{-3}}{6.0 \times 10^{-5}} \approx 72\).

Step 1: Analyze the given equilibria and constants

The first equilibrium represents the reaction where the amino acid glycine loses a proton from the carboxylic acid group to form a negatively charged carboxylate ion. The equilibrium constant for this reaction is Ka = 4.3 x 10^(-3). The second equilibrium represents the reaction where glycine gains a proton at the amino group, forming a positively charged ammonium ion. The equilibrium constant for this reaction is Kb = 6.0 x 10^(-5).

Step 2: Combine the two equilibria to form the desired reaction and equilibrium constant

In order to estimate the equilibrium constant for the intramolecular proton transfer to form a zwitterion from glycine, we can combine the first equilibrium with the reverse of the second equilibrium. The desired reaction can be represented as follows: $$ \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH} \rightleftharpoons{ }^{+} \mathrm{H}_{3} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COO}^{-} $$ If we reverse the second equilibrium, the reaction becomes: $$ { }^{+} \mathrm{H}_{3} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH} + \mathrm{OH}^{-} \rightleftharpoons \mathrm{H}_{2} \mathrm{~N}-\mathrm{CH}_{2}-\mathrm{COOH} + \mathrm{H}_{2}\mathrm{O} $$ Now, we can sum the first equilibrium and the reversed second equilibrium to obtain the desired reaction. The equilibrium constant for the desired reaction is the quotient of the equilibrium constants of the first reaction and the reversed second reaction: $$ K = \frac{K_{a}}{K_{b}} $$

Step 3: Calculate the equilibrium constant for the desired reaction

Now, we can calculate the equilibrium constant using the values of Ka and Kb: $$ K = \frac{4.3 \times 10^{-3}}{6.0 \times 10^{-5}} = 71.67 \approx 72 $$ Thus, the equilibrium constant for the intramolecular proton transfer to form a zwitterion from glycine is approximately 72.

Key Concepts

Understanding Acid-Base Equilibria

Acid-base equilibria involve the reversible reactions of acids and bases in aqueous solutions. When we discuss acids and bases, we often refer to the Brønsted-Lowry definition, where an acid is a substance that can donate a proton (hydrogen ion, H⁺), and a base is one that can accept a proton.

For the amino acid glycine, this balance of donating and accepting protons is crucial because it directly affects its behavior in water. Glycine can act as an acid by donating a proton from its carboxyl group, resulting in a carboxylate ion with a negative charge. Conversely, it can act as a base by accepting a proton at its amino group, forming an ammonium ion with a positive charge. These processes can be represented by their respective equilibrium constants, Ka and Kb, which signify the strength of glycine as an acid or a base.

Memorizing values for these constants isn't necessary if one understands that a higher Ka or Kb value indicates a stronger acid or base, respectively. When tackling equilibria problems, it's vital to pay attention to these values and consider the direction of the equilibrium, as this will affect the final calculation for the system's equilibrium constant.

Zwitterion Formation

Zwitterions are intriguing molecules that contain both a positive and a negative charge, but which balance out to make the molecule neutral overall. For amino acids such as glycine, the formation of a zwitterion involves the transfer of a proton within the molecule itself—a migration from the carboxyl group to the amino group.

In biological systems, zwitterions are of tremendous importance because they influence amino acids' solubility and interactions. Understanding zwitterions helps explain how amino acids behave in different environments, such as in varying pH levels within the human body. Glycine, being the simplest amino acid, serves as an excellent example to illustrate this; it can form a zwitterion at neutral pH levels, which then can affect both its structural characteristics and its biological function. The equilibrium between the zwitterionic form and the non-ionic form depends on the environmental conditions and can be calculated using the given values of Ka and Kb.

Equilibrium Constant Calculation

An equilibrium constant is a number that describes the ratio of the concentration of products to reactants at equilibrium for a reversible chemical reaction. It's a pivotal concept when dealing with chemical equilibria, as it offers insights into the extent of a reaction under a given set of conditions.

To calculate this constant when combining equilibria, we use the properties of logarithms—a concept known from mathematics that allows us to divide or multiply individual equilibria constants. In the context of the example given for glycine, we take the Ka value for the acid dissociation and divide it by the Kb value (which must be inverted since we are considering the reverse reaction). By doing this, we establish a new equilibrium constant that represents the formation of the zwitterion.

Through this calculation, we learn how likely it is for glycine to exist in the zwitterionic form within its environment—a vital piece of information for understanding the physical and chemical properties of amino acids in various biological and chemical systems.