Question

The odor of fish is due primarily to amines, especially methylamine \(\left(\mathrm{CH}_{3} \mathrm{NH}_{2}\right)\). Fish is often served with a wedge of lemon, which contains citric acid. The amine and the acid react forming a product with no odor, thereby making the less-than-fresh fish more appetizing, Using data from Appendix D, calculate the equilibrium constant for the reaction of citric acid with methylamine, if only the first proton of the citric acid \(\left(K_{a 1}\right)\) is important in the neutralization reaction.

Short answer

The equilibrium constant for the reaction between citric acid and methylamine can be calculated using the formula \(K = \frac{K_{a1}^{-1}}{K_{b}}\), where \(K_{a1}\) is the first proton dissociation constant of citric acid and \(K_b\) is the basicity constant of methylamine. Upon finding the values of \(K_{a1}\) and \(K_b\) in the Appendix, substitute them into the formula and solve for the equilibrium constant \(K\). This calculated value represents the equilibrium constant for the reaction between citric acid and methylamine.

Step-by-step solution

Write the given dissociation constant

We are given the first proton dissociation constant for citric acid, \(K_{a1}\). Write the equation for the dissociation reaction: \(\mathrm{H}_{3}\mathrm{C}_{6}\mathrm{H}_{5}\mathrm{O}_{7} \longleftrightarrow \mathrm{H}^{+} + \mathrm{HC}_{6}\mathrm{H}_{5}\mathrm{O}_{7}^{-}\)

Write the reaction for the acid-base reaction

Write the equation for the reaction between methylamine and citric acid: \(\mathrm{CH}_{3} \mathrm{NH}_{2} + \mathrm{H}_{3}\mathrm{C}_{6}\mathrm{H}_{5}\mathrm{O}_{7} \longleftrightarrow \mathrm{CH}_{3}\mathrm{NH}_{3}^{+} + \mathrm{HC}_{6}\mathrm{H}_{5}\mathrm{O}_{7}^{-}\)

Calculate the equilibrium constant for the reaction

The reaction is a combination of the dissociation of citric acid and the reaction between the methylamine and the produced hydrogen ion. To find the equilibrium constant, \(K\), for the entire reaction, we need to divide the equilibrium constants of the reverse reaction of the dissociation (i.e., \(K_{a1}^{-1}\)) by the constant of the reaction between the methylamine and hydrogen ion. The reaction between methylamine and hydrogen ion is a protonation reaction, and its equilibrium constant is the reciprocal of the basicity constant, \(K_b\), of methylamine, which can be found in the Appendix. \(K = \frac{K_{a1}^{-1}}{K_{b}}\)

Look up values for \(K_{a1}\) and \(K_b\)

In the Appendix, find the values for the first proton dissociation constant of citric acid, \(K_{a1}\), and the basicity constant of methylamine, \(K_b\). For this problem, we assume only the first proton of citric acid is important in the neutralization reaction.

Calculate the equilibrium constant

Using the values of \(K_{a1}\) and \(K_b\) found, calculate the equilibrium constant, \(K\), for the reaction between citric acid and methylamine: \(K = \frac{K_{a1}^{-1}}{K_{b}}\)

Report the calculated equilibrium constant

The calculated equilibrium constant, \(K\), represents the equilibrium constant for the reaction between citric acid and methylamine. This is the final answer to the problem.

Key concepts

Acid-Base Reactions

Acid-base reactions are fundamental chemical processes where an acid donates a proton ( H^+ ) to a base. These reactions often result in the formation of water and a salt. In the context of the exercise, citric acid acts as the proton donor, while methylamine serves as the proton acceptor.

This type of reaction is crucial in neutralizing odors, as seen with the use of lemon (containing citric acid) to combat the fishy smell of amines. When citric acid interacts with methylamine, the acid donates a proton to the amine, leading to the formation of a non-odorous product. This proton transfer effectively reduces the concentration of smelly amine compounds.

Key characteristics of acid-base reactions include:

• Proton transfer from acid (donor) to base (acceptor).
• Formation of water and appropriate salts from the neutralization.
• Changes in pH, often resulting in products that are more neutral in pH.

These reactions have vast applications in real life, from culinary practices to industrial processes.

Equilibrium Constant Calculation

Understanding equilibrium constant calculations is essential when predicting the extent of a chemical reaction. The equilibrium constant, represented as K, indicates whether a reaction favors the formation of products or reactants under a set of conditions. For our exercise, considering citric acid and methylamine, this constant helps in understanding the completeness of their reaction.

In the given task, the equilibrium constant ( K ) of the reaction is derived by combining the individual constants of the constituent reactions.

• The dissociation constant K_{a1} for citric acid represents its tendency to release a proton.
• The basicity constant K_b for methylamine reflects its capacity to accept a proton.

To arrive at the answer, the reciprocal of K_{a1} is taken because we're interested in the reverse reaction of dissociation, and then it is divided by K_b to resolve the final equilibrium constant. This provides insight into the reaction favorability at standard conditions, helping in practical applications such as fish odor neutralization.

Citric Acid

Citric acid is a weak, naturally occurring acid found in citrus fruits like lemons, which is what gives them their sour taste. It is widely used in food flavoring, preserving, and creating a balanced taste. In chemical reactions, particularly acid-base interactions, it is valued for its ability to donate protons.

In the context of the exercise at hand, citric acid plays an integral role in removing the fishy stench caused by amines. By donating its first proton, represented by its dissociation constant K_{a1} , it initiates the neutralization reaction with methylamine. This process not only masks unpleasant odors but also highlights citric acid's versatility beyond culinary applications.

Some key features of citric acid include:

• Tri-basic nature, meaning it can donate three protons sequentially, though in this reaction, only the first is of primary concern.
• Presence in metabolic processes, like the citric acid cycle, essential for energy production in living organisms.
• Common use as an acidulant due to its sharp taste and preservative characteristics in the food industry.

Citric acid thus serves multiple purposes, not just in cooking but also as a meaningful agent in chemical and industrial processes.