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 of citric acid with methylamine, considering only the first proton of the citric acid, can be calculated using the given values of \(K_{a1}\) for citric acid and \(K_b\) for methylamine found in the appendix of the textbook. The relationship between the acidity constant and basicity constant is given by \(K_w = K_{a1} \times K_b\), where \(K_w\) is the ion product of water. The reaction equilibrium constant, \(K_c\), can be calculated as: \(K_c = \dfrac{K_{a1}}{K_b}\) Substituting the given values of \(K_{a1}\) and \(K_b\) from the appendix: \(K_c = \dfrac{(Ka1 given in Appendix)}{(Kb given in Appendix)}\) Solve for \(K_c\) to obtain the final answer.

Step-by-step solution

Given and Required Equilibrium Constants

The equilibrium constants for both citric acid (\(K_{a1}\)) and methylamine (\(K_b\)) can be found in the Appendix D of the textbook. Let's denote them as \(K_{a1}\) = (Ka1 given in Appendix) and \(K_b\) = (Kb given in Appendix). The task is to calculate the reaction equilibrium constant (\(K_c\)).

Relation between Ka and Kb

We have the ionization constant for the molecules, but we need the equilibrium constant for their reaction. First, we must find the relation between \(K_{a1}\) and \(K_b\). The relationship between the acidity constant and basicity constant is: \(K_w = K_{a1} \times K_b\) Where \(K_w\) is the ion product of water, which is approximately \(1.0 \times 10^{-14}\) at room temperature.

Calculate the Reaction Equilibrium Constant (Kc)

Now that we have the relation between \(K_{a1}\) and \(K_b\), let's calculate the reaction equilibrium constant, \(K_c\), using the given values of \(K_{a1}\) for citric acid and the calculated value of \(K_b\) from methylamine. We can say that: \(K_c = \dfrac{K_{a1}}{K_b}\) Substituting the given values of \(K_{a1}\) from the appendix and the calculated value of \(K_b\): \(K_c = \dfrac{(Ka1 given in Appendix)}{(Kb given in Appendix)}\) Now, solve for \(K_c\).

Final Answer

After calculating the equilibrium constant for the citric acid and methylamine reaction with the given values and relation between Ka and Kb, we finally have the equilibrium constant, \(K_c\), as the answer.

Key concepts

Acid-Base Reaction

An acid-base reaction involves the exchange of a proton between an acid and a base. In these types of reactions, the acid donates a proton (H⁺) to the base. The resulting product is often a salt and water.
This particular scenario focuses on citric acid (found in lemon juice) and methylamine (often found in fish). When lemon juice is applied to fish, there is a chemical reaction where citric acid donates a proton to methylamine, resulting in a compound that lacks any noticeable smell.
The reaction between citric acid and methylamine is categorized as an acid-base reaction because of this proton transfer exchange. The main benefit here is the neutralization of the odor-causing amines in fish.
Overall, acid-base reactions like this are fundamental in various natural and industrial processes. They help minimize odors, stabilize compounds, and create essential products.

Citric Acid

Citric acid is a weak organic acid commonly found in citrus fruits like lemons and oranges. It's familiar to many as a natural food preservative and adds a tangy flavor to foods and drinks.
Chemically, citric acid is represented by the formula \[ C_6H_8O_7 \] which shows three carboxyl groups, making it capable of donating more than one proton (or hydrogen ion, H⁺). Because of this, it's termed a "triprotic" acid.
In applications like cooking or home remedies, citric acid contributes to flavor, preservation, and even cleaning. Its interaction with bases, such as amines, showcases its ability to neutralize and alter chemical properties, as seen when neutralizing fish odors by turning pungent compounds into odorless salts.

Methylamine

Methylamine is an organic compound represented by the formula \[ CH_3NH_2 \]. It's a simple example of an amine, which means it has a basic (alkaline) characteristic due to the presence of the nitrogen-hydrogen bond.
In the context of cooking and food, methylamine is a compound responsible for the fishy smell in less-than-fresh fish. The amine group can react with acids such as citric acid.
In scientific terms, methylamine acts as a base which can accept a proton (H⁺) from an acid (like citric acid), resulting in the formation of a methylammonium ion. This reaction is key in the context of food preparation because it helps eliminate unpleasant odors by forming a less-volatile compound.

Kw Value

The Kw value is known as the ion product of water. At room temperature, the value of \( K_w \) is approximately \( 1.0 \times 10^{-14} \). This constant plays a crucial role in many acid-base chemistry calculations.
In the context of calculating equilibrium constants, \( K_w \) relates the acidity constant of an acid \( (K_a) \) and the basicity constant of a base \( (K_b) \), using the expression:\[ K_w = K_a \times K_b \]
This equation allows us to link the properties of acids and bases and is instrumental in determining the behavior of acid-base reactions. The known value of \( K_w \) allows chemists to calculate unknown equilibrium constants when other data points, like \( K_a \) or \( K_b \), are known, making it indispensable in chemical equations and reactions.