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 \(\mathrm{D},\) 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 \((K_{a1})\), can be calculated using the relation: \[ K_{eq} = \frac{K_{a1} K_{b}}{K_w} \] Using the given values of \(K_{a1}\) and \(K_{b}\) from Appendix D, and considering \(K_w = 1.0 \times 10^{-14}\) at \(25°C\), we can calculate \(K_{eq}\) for this reaction.

Step-by-step solution

Write the chemical equation for the reaction of citric acid with methylamine under equilibrium conditions.

The chemical reaction between citric acid \((H_3A)\) and methylamine \((CH_3NH_2)\) can be represented, considering only the first proton, as: \[ H_{3}A + CH_{3} NH_{2} \rightleftharpoons H_{2}A^{-} + CH_{3} NH_{3}^{+} \]

Write the expression for the equilibrium constant for the reaction.

We can write the expression for the equilibrium constant \((K_{eq})\) for the given reaction as follows: \[ K_{eq} = \frac{[H_{2}A^{-}][CH_{3} NH_{3}^{+}]}{[H_{3}A][CH_{3} NH_{2}]} \]

Relate \(K_{eq}\) to the ionization constant, \(K_{a}\) of citric acid.

In order to find \(K_{eq}\), we need to relate it to the ionization constant of citric acid for the first proton, also known as \(K_{a1}\). When the first proton is removed from the citric acid molecule \((H_3A)\), we have the following equilibrium: \[ H_{3}A \rightleftharpoons H_{2}A^{-} + H^{+} \] \[ K_{a1} = \frac{[H_{2}A^{-}][H^{+}]}{[H_{3}A]} \]

Relate \(K_{eq}\) to the basicity constant, \(K_{b}\) of methylamine.

Similarly, we need to relate \(K_{eq}\) to the basicity constant of methylamine, \(K_{b}\). We can represent the base ionization constant of methylamine with the following equilibrium: \[ CH_{3} NH_{2} + H_{2}O \rightleftharpoons CH_{3} NH_{3}^{+} + OH^{-} \] \[ K_{b} = \frac{[CH_{3} NH_{3}^{+}][OH^{-}]}{[CH_{3} NH_{2}]} \]

Calculate \(K_{eq}\) using \(K_{a1}\) and \(K_{b}\).

We know that for a reaction between an acid and a base, the equilibrium constant \(K_{eq}\) is related to \(K_a\) and \(K_b\) as follows: \[ K_{eq} = \frac{K_{a1} K_{b}}{K_w} \] where \(K_w = [H^{+}][OH^{-}]\) is the ion product of water and is equal to \(1.0 \times 10^{-14}\) at \(25°C\). Now we can use the given values of \(K_{a1}\) and \(K_{b}\) from Appendix D, and use the equation above to calculate \(K_{eq}\) for the reaction between citric acid and methylamine at equilibrium.

Key concepts

Ionization Constant

The ionization constant, often denoted as \( K_a \), is a measure of the strength of an acid in solution. It tells us how well an acid donates its protons to the solvent, typically water. For every acid, the ionization reaction can be represented as follows: <[CH3]>C=C<

• The acid, represented as \( HA \), donates a proton to water, forming \( A^- \) and \( H^+ \).
• The ionization constant formula \( K_a \) is: \[ K_a = \frac{[A^-][H^+]}{[HA]} \] where square brackets indicate the concentration of each species.
• The larger the value of \( K_a \), the stronger the acid, meaning it donates protons more readily.

In our exercise, we focus on the citric acid's first proton, whose ionization constant is \( K_{a1} \). This value is crucial, as it defines the acid's reactivity with methylamine in reaching chemical equilibrium.

Base Ionization Constant

The base ionization constant, \( K_b \), provides a measure of the strength of a base. It specifically quantifies how well the base picks up protons. Methylamine serves as a classic example in our context.

• Bases like methylamine \( (CH_3NH_2) \) accept protons from water, producing \( CH_3NH_3^+ \) and \( OH^- \).
• The expression for the base ionization constant \( K_b \) is: \[ K_b = \frac{[CH_3NH_3^+][OH^-]}{[CH_3NH_2]} \]
• A high \( K_b \) means the base easily attracts protons, indicating strong basicity.

Grasping \( K_b \) alongside \( K_a \) helps us to calculate the equilibrium constant in reactions involving both acids and bases, such as citric acid and methylamine in the given problem.

Chemical Equilibrium

Chemical equilibrium represents the state in a reaction where the rate of the forward reaction equals the rate of the reverse reaction, and no net change occurs in the concentrations of reactants and products. The system appears static but is dynamic in nature.

• An equilibrium constant \( K_{eq} \) quantifies this balance, showing the ratio of product concentrations to reactant concentrations, each raised to the power of their stoichiometric coefficients.
• For the citric acid and methylamine reaction, \( K_{eq} \) is given by: \[ K_{eq} = \frac{[H_2A^-][CH_3NH_3^+]}{[H_3A][CH_3NH_2]} \]
• A change in conditions, like concentration or temperature, can shift the equilibrium position, a concept known as Le Chatelier's principle.

For our scenario, understanding how \( K_{eq} \) relates to \( K_a \) and \( K_b \) is essential to predict the extent of the reaction between citric acid and methylamine at equilibrium.

Citric Acid

Citric acid is a weak organic acid found naturally in citrus fruits like lemons. It is characterized by its three carboxylic acid groups, making it triprotic.

• This chemical property enables citric acid to donate up to three protons, though for our exercise, we consider only the first donation with \( K_{a1} \).
• The equation for its first proton donation is: \[ H_3A \rightleftharpoons H_2A^- + H^+ \]
• Citric acid in lemons helps neutralize fishy odors by reacting with methylamines, forming non-odorous compounds.

Leveraging its ionization properties helps us calculate the equilibrium constant in reactions where only the initial proton dissociation is significant.

Methylamine

Methylamine \( (CH_3NH_2) \) is a simple amine derived from ammonia, where one hydrogen atom is replaced by a methyl group \( (CH_3) \). It is a weak base and often exists in equilibrium with its protonated form.

• Methylamine plays a key role in the odor of fish, which is why it's pivotal in our equilibrium calculations.
• It acts as a base in water, as shown in the reaction: \[ CH_3NH_2 + H_2O \rightleftharpoons CH_3NH_3^+ + OH^- \]
• This equilibrium reaction helps us derive the \( K_b \) value important for calculating \( K_{eq} \).

Understanding methylamine's function and reactivity assists in comprehending the neutralization of odors through its reaction with citric acid, beneficial in culinary contexts to freshen seafood dishes.