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.

First, let's write down the reaction equations of citric acid and methylamine. For citric acid: \[ \text{H}_3 \text{A} \rightleftharpoons \text{H}^+ + \text{H}_2 \text{A}^{-} \] Here, \(\text{H}_3 \text{A}\) represents citric acid, and we are given \(K_{a1}\) for this equation. For methylamine: \[ \text{B} + \text{H}^+ \rightleftharpoons \text{BH}^{+} \] Here, \(\text{B}\) represents methylamine, and its equilibrium constant is given by \(K_b\). When a citric acid molecule reacts with a methylamine molecule, the following reaction occurs: \[ \text{H}_3 \text{A} + \text{B} \rightleftharpoons \text{H}_2 \text{A}^{-} + \text{BH}^{+} \] We need to find the equilibrium constant for this reaction, which can be represented by \(K_\text{eq}\).

To connect the given \(K_{a 1}\) and \(K_b\) values to find the desired \(K_\text{eq}\), we will use the ion product of water, which is given by \[ K_w = [\text{H}^+] [\text{OH}^-] = 1.0 \times 10^{-14} \]

The relationship between the equilibrium constants of the reaction is given by \[ K_b = \frac{[\text{BH}^{+}] [\text{OH}^{-}]}{[\text{B}] [\text{H}^{+}]} \Rightarrow K_w = K_{a 1} K_b \]

Given that only the first proton of the citric acid is important in the neutralization reaction, we can now use the relationship established in Step 3 to find the equilibrium constant for the reaction between citric acid and methylamine. \[ K_\text{eq} = \frac{[\text{H}_2 \text{A}^{-}] [\text{BH}^{+}]}{[\text{H}_3 \text{A}] [\text{B}]} \] Next, use the relationship established in Step 3 to substitute \(K_b\) in terms of \(K_{a1}\) and \(K_w\): \[ K_\text{eq} = \frac{K_w}{K_{a1}} \] Now, use the values from the Appendix D to substitute in the expression for \(K_\text{eq}\) and calculate the value for the reaction between citric acid and methylamine: \[ K_\text{eq} = \frac{1.0 \times 10^{-14}}{K_{a 1}} \] With this expression, we can now find the equilibrium constant for the reaction between citric acid and methylamine by substituting the given \(K_{a 1}\) value.