Question

The \(\mathrm{p} K_{\mathrm{a}}\) of the conjugate acid of morpholine is 8.33. (a) Calculate the ratio of morpholine to morpholinium ion in aqueous solution at \(\mathrm{pH}\) 7.0. (b) At what \(\mathrm{pH}\) are the concentrations of morpholine and morpholinium ion equal?

Short answer

Answer: The ratio of morpholine to morpholinium ion at pH 7.0 is approximately 0.046. The concentrations of morpholine and morpholinium ion are equal at a pH of 8.33.

Step-by-step solution

(Step 1: Write down the Henderson-Hasselbalch equation)

The Henderson-Hasselbalch equation is given by: pH = pKa + log10([A-]/[HA]) where pH is the pH of the solution, pKa is the acid dissociation constant, [A-] is the concentration of the conjugate base (morpholine), and [HA] is the concentration of the acid (morpholinium ion).

(Step 2: Calculate the ratio of morpholine to morpholinium ion at pH 7.0)

We are asked to find the ratio of morpholine to morpholinium ion at pH 7.0. We can use the Henderson-Hasselbalch equation with the given pH and pKa values to find this ratio: 7.0 = 8.33 + log10([morpholine]/[morpholinium ion]) Now, we can solve for the ratio ([morpholine]/[morpholinium ion]): log10([morpholine]/[morpholinium ion]) = 7.0 - 8.33 = -1.33 To find the actual ratio, we need to take the anti-log of both sides: [morpholine]/[morpholinium ion] = 10^(-1.33) So, the ratio of morpholine to morpholinium ion at pH 7.0 is approximately 0.046.

(Step 3: Determine the pH at which the concentrations of morpholine and morpholinium ion are equal)

When the concentrations of morpholine and morpholinium ion are equal, the ratio becomes 1. We can plug this value into the Henderson-Hasselbalch equation and solve for the pH: pH = pKa + log10([morpholine]/[morpholinium ion]) = pKa + log10(1) Since log10(1) = 0, the equation simplifies to: pH = pKa Therefore, the pH at which the concentrations of morpholine and morpholinium ion are equal is the same as the pKa of the conjugate acid of morpholine, which is 8.33.

Key concepts

pKa and pH Relationship

Understanding the relationship between the pKa of a substance and the pH of its environment is critical for analyzing acid-base behavior in solutions. The pKa value is a measure of the acidity of a substance; it indicates the pH at which half of the molecules of that substance are dissociated.

In the context of the Henderson-Hasselbalch equation, the pKa represents the pH at which the concentrations of the acid (HA, or in our example, morpholinium ion) and its conjugate base (A-, or morpholine) are equal. When the pH of the solution equals the pKa, we observe a 1:1 ratio of acid to base.

For environments where the pH is lower than the pKa, the acid form predominates, while at pH higher than the pKa, the conjugate base form is more prevalent. Thus, the Henderson-Hasselbalch equation helps us calculate the exact ratio of acid to base at any given pH, further illustrating the delicate balance between pKa and pH in a solution.

Acid-Base Equilibria

Acid-base equilibria refer to the state of balance between the protonated (acid) and deprotonated (base) forms of a molecule in a solution. This concept is crucial for the buffering capacity of biological systems and for understanding the solubility and reactivity of molecules in various pH conditions.

The equilibrium is represented by the dissociation of an acid into its conjugate base and a hydrogen ion (proton), a reversible chemical reaction. The Henderson-Hasselbalch equation applies to this equilibrium by offering a way to calculate the pH of a buffered solution, based on the concentration ratio of the conjugate acid-base pair.

In the exercise example with morpholine, the equation enables the calculation of the ratio of the base to its conjugate acid and vice versa, shedding light on the prevailing form under specific pH conditions, which directly impacts the behavior and reactivity of the compound in the solution.

Conjugate Acid-Base Pairs

A conjugate acid-base pair consists of two species that differ by the presence or absence of a proton. Acids donate protons whereas bases accept them. When an acid loses a proton, it forms its conjugate base, and when a base gains a proton, it forms its conjugate acid. These pairs are in constant dynamic equilibrium in solution, with the direction of the equilibrium depending on the pH of the environment.

The Henderson-Hasselbalch equation comes into play by quantifying the concentration ratio between a conjugate acid and its base at a given pH. This relationship is essential for designing buffer solutions that can maintain a stable pH, which is especially important in biochemical processes where enzymes and other macromolecules are sensitive to changes in pH.

By comparing the pH to the pKa value of the conjugate acid, you can predict how the equilibrium will shift and which form of the substance will be more abundant. This is beautifully demonstrated in our example exercise, where the pH dictates the form in which morpholine exists in the solution, as a base or as its conjugate acid morpholinium ion.