Indicate whether an alcohol (ROH) with a pK_a value of 15 has more charged or more neutral molecules in a solution with the pH values given in Problem 41.
3. pH = 7
4. pH = 10
5. pH = 13

Verified step by step guidance

1. Step 1: Recall the relationship between pKa, pH, and the ratio of charged to neutral species. The Henderson-Hasselbalch equation is used: $\mathrm{pH}=\mathrm{pKa}+\log \left(\frac{\mathrm{[A⁻]}}{\mathrm{[HA]}}\right)$, where [A⁻] is the concentration of the deprotonated (charged) form and [HA] is the concentration of the neutral form.
2. Step 2: For each pH value, calculate the ratio of [A⁻] to [HA] using the equation: $\frac{\mathrm{[A⁻]}}{\mathrm{[HA]}}=\; 10^\{\mathrm{pH}-\mathrm{pKa}\}$. Substitute the given pKa value of 15 and the respective pH values (7, 10.5, and 13) into the equation.
3. Step 3: Analyze the ratio for each pH value. If the ratio $\frac{\mathrm{[A⁻]}}{\mathrm{[HA]}}$ is greater than 1, the solution has more charged molecules ([A⁻]). If the ratio is less than 1, the solution has more neutral molecules ([HA]). If the ratio equals 1, the concentrations of charged and neutral molecules are equal.
4. Step 4: For pH = 7, calculate $\frac{\mathrm{[A⁻]}}{\mathrm{[HA]}}=\; 10^\{7\; -\; 15\}$. Since the pH is much lower than the pKa, expect more neutral molecules ([HA]).
5. Step 5: For pH = 10.5 and pH = 13, repeat the calculation using $\frac{\mathrm{[A⁻]}}{\mathrm{[HA]}}=\; 10^\{10.5\; -\; 15\}$ and $\frac{\mathrm{[A⁻]}}{\mathrm{[HA]}}=\; 10^\{13\; -\; 15\}$. Compare the results to determine whether the solution has more charged or neutral molecules at these pH values.

Key Concepts

pKa and pH Relationship

The pKa value of a compound indicates the acidity of that compound, with lower pKa values corresponding to stronger acids. In a solution, the pH reflects the concentration of hydrogen ions (H+). When the pH is lower than the pKa, the compound is more likely to be protonated (neutral), while at a higher pH, it tends to lose a proton and become deprotonated (charged).

Protonation and Deprotonation

Protonation refers to the addition of a proton (H+) to a molecule, which can change its charge state. For alcohols, when the pH is lower than the pKa, the alcohol remains mostly in its neutral form (ROH). Conversely, at higher pH levels, the alcohol can lose a proton, resulting in a negatively charged alkoxide ion (RO-), thus affecting the balance between charged and neutral species in solution.

Equilibrium in Acid-Base Chemistry

In acid-base chemistry, equilibrium describes the balance between protonated and deprotonated forms of a compound in solution. The Henderson-Hasselbalch equation can be used to predict the ratio of charged to neutral species based on pH and pKa. Understanding this equilibrium is crucial for determining the predominant form of the alcohol at different pH levels, which directly influences its behavior in chemical reactions.