Find the \(\mathrm{pH}\) of a buffer that consists of \(1.3 \mathrm{M}\) sodium phenolate \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{ONa}\right)\) and \(1.2 M\) phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right)\left(\mathrm{p} K_{\mathrm{a}}\right.\) of phenol \(\left.=10.00\right)\)

The Henderson-Hasselbalch equation is a critical formula in acid-base chemistry that allows us to calculate the pH of a buffer solution. This equation bridges the relationship between the pH (a measure of the acidity or basicity of a solution) and the concentrations of the acid and its conjugate base in the buffer. The equation is given by: \[ \text{pH} = \text{p}K_{\text{a}} + \text{log} \frac{[\text{A}^-]}{[\text{HA}]} \]
Here, \text{p}K_{\text{a}} refers to the acid dissociation constant expressed in logarithmic form, [\text{A}^-] is the concentration of the conjugate base (the species formed after the acid donates a proton), and [\text{HA}] is the concentration of the weak acid.
For our given problem:
- The \text{p}K_{\text{a}} of phenol is 10.00
- Sodium phenolate, which is the conjugate base, has a concentration of 1.3M.
- Phenol, the weak acid, has a concentration of 1.2M.
Plugging these values into the Henderson-Hasselbalch equation will allow us to solve for the pH of the buffer solution.

Acid-base chemistry is a fundamental area of chemistry that deals with the properties of acids and bases, their reactions, and concepts such as pH, pKa, and buffer solutions. In the realm of acid-base chemistry, acids are substances that donate protons (H^+), whereas bases are substances that accept protons. The strength of an acid is often measured by its dissociation constant, K_a, which quantifies its ability to donate protons in an aqueous solution. For phenol, the acid in our problem, it has a pK_a of 10.00.

Furthermore, buffer solutions play an essential role in maintaining the pH at a nearly constant value, even when small amounts of acid or base are added. Understanding how buffers work involves recognizing that they contain both a weak acid and its conjugate base. In the context of this problem, phenol acts as the weak acid, and sodium phenolate acts as its conjugate base.
By knowing the concentrations of both the acidic form (HA) and the basic form (A^-), we can use these values, along with the location of equilibrium expressed by the pK_a, to determine the overall pH of the solution.

Logarithmic functions are heavily utilized in chemistry, particularly within the field of acid-base chemistry. They help to condense large ranges of values into more manageable numbers. In the context of the Henderson-Hasselbalch equation, logarithmic functions allow for the straightforward addition of values rather than multiplication or division.

For example, in our buffer problem, once we have the ratio of the conjugate base to the acid, we take the log of this ratio. Let's break down the calculations:
1. Divide the concentrations: \( \frac{1.3}{1.2} \approx 1.0833 \)
2. Find the logarithm of the ratio: \( \text{log} 1.0833 \approx 0.0356\)
3. Add this logarithmic value to the pKa: \(10.00 + 0.0356 \approx 10.04\)
By understanding logarithmic concepts, students can grasp how these small numerical ratios influence the resultant pH value in practical scenarios.