Find the \(\mathrm{pH}\) during the titration of \(20.00 \mathrm{~mL}\) of \(0.1000 \mathrm{M}\) butanoic acid, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{COOH}\left(K_{\mathrm{a}}=1.54 \times 10^{-5}\right),\) with \(0.1000 \mathrm{M}\) \(\mathrm{NaOH}\) solution after the following additions of titrant: (a) \(0 \mathrm{~mL}\) (b) \(10.00 \mathrm{~mL}\) (c) \(15.00 \mathrm{~mL}\) (d) \(19.00 \mathrm{~mL}\) (e) \(19.95 \mathrm{~mL}\) (f) \(20.00 \mathrm{~mL}\) (g) \(20.05 \mathrm{~mL}\) (h) \(25.00 \mathrm{~mL}\)

Understanding how to calculate pH is crucial in various chemical contexts, especially during titration. The term 'pH' refers to the concentration of hydrogen ions in a solution and is expressed by the equation: \[ \text{pH} = -\text{log}[H^+] \] Here, \[ [H^+] \] represents the molarity (moles per liter) of hydrogen ions in the solution. This calculation uses logarithmic functions, so having a strong grasp of logarithms can help. When the initial concentration of an acid is given, we often begin with an ICE table to find out the change in concentration as the acid dissociates. Then, we use the equilibrium concentrations to find the \[ [H^+] \] and subsequently calculate the pH. This process will be repeatedly followed at various steps in acid-base titration to determine the changing pH levels.

The Henderson-Hasselbalch Equation is a remarkably useful tool in acid-base chemistry, especially for buffer solutions. It’s used to find the pH of a solution when you know the concentration of an acid and its conjugate base. The equation is given by: \[ \text{pH} = \text{p}K_{\text{a}} + \text{log} \frac{[A^-]}{[HA]} \] Here, \[ \text{p}K_{\text{a}} \] is the negative logarithm of the acid dissociation constant \[ K_{\text{a}} \], \[ [A^-] \] is the concentration of the conjugate base, and \[ [HA] \] is the concentration of the acid. This equation simplifies pH calculations when dealing with buffer systems because it directly relates the pH to the ratio of the concentrations of the conjugate base and the acid. In the context of titration, as the base (\text{NaOH}) is added to the acid (\text{butanoic acid}), the relative amounts of \[ [A^-] \] and \[ [HA] \] change, allowing us to use the Henderson-Hasselbalch equation at each step to find the corresponding pH.

ICE tables are a systematic way to keep track of the changing concentrations of reactants and products during a chemical reaction, especially useful in acid-base titrations. The acronym ICE stands for Initial, Change, and Equilibrium.
Initial: Start with the initial concentrations of all species before any reaction takes place.
Change: Deduce how these concentrations change as the reaction proceeds to equilibrium. This is often represented using variable ‘x’ for the change.
Equilibrium: Sum the initial concentrations with the changes to find the equilibrium concentrations.

For example, if you start with 0.1000 M butanoic acid, you might find:
\[ \text{Initial:} [\text{CH}_3\text{CH}_2\text{CH}_2\text{COOH}] = 0.1000 \text{ M}, [\text{H}^+] = 0 \text{ M}, [\text{CH}_3\text{CH}_2\text{CH}_2\text{COO}^-] = 0 \text{ M} \] \[ \text{Change:} -x, +x, +x \] \[ \text{Equilibrium:} [\text{CH}_3\text{CH}_2\text{CH}_2\text{COOH}] = 0.1000 - x, [\text{H}^+] = x, [\text{CH}_3\text{CH}_2\text{CH}_2\text{COO}^-] = x \]
This makes it simpler to plug these values into the corresponding equilibrium constant expressions or the Henderson-Hasselbalch equation to find the pH.