Question

Calculation of Original \(\mathrm{pH}\) from Final \(\mathrm{pH}\) after Titration A biochemist has \(100 \mathrm{~mL}\) of a \(0.100 \mathrm{~m}\) solution of a weak acid with a \(\mathrm{p} K_{\mathrm{a}}\) of \(6.3\). He adds \(6.0 \mathrm{~mL}\) of \(1.0 \mathrm{M} \mathrm{HCl}\), which changes the \(\mathrm{pH}\) to \(5.7\). What was the \(\mathrm{pH}\) of the original solution?

Short answer

The original pH was approximately 6.3.

Step-by-step solution

Determine Moles of Acid and HCl

First, calculate the moles of the weak acid. Given a 0.100 M solution in 100 mL, the moles of weak acid are \(0.100\, \text{mol/L} \times 0.100\, \text{L} = 0.010\, \text{mol}\). For HCl, given a 1.0 M solution in 6.0 mL, the moles of HCl are \(1.0\, \text{mol/L} \times 0.006\, \text{L} = 0.006\, \text{mol}\).

Establish the Reaction and Changes

HCl reacts with the weak acid to form its conjugate base. The equation is: \( \text{HA} + \text{HCl} \rightarrow \text{A}^- + \text{H}_2\text{O} \). Initially, there are 0.010 mol of HA and 0.006 mol of HCl. All 0.006 mol of HCl react, reducing HA to 0.004 mol and forming 0.006 mol of \(\text{A}^-\).

Calculate the Concentration of HA and A⁻

After the reaction, the concentrations in the total volume of 106 mL make the concentrations: \( [\text{HA}] = \frac{0.004\, \text{mol}}{0.106\, \text{L}} \approx 0.0377\, \text{M} \) and \( [\text{A}^-] = \frac{0.006\, \text{mol}}{0.106\, \text{L}} \approx 0.0566\, \text{M} \).

Use Henderson-Hasselbalch Equation

The final pH is given as 5.7. Use the Henderson-Hasselbalch equation: \( \text{pH} = \text{p}K_a + \log \left(\frac{[\text{A}^-]}{[\text{HA}]}\right) \). Substitute values: \( 5.7 = 6.3 + \log \left(\frac{0.0566}{0.0377}\right) \). Solve for the log term: \( \log \left(\frac{0.0566}{0.0377}\right) = 5.7 - 6.3 = -0.6 \).

Calculate Original pH

For the initial solution, before HCl addition, assume the solution is at equilibrium. The original \(\text{pH} = \text{p}K_a + \log \left( \frac{\text{x}}{[\text{HA}] - \text{x}}\right) \) simplifies for low ionization of weak acids: \( \text{pH} = \text{p}K_a = 6.3 \) since initial \([\text{A}^-] \approx 0\).

Key concepts

pH calculation

When calculating pH, it's crucial to understand the pH scale, which ranges from 0 to 14. This scale is used to express the acidity or alkalinity of a solution. The pH is defined as the negative logarithm of the hydrogen ion concentration: \[ \text{pH} = -\log[\text{H}^+]\]For solutions involving weak acids, calculating pH can get a bit tricky, as it involves considering both the acid dissociation constant \(\text{p}K_a\) and the concentrations of the acid and its conjugate base.
In the given problem, the final pH after an acid-base reaction is calculated using the Henderson-Hasselbalch equation. This approach provides a practical way to determine the pH of buffer solutions by accounting for the mixture of a weak acid (HA) and its conjugate base (A\(^-\)).
Understanding the pH calculation helps in assessing the change in acidity as more acid or base is added to a solution.

titration

Titration is a lab technique used to determine the concentration of an unknown solution by reacting it with a solution of known concentration. In a typical titration, a titrant (the solution with known concentration) is slowly added to a solution of analyte (the unknown) until the reaction reaches its equivalence point.
For a weak acid, the equivalence point is reached when the amount of titrant added neutralizes the analyte. This results in a solution containing only the conjugate base of the weak acid plus any ensuing hydrolysis reaction products.
In this exercise, HCl titrates the weak acid, converting it to its conjugate base. This change can be visualized through the calculated pH shift from the addition of HCl, moving the solution from an acidic state to a more neutral one, depending on the strength of the conjugate base.

weak acid dissociation

Weak acids partially dissociate in water, meaning not all of the acid molecules release their hydrogen ions into solution. This partial dissociation is characterized by the acid dissociation constant \(\text{K}_a\), a measure of the strength of the weak acid.
The smaller the \(\text{K}_a\), the weaker the acid, and the less it dissociates. For convenience, \(\text{K}_a\) is often converted to \(\text{p}K_a\), using the formula \(\text{p}K_a = -\log(\text{K}_a)\).
In problems involving weak acids, assumptions about minimal dissociation simplify calculations and allow the use of approximations like in the Henderson-Hasselbalch equation.
For the original solution, the statement that initial \([\text{A}^-] \approx 0\) indicates negligible dissociation, allowing the \(\text{pH}\) to approximate \(\text{p}K_a\).

acid-base reaction

Acid-base reactions are chemical reactions that occur between an acid and a base. The essence of these reactions is the transfer of a proton (H\(^+\)) from the acid to the base.
These reactions can dramatically change the pH of a solution, thereby affecting the overall equilibria. It's important to note that strong acids like HCl fully dissociate in water, meaning every molecule releases an H\(^+\) ion. This is in contrast to the weak acids, which do not fully dissociate.
In the provided exercise, the reaction between the weak acid (HA) and hydrochloric acid (HCl) is examined. The HCl donates protons that react with the conjugate base A\(^-\) derived from HA, forming more HA and altering concentrations. Understanding these reactions is key to calculating pH changes, as each new addition of an acid or a base can shift the balance between HA and A\(^-\), thus impacting the overall pH of the solution.