Question

Find the \(\mathrm{pH}\) of \(0.10 \mathrm{M}\) HOAc solution that has \(0.20 \mathrm{M} \mathrm{NaOAc}\) dissolved in it. The dissociation constant of HOAc is \(1.75 \times 10^{-5}\)

Short answer

The pH of the given solution can be found using the Henderson-Hasselbalch equation: pH = pKa + log(\(\frac{[A^-]}{[HA]}\)). First, calculate the pKa from the given Ka: pKa = -log(\(1.75 \times 10^{-5}\)) ≈ 4.76. Then substitute the given concentrations and pKa into the equation: pH = 4.76 + log(\(\frac{0.20}{0.10}\)). Finally, solve for pH: pH ≈ 4.76 + 0.30 = 5.06.

Step-by-step solution

Write down the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by the formula: pH = pKa + log(\(\frac{[A^-]}{[HA]}\)) where - \(pH\) is the pH of the solution - \(pKa\) is the negative base-10 logarithm of the dissociation constant, Ka - \([A^-]\) is the concentration of the conjugate base (NaOAc in this case) - \([HA]\) is the concentration of the weak acid (HOAc in this case)

Calculate pKa from Ka

The dissociation constant of HOAc, Ka, is given as \(1.75 \times 10^{-5}\). We need to find the pKa using the formula: \(pKa = -\log_{10}(Ka)\) Plug in the given Ka value: \(pKa = -\log_{10}(1.75 \times 10^{-5}) \approx 4.76\)

Substitute values into the Henderson-Hasselbalch equation

Now we have the pKa value, and we are given the concentrations of NaOAc (\([A^-]\) = 0.20 M) and HOAc (\([HA]\) = 0.10 M). We can substitute these values into the Henderson-Hasselbalch equation: pH = 4.76 + log(\(\frac{0.20}{0.10}\))

Solve for pH

Now we can simplify the equation and solve for the pH of the solution: pH = 4.76 + log(\(2\)) pH ≈ 4.76 + 0.30 = 5.06 The pH of the 0.10 M HOAc solution with 0.20 M NaOAc dissolved in it is approximately 5.06.

Key concepts

pH Calculation

Understanding the concept of pH is crucial for many fields in science, especially chemistry and biology. The pH scale is a measure of how acidic or basic a solution is. It ranges from 0 to 14 with 7 being neutral, below 7 acidic, and above 7 basic. The pH of a solution is calculated using the negative logarithm of the hydrogen ion concentration (\textbf{H⁺}), represented mathematically as \( pH = -\log_{10}[H⁺] \).

In the case of buffer solutions, such as a mixture of acetic acid and its conjugate base, sodium acetate (NaOAc), the pH calculation simplifies with the Henderson-Hasselbalch equation. This equation provides a direct relationship between the pH and the ratio of the concentrations of the conjugate acid-base pair. This allows students to calculate the pH of buffer systems with ease, once the pKa and the concentrations of the acid and its conjugate base are known.

Acetic Acid and Acetate Buffer

A buffer solution resists changes in pH when small amounts of acid or base are added. This property is vital in many biological and chemical processes where a stable pH is required. An acetic acid and acetate buffer is one such system, consisting of acetic acid (HOAc) and its conjugate base, acetate (CH₃COO^- or A^-).

This type of buffer operates on the principle of Le Chatelier's Equilibrium, where adding more acid or base shifts the equilibrium to counteract the pH change. In a practical sense, when acetic acid dissociates, it releases H⁺ ions. However, if the solution also contains acetate ions, these will react with any additional H⁺ ions added to the system, thereby 'buffering' against pH change.

Weak Acid Dissociation Constant

The dissociation constant (Ka) of a weak acid is a measure of its ability to donate protons (H⁺ ions) in solution. Unlike strong acids that dissociate completely, a weak acid only partially dissociates in water. The value of Ka is determined experimentally and is crucial for predicting the extent of dissociation.

The lower the Ka, the weaker the acid and the less it dissociates. In the example given with acetic acid (HOAc), the Ka is \(1.75 \times 10^{-5}\), indicating it is a weak acid. Calculating the pKa, the negative logarithm of the Ka, provides a more convenient number to work with in calculations, especially when using the Henderson-Hasselbalch equation for pH calculations.

Acid-Base Equilibrium

Acid-base equilibrium refers to the state where the rate of the forward reaction (acid dissociating to produce H⁺ and its conjugate base) equals the rate of the reverse reaction (conjugate base accepting H⁺ to form the acid). At this point, the concentrations of all species remain constant over time, though they are not necessarily equal.

The equilibrium constants (Ka for acids and Kb for bases) quantify the position of this equilibrium. Understanding this equilibrium concept is necessary for grasping why the pH doesn't change significantly when small amounts of acids or bases are added to buffer solutions. It is the essence of how buffers function and is critical in the context of many chemical reactions and biological systems.