Question

The \(\mathrm{pH}\) value of a solution obtained by mixing \(5 \mathrm{~g}\) of acetic acid and \(7.5 \mathrm{~g}\) of sodium acetate and making the volume to \(500 \mathrm{~mL}\left(\mathrm{~K}_{\mathrm{CH}_{3} \mathrm{COOH}}=1.8 \times 10^{-5}\right)\) is (a) \(4.78\) (b) \(3.78\) (c) \(5.78\) (d) zero

Short answer

The pH value of the solution is 4.78

Step-by-step solution

Understanding the concept of buffer solution

Buffers are aqueous systems that tend to resist changes in the pH when small amounts of an acid or an alkali are added to them. A buffer solution typically contains a weak acid and its conjugate base. In this case, acetic acid (CH3COOH) is a weak acid and sodium acetate (NaCH3COO) is its conjugate base.

Calculating the concentration of the acid and conjugate base

We know that molarity (M) = moles/volume (in Liters). So, to calculate the molarity we need to find the moles of each compound first.\nMoles of acetic acid (CH3COOH) = mass (g) / molar mass (g/mol). The molecular weight of CH3COOH is approximately 60, so the moles = 5/60 = 0.0833 mol.\nMoles of sodium acetate (NaCH3COO) = mass (g) / molar mass (g/mol). The molecular weight of NaCH3COO is approximately 82, so the moles = 7.5/82 = 0.09 mol. \nThen we calculate the molarity.\nConcentration of acetic acid [CH3COOH] = moles/volume (L) = 0.0833/0.5 = 0.167M.\nConcentration of sodium acetate [NaCH3COO] = moles/volume(L) = 0.09/0.5 = 0.18M.

Applying Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by: \(pH = pKa + log ([A-]/[HA])\), where [A-] is the concentration of the conjugate base and [HA] is the concentration of the weak acid. In this problem, KA (acid dissociation constant for CH3COOH) = 1.8 x 10^-5. Therefore, pKa = -log(1.8 x 10^-5) = 4.74. \nSubstituting [A-] = 0.18 and [HA] = 0.167 into the Henderson-Hasselbalch equation gives: \(pH = 4.74 + log (0.18/0.167) = 4.74 + 0.029 = 4.769.

Rounding off the result

Upon rounding off, the pH of the solution obtained by mixing 5g of acetic acid and 7.5g of sodium acetate in 500mL of water is 4.77, which is not an available choice in the given answers. Therefore, the closest and most accurate answer would be (a) 4.78.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a critical formula used to estimate the pH of buffer solutions, which consist of a weak acid and its conjugate base. The equation is given by:\[\begin{equation} pH = pKa + \log\left(\frac{[A^-]}{[HA]}\right) \end{equation}\]where:

• \( pH \) is the measure of the acidity or basicity of the solution.
• \( pKa \) is the negative logarithm of the acid dissociation constant (Ka), reflecting the acid's strength.
• \( [A^-] \) is the molarity of the conjugate base.
• \( [HA] \) is the molarity of the weak acid.

This equation is a rearrangement of the acid dissociation constant formula and assumes that the pH does not change substantially with the addition of small quantities of acid or base. Utilizing the Henderson-Hasselbalch equation simplifies the process of calculating the pH in buffering systems, like the acetic acid and sodium acetate mixture in our example. For a buffer solution, where the concentration of the weak acid and its conjugate base are similar, the pH is close to the pKa value. This stability is what makes buffer solutions so important in chemical and biological systems.

Acid-Base Equilibrium

Acid-base equilibrium refers to the state of balance between a conjugate acid-base pair in solution. It is a concept of paramount importance to understand how buffer solutions work. In any solution where a weak acid and its conjugate base are present, they will establish an equilibrium:\[\begin{equation} HA \leftrightarrow H^+ + A^- \end{equation}\]The position of this equilibrium is determined by the acid dissociation constant \(Ka\), which quantifies the extent of dissociation of the weak acid into its ions. A small \(Ka\) value indicates a stronger conjugate base and a weaker acid, meaning the acid will not dissociate much. In buffer solutions, the presence of both the acid and its conjugate base enables the solution to resist pH changes through this equilibrium. When an acid is added, it is neutralized by the conjugate base; when a base is added, it is neutralized by the weak acid. This dynamic maintains a relatively stable pH, which is crucial for processes requiring consistent pH levels, such as enzymatic reactions in the body.

Molarity Calculation

Molarity, represented as \(M\), is the measure of concentration of a solute within a solution, defined as the number of moles of solute per liter of solution. The formula for molarity is straightforward:\[\begin{equation} M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \end{equation}\]To calculate moles, you divide the mass of the substance by its molar mass:\[\begin{equation} \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \end{equation}\]This calculation is essential in preparing solutions with precise concentrations, such as in our example exercise where we determined the moles of acetic acid and sodium acetate needed to make a buffer solution. Understanding molarity allows us to mix these components in exact ratios to achieve a specific pH according to the Henderson-Hasselbalch equation.

Conjugate Acid-Base Pair

A conjugate acid-base pair is defined by the presence of two compounds in a solution, which differ by one proton. The acid (HA) donates the proton and becomes its conjugate base (A-), while the base (B) accepts a proton to become its conjugate acid (HB+). Here’s a simple representation of this concept:\[\begin{equation} HA + B \leftrightarrow A^- + HB^+ \end{equation}\]An easy way to remember this relationship: acids lose a proton to become bases, and bases gain a proton to become acids. In buffer solutions, working with conjugate acid-base pairs is crucial because they can neutralize added acids or bases, allowing the buffer to maintain a nearly constant pH. When selecting a buffer, it is important to choose an acid and a conjugate base with sufficient capacity to handle the expected amounts of acids or bases that might be introduced to the system, ensuring that the buffer’s pH remains within the desired range.