Question

You are asked to prepare a pH \(=4.00\) buffer starting from \(1.50 \mathrm{~L}\) of \(0.0200 \mathrm{M}\) solution of benzoic acid \(\left(\mathrm{C}_{4} \mathrm{H}_{4} \mathrm{COOH}\right)\) and any amount you need of sodium benzoate \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COONa}\right)\). (a) What is the \(\mathrm{pH}\) of the benzoic acid solution prior to adding sodium benzoate? (b) How many grams of sodium benzoate should be added to prepare the buffer? Neglect the small volume change that occurs when the sodium benzoate is added.

Short answer

The pH of the benzoic acid solution prior to adding sodium benzoate is approximately 3.04. To prepare the pH 4.00 buffer, you need to add approximately 3.27 grams of sodium benzoate to the 1.50 L of 0.0200 M benzoic acid solution.

Step-by-step solution

Calculate the pH of the benzoic acid solution

To calculate the pH of the benzoic acid solution, we first need to determine the weak acid equilibrium constant, Ka, for benzoic acid. The Ka of benzoic acid is \(6.5 \times 10^{-5}\). Now, we can use the following formula relating the Ka and the concentration of H+ ions: Ka = [H+][C6H5COO-] / [C6H5COOH] Since the solution is initially only benzoic acid (C6H5COOH), we can assume that the concentrations of H+ and C6H5COO- are equal, x. Therefore: Ka = x^2 / (0.0200 - x) Now we can solve for x (concentration of H+ ions): \(6.5 \times 10^{-5}\) = x^2 / (0.0200 - x) To approximate, we can assume x << 0.0200, so: x^2 ≈ \(6.5 \times 10^{-5}\) * 0.0200 x ≈ \(9.1 \times 10^{-4}\) Therefore, the pH of the benzoic acid solution is: pH = -log([H+]) = -log(\(9.1 \times 10^{-4}\)) ≈ 3.04

Use the Henderson-Hasselbalch equation

Now that we have the initial pH of the benzoic acid solution, we can use the Henderson-Hasselbalch equation to determine the amount of sodium benzoate needed to achieve a final pH of 4.00: pH = pKa + log([C6H5COO-] / [C6H5COOH]) In this equation, pH is the desired final pH, pKa is the negative logarithm of the Ka, and [C6H5COO-] and [C6H5COOH] are the concentrations of the conjugate base (sodium benzoate) and conjugate acid (benzoic acid), respectively. We know the pKa = -log(6.5 × 10^{-5}) ≈ 4.19, the desired pH is 4.00, and the concentration of benzoic acid is 0.0200 M. 4.00 = 4.19 + log([C6H5COO-] / 0.0200) Now we can solve for the concentration of sodium benzoate: -0.19 = log([C6H5COO-] / 0.0200) [C6H5COO-] = 0.0200 * 10^{-0.19} ≈ 0.0151 M Since we know the volume of the benzoic acid solution is 1.50 L, we can find the moles of sodium benzoate needed: moles of sodium benzoate = 0.0151 M * 1.50 L = 0.0227 moles Now we can convert moles of sodium benzoate to grams using its molar mass (144.11 g/mol): grams of sodium benzoate = 0.0227 moles * 144.11 g/mol ≈ 3.27 g So, to achieve a pH of 4.00, we need to add approximately 3.27 grams of sodium benzoate to the 1.50 L of 0.0200 M benzoic acid solution.

Key concepts

Henderson-Hasselbalch Equation

Preparing a buffer solution requires an understanding of the Henderson-Hasselbalch equation, a mathematical formula crucial to biochemistry and chemistry. This equation provides the relationship between the pH of a buffer system, its acid dissociation constant (\text{Ka}), and the concentrations of the weak acid and its conjugate base.

The generic form of the equation is:
\[\text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\]
Here, \(\text{pH}\) is the measure of acidity, \(\text{pKa}\) is the acid dissociation constant expressed in logarithmic form, \([\text{A}^-]\) is the concentration of the conjugate base, and \([\text{HA}]\) is the concentration of the acid. The equation is derived from the Ka expression for a weak acid and shows the direct logarithmic relationship between the ratio of ionized and unionized species and the pH of the solution. To solve for buffer preparation, rearrange the equation to find the required concentration of the conjugate base or acid necessary to achieve a desired pH.

It's important to keep in mind that this equation assumes the concentration of acid and conjugate base are not significantly changed by dilution and that they are the major species in the solution contributing to pH.

Weak Acid Equilibrium Constant (Ka)

A key concept when working with buffers is the weak acid equilibrium constant, more commonly known as Ka. This value represents the strength of a weak acid, indicating how completely it dissociates in water.

The equilibrium expression for a weak acid dissociating in water is: \[\text{Ka} = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}.\]
In this expression, \([\text{H}^+]\) is the concentration of hydrogen ions, \([\text{A}^-]\) is the concentration of the conjugate base, and \([\text{HA}]\) is the concentration of the undissociated acid. For the exercise involving benzoic acid, we use the acid's Ka value to calculate the initial pH before any sodium benzoate is added.

Understanding and applying Ka is crucial for determining the proportions of acid and conjugate base needed in a buffer solution to attain the target pH. Ka also indirectly influences the buffering capacity, which is the buffer's resistance to changes in pH upon addition of acid or base. Higher Ka values correspond to stronger acids, which dissociate more in aqueous solution, affecting the acid-base equilibrium.

Acid-Base Conjugate Pairs

Buffer solutions are typically composed of acid-base conjugate pairs, which consist of a weak acid and its corresponding conjugate base (or a weak base and its conjugate acid). These pairs are molecules or ions that differ by one proton (H+ ion).

In our buffer preparation exercise, benzoic acid \((\text{C}_6\text{H}_5\text{COOH})\) acts as the weak acid, and its conjugate base is the ion created by losing a proton, specifically, the benzoate ion \((\text{C}_6\text{H}_5\text{COO}^-)\) from sodium benzoate \((\text{C}_6\text{H}_5\text{COONa})\).

The equilibrium between a weak acid and its conjugate base in water allows the buffer to resist changes in pH. When an acid is added to the buffer, it reacts with the conjugate base, thus limiting the increase in the hydrogen ion concentration. Conversely, when a base is added, the weak acid provides hydrogen ions to combine with the excess hydroxide ions, thereby reducing the pH change. This conjugate pair mechanism is the fundamental principle behind buffer solutions and is essential for maintaining stable pH conditions in various chemical and biological applications.