Question

Preparation of Buffer of Known \(\mathrm{pH}\) and Strength You have \(0.10 \mathrm{~m}\) solutions of acetic acid \(\left(\mathrm{p} K_{\mathrm{n}}=4.76\right)\) and sodium acetate. If you wanted to prepare \(1.0 \mathrm{~L}\) of \(0.10 \mathrm{~m}\) acetate buffer of \(\mathrm{pH}\) 4.00, how many milliliters of acetic acid and sodium acetate would you mix together?

Short answer

Mix 85.17 mL of acetic acid and 14.83 mL of sodium acetate.

Step-by-step solution

Understand the Henderson-Hasselbalch Equation

To prepare a buffer solution, you can use the Henderson-Hasselbalch equation: \[pH = pK_a + \log_{10}\left(\frac{[A^-]}{[HA]}\right) \] Where \(pH\) is the desired pH of the buffer, \(pK_a\) is the negative logarithm of the acid dissociation constant, \([A^-]\) is the concentration of the base (sodium acetate), and \([HA]\) is the concentration of the acid (acetic acid).

Substitute Known Values

You have the following values: - Desired \(pH = 4.00\)- \(pK_a = 4.76\)- Concentration of buffer needed is \(0.10 \text{ M}\) Substitute into the Henderson-Hasselbalch equation: \[4.00 = 4.76 + \log_{10}\left(\frac{[A^-]}{[HA]}\right) \]

Solve for the Ratio \(\frac{[A^-]}{[HA]}\)

Rearrange the equation to solve for \(\log_{10}\left(\frac{[A^-]}{[HA]}\right)\): \[4.00 - 4.76 = \log_{10}\left(\frac{[A^-]}{[HA]}\right) \] Simplify: \[-0.76 = \log_{10}\left(\frac{[A^-]}{[HA]}\right) \]Now, solve for \(\frac{[A^-]}{[HA]}\): \[\frac{[A^-]}{[HA]} = 10^{-0.76} \approx 0.17378 \]

Find Individual Molarities for Acetic Acid and Sodium Acetate

Set up equations for the individual molarities of acetic acid \( [HA] \) and sodium acetate \( [A^-] \):\[[HA] + [A^-] = 0.10 \]\[\frac{[A^-]}{[HA]} = 0.17378\]

Solve the System of Equations

We have:\[[HA] + [A^-] = 0.10 \]\[[A^-] = 0.17378 \, [HA] \]Substitute \([A^-]\) in the first equation:\[[HA] + 0.17378 \, [HA] = 0.10 \]Combine terms:\[1.17378 \, [HA] = 0.10 \]Solve for \([HA]\):\[[HA] = \frac{0.10}{1.17378} \approx 0.08517 \, \text{M} \]Then \([A^-] = 0.10 - [HA] = 0.10 - 0.08517 = 0.01483 \, \text{M} \)

Calculate Volumes

Calculate the volumes from the molarities to use in a 1 L buffer solution:Volume of acetic acid (\(HA\)) needed: \[V_{HA} = 0.08517 \, \text{M} \times 1.0 \, \text{L} = 0.08517 \, \text{L} \approx 85.17 \, \text{mL} \]Volume of sodium acetate (\([A^-]\)) needed:\[V_{A^-} = 0.01483 \, \text{M} \times 1.0 \, \text{L} = 0.01483 \, \text{L} \approx 14.83 \, \text{mL} \]

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a crucial tool in chemistry for buffer preparation. This equation, \[pH = pK_a + \log_{10}\left(\frac{[A^-]}{[HA]}\right)\] helps us understand the relationship between pH, the concentration of an acid's conjugate base \([A^-]\), and the acid \([HA]\) itself.
It's essential to know that

• \(pH\) indicates how acidic or basic a solution is,
• \(pK_a\) is the acid's strength, reflecting how easily it donates protons,
• \([A^-]\) represents the base form, like sodium acetate in an acetic acid buffer, and
• \([HA]\) is the acid form, such as acetic acid in this scenario.

By rearranging this equation, you can determine how much acid or base is needed for a buffer with a specific pH. Thus, the equation is fundamental to predicting and maintaining a steady pH in various chemical systems.

Acetic Acid Buffer

An acetic acid buffer solution is composed of acetic acid (a weak acid) and its conjugate base, sodium acetate. This combination creates a solution that resists pH changes when small amounts of strong acid or base are added.
This resistance to change is crucial for many biological and chemical processes. Acetic acid buffers work by neutralizing added acids or bases with the help of their components:

• The acetic acid can react with added bases, limiting pH increase, while
• sodium acetate can react with added acids, limiting pH decrease.

Thus, an acetic acid buffer is an effective way to maintain pH stability in reactions.

pH Calculation

Calculating the pH of a buffer solution is an important skill in chemistry. With the Henderson-Hasselbalch equation, you are able to find how concentrations of acid and base components affect the pH.
When the pH of a buffer solution is known, like 4.00 in our example, you can substitute values into the formula to find the ratio between the amounts of acid and base needed. After solving equations derived from the formula, you can determine each component's concentration, such as identifying that:

• acetic acid makes up approximately 0.08517 M, and
• sodium acetate makes up approximately 0.01483 M.

Knowing these values helps you calculate the precise volumes required to achieve the desired pH in your buffer solution. This understanding is key for laboratory practice and precise chemical manipulations.