Problem 34 - Chemistry: Principles and Reactions

Question

A buffer is prepared using the propionic acid/propionate \(\left(\mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{2} /\right.\) \(\left.\mathrm{C}_{3} \mathrm{H}_{5} \mathrm{O}_{2}^{-}\right)\) acid-base pair for which the ratio \(\left[\mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{2}\right] /\left[\mathrm{C}_{3} \mathrm{H}_{5} \mathrm{O}_{2}^{-}\right]\) is \(4.50 .\) \(K_{\mathrm{a}}\) for propionic acid is \(1.4 \times 10^{-5}\). (a) What is the \(\mathrm{pH}\) of this buffer? (b) Enough strong base is added to convert \(27 \%\) of \(\mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{2}\) to \(\mathrm{C}_{3} \mathrm{H}_{5} \mathrm{O}_{2}^{-} .\) What is the \(\mathrm{pH}\) of the resulting solution? (c) Strong base is added to increase the \(\mathrm{pH}\). What must the acid/base ratio be so that the \(\mathrm{pH}\) increases by exactly one unit (e.g., from 2 to 3 ) from the answer in (a)?

Step-by-Step Solution

Answer

Question: Calculate the initial pH of the buffer, the new pH after adding a strong base that converts 27% of propionic acid to propionate, and the new acid/base ratio required to increase the pH by 1 unit. Given the ratio of acid to conjugate base as 4.50 and the \(K_a\) value for propionic acid as \(1.4 \times 10^{-5}\).

Step 1: Calculate the pKa of propionic acid

The pKa can be calculated as follows: pKa = -log(Ka) pKa = -log(\(1.4 \times 10^{-5}\))

Step 2: Calculate the initial pH of the buffer

Using the Henderson-Hasselbalch equation: pH = pKa + log\(\frac{[\text{A}^-]}{[\text{HA}]}\) Since the ratio is given as 4.5, pH = pKa + log(4.50) Calculate the pH.

Step 3: Calculate the amount of propionic acid converted to propionate

Given that 27% of propionic acid is converted to propionate when a strong base is added, let x be the amount of propionic acid (in moles), then 0.27x moles of propionic acid will be converted into propionate.

Step 4: Calculate the new concentrations of propionate and propionic acid

After the reaction, both concentrations change: New concentration of propionic acid [\(\text{HA}\)] = \([4.5 * x - 0.27x]\) New concentration of propionate [\(\text{A}^-\)] = \([x + 0.27x]\)

Step 5: Calculate the new pH of the solution after base addition

Using the Henderson-Hasselbalch equation with the new concentrations: pH = pKa + log\(\frac{[x + 0.27x]}{[4.5 * x - 0.27x]}\) Calculate the new pH.

Step 6: Calculate the new acid/base ratio to increase the pH by 1 unit

To increase the pH by one unit, let the new pH be pH_initial + 1. Use the Henderson-Hasselbalch equation again: pH_initial + 1 = pKa + log\(\frac{[\text{A}_{new}^-]}{[\text{HA}_{new}]}\) Find the appropriate ratio to get the desired pH value.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a staple in acid-base chemistry, especially when dealing with buffer solutions. This equation is elegantly simple, providing a way to calculate the pH of a buffer solution. It is given by: \[pH = pKa + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\]Where:

• \([\text{A}^-]\) is the concentration of the base form (anion).
• \([\text{HA}]\) is the concentration of the acid form.
• \(pKa\) is the negative logarithm of the acid's dissociation constant \(K_a\).

This equation is essential when predicting how the addition of acids or bases will affect a buffer system's pH. It assumes that the concentrations of the acid and its conjugate base are high enough to suppress their complete dissociation or association.

Acid-Base Chemistry

Acid-base chemistry is the study of proton transfer reactions, which are foundational in understanding how buffers function. Propionic acid, in this case, serves as a weak acid, and its conjugate base, propionate, interacts in solution to resist changes in pH. A key aspect of acid-base chemistry is the concept of equilibrium, described by an acid's dissociation constant, \(K_a\). The strength of an acid is determined by its \(K_a\) value. The equation \(K_a = 1.4 \times 10^{-5}\) for propionic acid indicates it doesn't completely dissociate, thus, making it a weak acid.
Understanding these fundamentals provides insight into how buffers maintain equilibrium. When strong acids or bases are added, the buffer system moderates these additions by shifting equilibrium, allowing the pH to remain relatively constant!

pH Calculation

Calculating pH, especially in buffered solutions, involves understanding the relationship between the concentration of hydrogen ions and the properties of the buffer components. The pH is defined as the negative logarithm of the hydrogen ion concentration:\[pH = -\log[\text{H}^+]\]In buffer solutions, the pH will change less drastically compared to pure acid or base solutions when acids or bases are added. Here's a quick way to think about it:

• Initial pH is determined using the Henderson-Hasselbalch equation.
• Upon adding a strong base, as in the exercise, propionic acid is converted to propionate, affecting the acid/base ratio.
• To find the final pH after this conversion, adjust the concentrations in the Henderson-Hasselbalch equation.

Accurate pH calculations are critical in various chemical processes and biological systems where maintaining a specific pH range is necessary.