Problem 32 - Chemistry: The Central Science

Question

You have to prepare a \(\mathrm{pH} 4.80\) buffer, and you have the following \(0.10 \mathrm{M}\) solutions available: formic acid, sodium formate, propionic acid, sodium propionate, phosphoric acid, and sodium dihydrogen phosphate. Which solutions would you use? How many milliliters of each solution would you use to make approximately a liter of the buffer?

Step-by-Step Solution

Answer

To prepare a pH 4.80 buffer using the available \(0.10 \mathrm{M}\) solutions, you would use approximately 454 mL of \(0.10 \mathrm{M}\) sodium propionate and approximately 546 mL of \(0.10 \mathrm{M}\) propionic acid. This is based on the pKa values of the provided weak acids and the application of the Henderson-Hasselbalch equation to determine the appropriate ratio of weak acid and conjugate base concentrations.

Step 1: 1. Identifying the appropriate solutions for the pH 4.80 buffer

First, we need to look at the pKa values of each available weak acid: Formic acid: pKa = 3.74 Propionic acid: pKa = 4.88 Phosphoric acid (first dissociation): pKa1 = 2.15 Our desired buffer pH is 4.80, so we should choose the weak acid with a pKa value closest to 4.80, which is propionic acid (pKa = 4.88). Therefore, we will use \(0.10 \mathrm{M}\) propionic acid and its salt, \(0.10 \mathrm{M}\) sodium propionate, to prepare the buffer.

Step 2: 2. Using the Henderson-Hasselbalch equation to determine the ratio

We will use the Henderson-Hasselbalch equation to find the ratio of the concentrations of sodium propionate ([A-]) and propionic acid ([HA]), which has the desired pH value: Henderson-Hasselbalch equation: \[pH = pKa + \log \frac{[A-]}{[HA]}\] Our desired pH is 4.80, and the pKa for propionic acid is 4.88. Plugging those values in: \(4.80 = 4.88 + \log \frac{[A-]}{[HA]}\)

Step 3: 3. Solving for the ratio of conjugate base to weak acid concentrations

First, we isolate the logarithm: \(\log \frac{[A-]}{[HA]} = 4.80 - 4.88\) \(\log \frac{[A-]}{[HA]} = -0.08\) Now, we can solve for the ratio of [A-] to [HA]: \(\frac{[A-]}{[HA]} = 10^{-0.08}\) \(\frac{[A-]}{[HA]} \approx 0.831\)

Step 4: 4. Calculating the volumes needed to prepare a 1-liter buffer

Now that we have established the ratio of sodium propionate to propionic acid, we can calculate the volumes of each solution that is needed. Let x be the volume (mL) of the sodium propionate solution, and y be the volume (mL) of the propionic acid solution. Our objective is to prepare a 1-liter buffer, so we can write the equation: \(x + y = 1000\) Now, we use the ratio \(\frac{[A-]}{[HA]} \approx 0.831\) for the sodium propionate and propionic acid concentrations: \(\frac{0.10x}{0.10y} \approx 0.831\) After simplification, we get: \(x \approx 0.831y\) Now we have a system of equations: \(x + y = 1000\) \(x \approx 0.831y\)

Step 5: 5. Solving the system of equations

We substitute the second equation into the first equation and solve for y: \(0.831y + y = 1000\) \(1.831y \approx 1000\) \(y \approx 546\, \mathrm{mL}\) Now we plug our y value back into our x equation: \(x \approx 0.831(546)\) \(x \approx 454\, \mathrm{mL}\)

Step 6: 6. Final answer for the required volumes

We have calculated that we need approximately 454 mL of \(0.10 \mathrm{M}\) sodium propionate and approximately 546 mL of \(0.10 \mathrm{M}\) propionic acid to make a pH 4.80 buffer with a total volume of approximately 1 liter.

Key Concepts

Henderson-Hasselbalch equation

Understanding the Henderson-Hasselbalch equation is crucial when it comes to preparing buffer solutions. The equation provides a relationship between the pH of a buffer, the pKa (which is a measure of the acid strength) of the buffering agent, and the ratio of the concentration of the conjugate base ([A^-]) to the concentration of the acid ([HA]). It is elegantly expressed as:

\[\begin{equation} pH = pKa + \log\left(\frac{[A^-]}{[HA]}\right)\end{equation}\]
To use this equation, you simply need the pKa of the buffering acid and the desired pH of the buffer. Then, you can calculate the necessary ratio of conjugate base to acid to achieve that specific pH.

pKa value

The pKa value is a decimal number that represents the acid dissociation constant (Ka) in logarithmic form. It is essential to know the pKa value of an acid to prepare a buffer solution because it indicates the pH at which half of the acid is dissociated. In a buffer solution, you ideally want to choose an acid that has a pKa value close to the target pH of the buffer. This ensures that the acid and its conjugate base will be present in approximately equal amounts, which is the optimal condition for a buffer to resist changes in pH. In the example given, propionic acid with a pKa of 4.88 is nearly ideal for making a pH 4.80 buffer because the pKa is very close to the desired pH.

acid-base conjugate pairs

In acids and bases chemistry, an acid-base conjugate pair consists of two species that transform into one another by the gain or loss of a proton (H+). When preparing a buffer solution, it is important to use a conjugate acid-base pair. This pair resists pH changes because the acid can neutralize added bases and the base can neutralize added acids. In the preparation of a pH 4.80 buffer, we use propionic acid and its conjugate base, sodium propionate. Their ability to maintain the pH at the desired level of 4.80 showcases the importance of selecting appropriate conjugate pairs in buffer preparation.