Problem 17 - Chemical Principles: The Quest for Insight

Question

(a) What must be the ratio of the concentrations of \(\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}\) and \(\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}\)ions in a buffer solution having a \(\mathrm{pH}\) of \(11 . \mathrm{O}\) ? (b) What mass of \(\mathrm{K}_{2} \mathrm{CO}_{3}\) must be added to \(1.00 \mathrm{~L}\) of \(0.100 \mathrm{M} \mathrm{KHCO}_{3}(\mathrm{aq})\) to prepare a buffer solution with a pH of \(11.0\) ? (c) What mass of \(\mathrm{KHCO}_{3}\) must be added to \(1.00 \mathrm{~L}\) of \(0.100 \mathrm{M} \mathrm{K}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) to prepare a buffer solution with a pH of \(11.0\) ? (d) What volume of \(0.200 \mathrm{M} \mathrm{K}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) must be added to \(100 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{KHCO}_{3}(\mathrm{aq})\) to prepare a buffer solution with a pH of \(11.0\) ?

Step-by-Step Solution

Answer

The ratio is \(10^{0.67}\). Mass and volume calculations require using this ratio along with the molar mass of the salts and the molarity of the given solutions to determine the precise amounts.

Step 1: Calculate the ratio using the Henderson-Hasselbalch equation

To find the ratio of the concentrations of \(\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}\) and \(\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}\) ions, the Henderson-Hasselbalch equation is used: \[\mathrm{pH} = \mathrm{p}K_{\mathrm{a}} + \log\left(\frac{[\mathrm{Base}]}{[\mathrm{Acid}]}\right)\]. In this case, base is \(\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}\) and acid is \(\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}\). The \(\mathrm{p}K_{\mathrm{a2}}\) for carbonic acid \(\mathrm{(H}_{2}\mathrm{CO}_{3})\) is typically taken to be approximately 10.33. Rearranging the equation to solve for the ratio \(\left(\frac{[\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}]}{[\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}]}\right)\), we get \[\left(\frac{[\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}]}{[\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}]}\right) = 10^{(\mathrm{pH} - \mathrm{p}K_{\mathrm{a}})}\].

Step 2: Solve for the ratio of concentrations

Substituting the given pH of 11.0 and \(\mathrm{p}K_{\mathrm{a2}}\) of 10.33 into the equation yields \[\left(\frac{[\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}]}{[\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}]}\right) = 10^{(11.0 - 10.33)} = 10^{0.67}\]. We calculate this value to find the ratio.

Step 3: Calculate the mass of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) needed

To calculate the mass of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) to add, we use the molar mass of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) (138.205 g/mol) and the ratio from Step 2. Since we have 1.00 L of 0.100 M \(\mathrm{KHCO}_{3}\), and the ratio is for \([\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}]\), we need to find the amount of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) that will give the correct amount of \(\mathrm{CO}_{3}{\underline{\phantom{xx}}}^{2-}\) ions. We will calculate the moles of \(\mathrm{KHCO}_{3}\) and use the ratio to find the moles of \(\mathrm{K}_{2}\mathrm{CO}_{3}\), then convert this to mass.

Step 4: Calculate the mass of \(\mathrm{KHCO}_{3}\) needed

Similar to step 3, we will use the molar mass of \(\mathrm{KHCO}_{3}\) (100.115 g/mol) and the ratio from Step 2 to find the mass of \(\mathrm{KHCO}_{3}\) to add to 1.00 L of 0.100 M \(\mathrm{K}_{2}\mathrm{CO}_{3}\). We will calculate the moles of \(\mathrm{K}_{2}\mathrm{CO}_{3}\), determine the moles of \(\mathrm{KHCO}_{3}\) needed using the ratio, and then find the corresponding mass.

Step 5: Calculate the volume of \(0.200 \mathrm{M} \mathrm{K}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) needed

To find the volume of \(0.200 \mathrm{M} \mathrm{K}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) required, we again use the ratio from Step 2. We will calculate the moles of \(\mathrm{KHCO}_{3}\) in 100 mL of 0.100 M solution, find the corresponding moles of \(\mathrm{K}_{2} \mathrm{CO}_{3}\) needed, and then use the molarity of the \(\mathrm{K}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) solution to find the volume to add.

Key Concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a key formula used to calculate the pH of buffer solutions. A buffer solution is one that resists changes in pH when small amounts of an acid or base are added.
The equation is expressed as:
\[\begin{equation}\text{pH} = \text{p}K_{\text{a}} + \log\left(\frac{[\text{Base}]}{[\text{Acid}]}\right)\end{equation}\]
In the equation, \( \text{p}K_{\text{a}} \) is the negative logarithm of the acid dissociation constant (Ka), which measures the strength of the acid. The terms \([\text{Base}]\) and \([\text{Acid}]\) represent the molar concentrations of the conjugate base and the acid, respectively.

The power of this equation lies in its ability to help us determine what the concentrations of these components should be to achieve a certain pH value. For instance, if a problem requires us to prepare a buffer with a specific pH, we can rearrange the equation to solve for the ratio of the base to the acid, which is precisely what we did in the original exercise.

Concentration ratio

The concentration ratio of the components in a buffer solution critically determines the pH of the solution. It refers to the molar concentration of the conjugate base (usually denoted as \([\text{A}^-]\)) to that of the weak acid (denoted as \([\text{HA}]\)).

From the original exercise, calculating this ratio was essential for determining how much of each substance was needed. By understanding the desired pH and the pKa value of the acid involved, you can use the Henderson-Hasselbalch equation to find this ratio. Once you have the ratio, you can then calculate the respective masses of acid and base needed to create the buffer solution with the desired pH.

The ratio is crucial, as it indicates the proportions in which acid and base are present and, therefore, their capacities to neutralize any added acids or bases, helping to maintain the buffer's pH.

Molar mass calculation

When dealing with buffer solutions, molar mass calculations become critical for converting between moles and grams. The molar mass is the weight in grams of one mole of a substance. One mole represents \(6.022 \times 10^{23}\) particles of the substance, known as Avogadro's number.

In our textbook exercise, we needed to determine the mass of \(\text{K}_2\text{CO}_3\) and \(\text{KHCO}_3\) to add to our solution. To do this, we first had to calculate the number of moles needed based on the molarity and volume of solution and then convert those moles to grams using the molar mass of these compounds. By knowing the molar mass, one can accurately prepare a solution with the precise amount of a substance required for the buffer.

pKa value

The pKa value plays a fundamental role in understanding the acidity of a substance in solution. It is the logarithmic expression of the acid dissociation constant (Ka) and is used to gauge the strength of an acid. Lower pKa values indicate stronger acids, which dissociate to a greater extent in water.

In relation to the Henderson-Hasselbalch equation, the pKa value is essential for calculating the pH of buffer solutions. As seen in our exercise, knowing the pKa of carbonic acid allowed us to determine the correct ratio of bicarbonate (\(\text{HCO}_3^-\)) to carbonate (\(\text{CO}_3^{2-}\)) ions to achieve a desired pH. It's a measure of how readily an acid will donate its proton (H+) in solution. In buffering systems, this value helps inform us how the solution can resist pH changes when acids or bases are added.