Problem 21 - Chemical Principles

Question

(a) What must be the ratio of the concentrations of \(\mathrm{CO}_{3}^{2-}\) and \(\mathrm{HCO}_{3}{\underline{\phantom{xx}}}^{-}\)ions in a buffer solution having a \(\mathrm{pH}\) of \(11.0\) ? (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 \(\mathrm{pH}\) 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 \(\mathrm{pH}\) 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 \(\mathrm{pH}\) of \(11.0\) ?

Step-by-Step Solution

Answer

First, use the Henderson-Hasselbalch equation to find the ratio of \(\mathrm{CO}_{3}^{2-}\) to \(\mathrm{HCO}_{3}^{-}\), and then use that ratio to calculate the required masses or volumes of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) or \(\mathrm{KHCO}_{3}\) to create the buffer solution with a pH of 11.0.

Step 1: Use the Henderson-Hasselbalch equation

To find the ratio of the concentrations of \(\mathrm{CO}_{3}^{2-}\) and \(\mathrm{HCO}_{3}^{-}\) ions, use the Henderson-Hasselbalch equation which relates pH, pKa, and the ratio of the concentrations of the acid and its conjugate base. The equation is \(\text{pH} = \text{pKa} + \log\left(\frac{[\text{Base}]}{[\text{Acid}]}\right)\). In this case, the acid is \(\mathrm{HCO}_{3}^{-}\) and the base is \(\mathrm{CO}_{3}^{2-}\).

Step 2: Calculate the ratio of \(\mathrm{CO}_{3}^{2-}\) to \(\mathrm{HCO}_{3}^{-}\) for pH 11.0

First, find the \(\text{pKa}\) value for the \(\mathrm{HCO}_{3}^{-}/\mathrm{CO}_{3}^{2-}\) buffer system, which corresponds to the equilibrium \(\mathrm{HCO}_{3}^{-} \rightleftharpoons \mathrm{CO}_{3}^{2-} + \mathrm{H}^{+}\) with \(\text{pKa} \approx 10.3\). Then, plug in the known pH of 11.0 into the Henderson-Hasselbalch equation and solve for the ratio \(\frac{[\mathrm{CO}_{3}^{2-}]}{[\mathrm{HCO}_{3}^{-}]}\).

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

Determine the molar mass of \(\mathrm{K}_{2}\mathrm{CO}_{3}\). Then, using the ratio from Step 2 and the initial concentration of \(\mathrm{KHCO}_{3}\), calculate the required concentration of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) and from that the mass of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) to add to 1 liter of \(0.100 \mathrm{M} \mathrm{KHCO}_{3}\).

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

To find the needed mass of \(\mathrm{KHCO}_{3}\) to add to 1.00 L of \(0.100 \mathrm{M} \mathrm{K}_{2} \mathrm{CO}_{3}\), use the molar mass of \(\mathrm{KHCO}_{3}\) along with the ratio from Step 2, to determine the buffer's desired concentration.

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

Using the molarity of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) stock solution and the required concentration derived from the ratio in Step 2, calculate how much volume of the stock solution should be added to 100 mL of \(0.100 \mathrm{M} \mathrm{KHCO}_{3}\) to achieve the desired buffer solution.

Key Concepts

Henderson-Hasselbalch Equation

Understanding the Henderson-Hasselbalch equation is a fundamental aspect of pH calculation in buffer solutions. This remarkable equation serves as a simplified method to estimate the pH of a buffer system and is based on the acid-base equilibrium principle. It is expressed as:
\[\text{pH} = \text{pKa} + \log\left(\frac{[\text{Base}]}{[\text{Acid}]}\right)\]
Here, \[\text{pKa}\] represents the negative logarithm of the acid dissociation constant (Ka) of the conjugate acid, and the ratio of \[\frac{[\text{Base}]}{[\text{Acid}]}\] refers to the molar concentrations of the conjugate base and the conjugate acid, respectively. This tool is invaluable when you need to determine the exact proportions of these two components to achieve a desired pH, especially when dealing with buffer solutions like carbonates in a bicarbonate system.

Acid-Base Equilibrium

At the core of buffer solution pH calculations lies the concept of acid-base equilibrium. This refers to the state wherein the rates of the forward and backward reactions of acids and bases are equal, leading to stable concentrations of the reactants and products. For a buffer system comprising a weak acid and its conjugate base, this equilibrium can be expressed as:
\[\text{Acid (HA)} \rightleftharpoons \text{Base (A}^-) + \text{H}^+\]
Carbonates and bicarbonates, such as those found in the original exercise, follow this equilibrium principle. The pH of the resulting solution is dependent on the concentration of hydrogen ions (\[\text{H}^+\]) as well as on the ratio of the concentrations of the conjugate base and acid. In a practical setting, manipulating this ratio alters the buffer's pH, which is particularly useful in many biological and chemical applications.

Molar Mass Calculation

To perform buffer solution calculations, knowing the molar mass of the compounds involved is crucial. Molar mass is the weight of one mole of a substance and is typically expressed in grams per mole (g/mol). It can be determined by summing the atomic masses of all atoms in a molecule. For instance, the molar mass of potassium carbonate \(\mathrm{K}_{2}\mathrm{CO}_{3}\) is calculated by adding together the molar masses of potassium, carbon, and oxygen individually.

Example Calculation:

The atomic masses are approximately 39.10 g/mol for potassium (\(\text{K}\)), 12.01 g/mol for carbon (\(\text{C}\)), and 16.00 g/mol for each oxygen (\(\text{O}\)), leading to:
\[\text{Molar Mass} = 2(39.10) + 12.01 + 3(16.00)\]
The calculated molar mass of \(\mathrm{K}_{2}\mathrm{CO}_{3}\) is necessary for determining how much to weigh out to achieve a specific concentration in the buffer solution.

Chemical Buffer Systems

Chemical buffer systems are solutions that resist significant changes in pH when small amounts of acid or base are added. A buffer is typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid. The carbonic acid (\(\mathrm{H}_{2}\mathrm{CO}_{3}\)) - bicarbonate (\(\mathrm{HCO}_{3}^{-}\)) buffer system used in blood to maintain pH is a classic example.
\(\mathrm{HCO}_{3}^{-}\) acts as the acid, which can donate an \(\text{H}^+\) ion, while \(\mathrm{CO}_{3}^{2-}\) serves as the base, accepting an \(\text{H}^+\). In the original exercise, the buffer system is created using potassium carbonate and potassium bicarbonate. Essential for maintaining the correct pH in various environments, these systems find their importance across biological systems, industrial processes, and environmental regulations.