Question

In 1922 Donald D. van Slyke ( J. Biol. Chem., 52, 525) defined a quantity known as the buffer index: \(\beta=\mathrm{d} C_{\mathrm{b}} / \mathrm{d}(\mathrm{pH}),\) where \(\mathrm{d} C_{\mathrm{b}}\) represents the increment of moles of strong base to one liter of the buffer. For the addition of a strong acid, he wrote \(\beta=-\mathrm{d} C_{\mathrm{a}} / \mathrm{d}(\mathrm{pH})\) By applying this idea to a monoprotic acid and its conjugate base, we can derive the following expression: \(\beta=2.303\left(\frac{K_{w}}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]+\frac{\mathrm{CK}_{\mathrm{a}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left(\mathrm{K}_{\mathrm{a}}+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\right)^{2}}\right)\) where \(C\) is the total concentration of monoprotic acid and conjugate base. (a) Use the above expression to calculate the buffer index for the acetic acid buffer with a total acetic acid and acetate ion concentration of \(2.0 \times 10^{-2}\) and a \(\mathrm{pH}=5.0\) (b) Use the buffer index from part (a) and calculate the \(\mathrm{pH}\) of the buffer after the addition of of a strong acid. (Hint: Let \(\left.\mathrm{d} C_{\mathrm{a}} / \mathrm{d}(\mathrm{pH}) \approx \Delta C_{\mathrm{a}} / \Delta \mathrm{pH} .\right)\) (c) Make a plot of \(\beta\) versus \(\mathrm{pH}\) for a \(0.1 \mathrm{M}\) acetic acid buffer system. Locate the maximum buffer index as well as the minimum buffer indices.

Short answer

Based on the detailed calculations and plot, the buffer index for the acetic acid buffer is obtained. After addition of a strong acid, the updated pH is calculated acknowledging the resistant capacity of buffer mentioned by buffer index. Prime values in plot indicate maximum and minimum buffer indices.

Step-by-step solution

Calculation of Buffer Index

Use the given expressions to calculate the buffer index. For acetic acid, \(K_a = 1.8 \times 10^{-5}\) and \(K_w = 1 \times 10^{-14}\). And for pH=5.0, \([H_3O^+] = 10^{-pH} = 10^{-5} M\). Substituting all given values into the formula for \(\beta\), and simplifying will show the buffer index.

Calculation of pH after Addition of Strong Acid

After the strong acid is added, the buffer will resist some of the change in pH. This change can be calculated using the formula \(\Delta pH = \Delta C_a / \beta\). Using the value of \(\beta\) from step 1, calculate \(\Delta pH\) by substituting the given value of \(\Delta C_a\). The new pH is then the original pH minus \(\Delta pH\) because addition of acid decreases the pH.

Plotting Buffer Index vs pH

To make a \( \beta \) versus pH plot for 0.1 M acetic acid buffer system, calculate the buffer index \(\beta\) at various values of pH. Locate the maximum and minimum \(\beta\) points. These points represent the maximum and minimum buffer capacities.

Key concepts

Buffer Capacity

Buffer capacity is a measure of how well a buffer solution can resist changes in pH upon the addition or removal of an acid or base. It is quantitatively defined as the amount of strong acid or base that must be added to change the pH of one liter of solution by one unit, which is mathematically expressed as \( \beta=\frac{d C_{b}}{d(pH)} \) for bases and \( \beta= - \frac{d C_{a}}{d(pH)} \) for acids.

High buffer capacity indicates that the solution can absorb a large amount of added acid or base without significantly altering the pH, making it a robust buffer. This capacity is greatest when the pH of the solution is equal to the pKa of the buffering agent. Outside this optimal pH range, the buffer capacity declines. Understanding buffer capacity is crucial for many chemical and biological processes, ensuring the environment remains stable even when potentially disruptive substances are introduced.

Remember, buffer capacity depends on both the concentration of the buffer components and the pH of the solution. Doubling the concentration of the buffer components will, in turn, roughly double the buffer capacity.

Acetic Acid Buffer

An acetic acid buffer is a mixture of acetic acid (CH3COOH) and its conjugate base, acetate (CH3COO-), in solution. This type of buffer is frequently used in biochemical experiments to maintain a stable pH environment.

Acetic acid is a weak acid, which means it doesn't fully dissociate in water. The presence of its conjugate base allows the buffer solution to neutralize added acids or bases. When a strong acid is added to the buffer, the acetate ions combine with the incoming protons to form more acetic acid, thus resisting a decrease in pH. Similarly, when a base is added, the acetic acid donates a proton to the base, forming more acetate and resisting an increase in pH.

The effectiveness of an acetic acid buffer is governed by the Henderson-Hasselbalch equation, which is a derived relation used to calculate the pH of buffer solutions. The pKa of acetic acid is 4.75, so the buffer capacity will be at its maximum when the pH is close to this value.

pH Calculation

pH calculation is vital to determine the acidity or basicity of a solution. The pH scale runs from 0 to 14, with 7 being neutral. Solutions with pH less than 7 are acidic, and those with pH greater than 7 are basic.

In the case of buffer systems, pH can be calculated using the Henderson-Hasselbalch equation: \( pH = pKa + \log \left( \frac{[A^-]}{[HA]} \right) \), where \( [A^-] \) is the concentration of the conjugate base, and \( [HA] \) is the concentration of the acid. This relationship becomes essential when working with buffered solutions, as it allows one to find the pH after the addition of an acid or base. When additional strong acid (or base) changes the concentrations of \( [A^-] \) and \( [HA] \) in the solution, the pH change can be predicted and often calculated more precisely, taking into account the buffer index \( \beta \) and the changes in concentration of acid/base added to the solution. Understanding the math behind pH calculations is paramount for students to predict the behavior of buffered solutions.

Strong Acid and Base Reactions

Strong acid and base reactions are characterized by their complete dissociation in water. This means that strong acids, like hydrochloric acid (HCl), and strong bases, like sodium hydroxide (NaOH), will fully ionize when dissolved. As a result, they can alter the pH of a solution more dramatically than weak acids or bases.

When adding a strong acid to a buffer, H+ ions are introduced and react with the conjugate base present in the buffer to minimize the pH change. Conversely, when adding a strong base, OH- ions react with the acid component of the buffer. The buffer solution's capacity to mitigate these pH changes without substantially altering its own pH is a reflection of both its buffer capacity and the concentration of buffer components.

Students must appreciate that the neutralization reactions in buffered solutions differ from those in non-buffered solutions. In a non-buffered solution, the pH would drastically change with the addition of strong acids or bases, while a buffered solution aims to maintain pH stability through these reactions, showcasing the importance of buffers in many biological and chemical systems.