Question

A very common buffer agent used in the study of biochemical processes is the weak base TRIS, \(\left(\mathrm{HOCH}_{2}\right)_{3} \mathrm{CNH}_{2},\) which has a \(\mathrm{pK}_{\mathrm{b}}\) of 5.91 at \(25^{\circ} \mathrm{C} . \mathrm{A}\) student is given a sample of the hydrochloride of TRIS together with standard solutions of \(10 \mathrm{M}\) NaOH and HCl. (a) Using TRIS, how might the student prepare 1 L of a buffer of \(\mathrm{pH}=7.79 ?\) (b) In one experiment, 30 mmol of protons are released into \(500 \mathrm{mL}\) of the buffer prepared in part (a). Is the capacity of the buffer sufficient? What is the resulting pH? (c) Another student accidentally adds \(20 \mathrm{mL}\) of \(10 \mathrm{M}\) HCl to 500 mL of the buffer solution prepared in part (a). Is the buffer ruined? If so, how could the buffer be regenerated?

Short answer

a) The buffer is prepared using a mix of TRIS and TRIS HCl in the ratio 0.631:1. b) The capacity of the buffer is sufficient and the resulting pH is calculated using the Henderson-Hasselbalch equation while considering the added protons. c) The buffer is ruined due to excess HCl, but could be regenerated by neutralizing the added HCl with an equivalent amount of NaOH. After this, pH should be checked and adjusted as needed.

Step-by-step solution

Preparation of the buffer

The Henderson-Hasselbalch equation should be used to calculate the ratio of TRIS/base to TRIS HCl/acid in the solution. This equation is \(pH = pKa + log ([base] / [acid])\). Given that \(pKb = 5.91\) for TRIS, the \(pKa\) for its conjugate acid TRIS HCl is \(14 - pKb = 8.09\). We are asked to prepare a buffer with \(pH = 7.79\), so we have \(7.79 = 8.09 + log ([base] / [acid])\). Solving for \([base] / [acid]\) yields \(0.631\). Therefore, to prepare 1 L of buffer, a mix of TRIS and TRIS HCl that are in the ratio 0.631:1 is required.

Buffer capacity

The buffer's capacity is its ability to neutralize added acid or base and thereby resist changes in pH. Here, 30 mmol of protons (\(H^+\)) are released into 500 mL of the buffer prepared in Step 1. As 1 L of buffer can absorb 1 mole (\(H^+\)) for each pH unit decrease, the buffer is sufficient as \(30 mmol / 0.5 L = 60 mmol/L = 0.06 mole/L < 1 mole/L\). The pH is now calculated using the Henderson-Hasselbalch equation once again. Since we're adding protons, the amount of TRIS decreases by 30 mmol and the amount of TRIS HCl increases by 30 mmol. Applying the changes to the equation will provide the resulting pH.

Buffer Regeneration

When \(20 mL\) of \(10 M\) HCl (which equals \(200 mmol\)) is added to 500 mL of buffer, it not only lowers the pH by consuming the base TRIS but also exceeds the buffer's capacity. Hence, the buffer is ruined. To regenerate the buffer, the added amount of \(HCl\) could be neutralized by adding an equivalent amount of \(NaOH\). The final pH should be checked and adjustments made until the desired pH of 7.79 is reclaimed.

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a vital tool in the preparation of buffer solutions. It allows one to relate the pH of a solution to the pKa (acid dissociation constant) and the concentration ratio of the conjugate base and acid. Expressed mathematically, the equation is:
\[\begin{equation} pH = pKa + \log\left(\frac{[\text{base}]}{[\text{acid}]}\right) \end{equation}\]
When preparing a buffer, as in our exercise, the first step involves determining the desired pH and then using this equation to find the right proportions of buffer components. In the context of TRIS, a weak base with a pKa of 8.09 (after converting pKb to pKa), to achieve a target pH of 7.79, one would calculate the necessary ratio of TRIS (base) to TRIS HCl (acid).
Following the steps correctly is crucial because the accuracy of the ratio directly affects the effectiveness of the buffer in maintaining the desired pH. Students should note that the logarithm in the equation denotes the log base 10, and the concentrations should be in moles per liter.

Buffer capacity

Buffer capacity is a measure of a buffer solution's ability to maintain its pH when acids or bases are added. It is highly dependent on the concentrations of the buffer components. A higher concentration means a larger capacity to neutralize added substances without significantly changing the pH.
In our exercise, buffer capacity is tested by adding protons to the solution. The question is whether the buffer can absorb the introduced protons without a major shift in pH. The ability of a buffer to absorb 1 mole of protons per liter for each pH unit is a common benchmark. In the given scenario, only 30 mmol of protons are added to 500 mL of buffer, which is well within the capacity of our 1 L buffer solution.
It's imperative for students to understand that buffer capacity is not infinite; it will be exhausted when the amounts of added acid or base surpass what the buffer can handle. Thus, being aware of the buffer's capacity is critical during experimental design, to ensure buffer effectiveness throughout the procedure.

Acid-base neutralization

Acid-base neutralization is a chemical reaction where an acid and a base react to form water and a salt. This is central to the function of buffers as well as the regeneration of a buffer after its capacity is exceeded. In the case where the buffer solution faces added HCl, which is an excess of acid, neutralization involves adding a strong base to counteract the excess acid.
The neutralization process is crucial in buffer regeneration as it restores the pH to its intended level. When a buffer's capacity is overcome, as seen when excess HCl was added to the TRIS buffer, adding an equivalent amount of NaOH can neutralize the additional acid, thereby regenerating the buffer system. After this addition, recalculating the pH and adjusting the concentrations of buffer components becomes necessary to steer the pH back to the desired value. Students must consider stoichiometry during this process to ensure equivalent amounts of acid and base are used to achieve complete neutralization.