Question

2 -amino- 2 -hydroxymethylpropane- 1,3 -diol, (commonly known as 'Tris') and imidazole are very commonly used as buffers in biochemistry and molecular biology laboratories. You make two buffer solutions: One is \(50 \mathrm{mM}\) Tris at \(\mathrm{pH} 7.0\), the other \(50 \mathrm{mM}\) imidazole/imidazolium at \(\mathrm{pH} 7.0\) For each solution, calculate the concentration of buffer molecules in the cationic protonation state.

Short answer

Answer: The concentrations of buffer molecules in the cationic protonation state for Tris and imidazole solutions are approximately 41.8 mM and 25 mM, respectively.

Step-by-step solution

Write the given information

We are given: - Total buffer concentration for both solutions: \(50\,\mathrm{mM}\) - \(\mathrm{pH}\) for both solutions: \(7.0\) - The pKa values for Tris and Imidazole are different.

Find the pKa values for Tris and Imidazole

Using a pKa table or your textbook, you can find the pKa values for Tris and imidazole as follows: - pKa of Tris: \(8.06\) - pKa of imidazole: \(7.00\)

Calculate the concentration of buffer molecules in the cationic protonation state for Tris solution

We'll use the Henderson-Hasselbalch equation: \(\mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log{\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}} \Rightarrow [\mathrm{HA}] = [\mathrm{A}^-] \times 10^{\mathrm{p}K_\mathrm{a} - \mathrm{pH}}\) For the Tris buffer: \([\mathrm{HA}]_\mathrm{Tris} = [\mathrm{A}^-] \times 10^{(8.06 - 7.00)}\) Since the total buffer concentration is \(50\,\mathrm{mM}\): \([\mathrm{HA}]_\mathrm{Tris} + [\mathrm{A}^-] = 50\,\mathrm{mM}\) We can substitute this into the previous equation: \([\mathrm{HA}]_\mathrm{Tris} = (50\,\mathrm{mM} - [\mathrm{HA}]_\mathrm{Tris}) \times 10^{(8.06 - 7.00)}\) This equation can be solved for \([\mathrm{HA}]_\mathrm{Tris}\) to give: \([\mathrm{HA}]_\mathrm{Tris} \approx 41.8\,\mathrm{mM}\)

Calculate the concentration of buffer molecules in the cationic protonation state for imidazole solution

Using the Henderson-Hasselbalch equation for the imidazole buffer: \([\mathrm{HA}]_\mathrm{imidazole} = [\mathrm{A}^-] \times 10^{(7.00 - 7.00)} = [\mathrm{A}^-]\) Since the total buffer concentration is \(50\,\mathrm{mM}\): \([\mathrm{HA}]_\mathrm{imidazole} + [\mathrm{A}^-] = 50\,\mathrm{mM}\) Thus, the concentration of buffer molecules in the cationic protonation state for the imidazole buffer is: \([\mathrm{HA}]_\mathrm{imidazole} = 50\,\mathrm{mM} - [\mathrm{A}^-] \approx 25\,\mathrm{mM}\) So, the concentrations of buffer molecules in the cationic protonation state for the Tris and imidazole solutions are approximately \(41.8\,\mathrm{mM}\) and \(25\,\mathrm{mM}\), respectively.

Key concepts

Henderson-Hasselbalch Equation

Understanding the Henderson-Hasselbalch equation is crucial for biochemistry students dealing with buffer systems. This equation helps to estimate the pH of a buffer solution based on the concentrations of the acid and its conjugate base. The equation is written as:

\[\begin{equation}\text{pH} = \text{p}K_{\text{a}} + \log{\frac{[\text{A}^-]}{[\text{HA}]}}\end{equation}\]

Where:\begin{itemize}\item \(\text{p}K_{\text{a}}\) is the acid dissociation constant, a measure of the strength of the acid.\item \([\text{A}^-]\) represents the concentration of the conjugate base.\item \([\text{HA}]\) represents the concentration of the acid.\end{itemize}
In the exercise, we applied this equation to calculate the concentration of buffer molecules in the protonated state for both Tris and imidazole buffers at pH 7.0.

pKa Values

pKa values are pivotal in biochemistry when dealing with acids and bases. They indicate the pH at which half of the molecules of a species are deprotonated. It's a logarithmic scale, and lower pKa values imply a stronger acid, capable of donating a proton more readily. When you know the pKa of a buffer, you can predict how the buffer will behave at a certain pH.

For example, in our exercise:

• The pKa value for Tris is 8.06.
• The pKa of imidazole is 7.00.

These values are critical for determining the protonation state of the buffering agents at the desired pH level.

Protonation State

The protonation state of a molecule in a buffer solution refers to whether it is in its protonated (acidic, HA) or deprotonated (basic, A-) form. This state is influenced by the pH of the solution and the pKa of the molecule. The protonation state matters because it determines the buffering capacity and the molecule's ability to interact with other compounds in the solution.

Using the Henderson-Hasselbalch equation and the pKa value, you can calculate the ratio of the protonated and deprotonated forms. This ratio is essential for buffer solutions used in biological experiments where precise pH control is necessary for enzyme activity and protein stability.

Tris Buffer

A Tris buffer, also known by its full name as Tris(hydroxymethyl)aminomethane, is widely used in biochemistry due to its effective pH range near physiological pH (7.4). It maintains a stable pH during experiments, which is vital for most biological processes. With a pKa around 8.06, Tris has an effective buffering range of pH 7.0 to 9.0, which covers most biological applications.

When preparing a Tris buffer at a specific pH, the ratio of Tris base to Tris-HCl is adjusted according to the pH and the buffer’s pKa using the Henderson-Hasselbalch equation. For pH 7.0, as in the exercise, the majority of Tris molecules will be in their protonated form.

Imidazole Buffer

Imidazole buffer is another important buffering agent in biochemical labs, particularly for enzyme reactions or protein purification procedures, as imidazole residues are part of histidine amino acids in proteins. The pKa of imidazole is around 7.00, making it particularly suitable for experiments requiring a pH close to neutral. Given that the pKa is equal to the desired pH in our exercise, the protonation state of imidazole at pH 7.0 will have equal amounts of protonated and deprotonated species, creating an effective buffering system at this pH.

In practice, adjusting the pH of an imidazole buffer to the desired level often involves adding either HCl or NaOH, while monitoring the pH until the correct balance is achieved, as per the Henderson-Hasselbalch equation.