Question

Tris (hydroxymethyl)aminomethane, commonly called TRIS or Trizma, is often used as a buffer in biochemical studies. Its buffering range is \(\mathrm{pH} 7\) to \(9,\) and \(K_{\mathrm{b}}\) is \(1.19 \times 10^{-6}\) for the aqueous reaction $$ \left(\mathrm{HOCH}_{2}\right)_{3} \mathrm{CNH}_{2}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\left(\mathrm{HOCH}_{2}\right)_{3} \mathrm{CNH}_{3}^{+}+\mathrm{OH}^{-} $$ a. What is the optimal pH for TRIS buffers? b. Calculate the ratio \([T R I S] /\left[T R I S H^{+}\right]\) at \(p H=7.00\) and at p H=9.00 c. A buffer is prepared by diluting 50.0 \(\mathrm{g}\) TRIS base and 65.0 \(\mathrm{g}\) TRIS hydrochloride (written as TRISHCl) to a total volume of 2.0 \(\mathrm{L}\) What is the pH of this buffer? What is the pH after 0.50 \(\mathrm{mL}\) of 12 \(\mathrm{MHCl}\) is added to a 200.0 -mL portion of the buffer?

Short answer

The optimal pH for TRIS buffers is found to be 8.07. The ratio \([TRIS]/[TRIS H^{+}]\) at pH 7.00 is 0.135 and at pH 9.00 is 7.40. The pH of the prepared buffer is 7.95, and after adding 0.50 mL of 12 M HCl to a 200.0 mL portion of the buffer, the pH becomes 7.91.

Step-by-step solution

To find the optimal pH for TRIS buffers, we can refer to the optimal buffering range of pH 7 to 9. The optimal pH would be the pH at which the buffer capacity is the highest. This is when the concentration of the weak base equals the concentration of its conjugate acid, which occurs at the pKa (pKb in this case) value. We can find the pKb from the given Kb value and then convert it to pKa using the relationship pKa + pKb = 14. First, find pKb: $$ pK_b = -\log_{10}(K_b) $$ Then, find pKa: $$ pK_a = 14 - pK_b $$ Finally, find the optimal pH, which is equal to pKa. #Step 2#: Calculate the ratio \([T R I S] /[TRIS H^{+}]\) at pH 7.00 and pH 9.00

To calculate the ratio of concentrations, we will use the Henderson-Hasselbalch equation: $$ pH = pK_a + \log_{10}\frac{[TRIS]}{[TRIS H^{+}]} $$ Rearrange the equation to find the ratio \(\frac{[TRIS]}{[TRIS H^{+}]}\) at the given pH values. #Step 3#: Find the pH of the prepared buffer

To find the pH of the buffer, first determine the molar concentrations of both TRIS and TRISH+ in the buffer. 1. Convert the masses of TRIS base (50.0 g) and TRISHCl (65.0 g) to moles 2. Use the total volume (2.0 L) to calculate the molar concentrations of TRIS and TRISH+ 3. Use the Henderson-Hasselbalch equation to find the pH of the buffer. #Step 4#: Calculate the pH after adding HCl

We know that 0.50 mL of 12 M HCl will react with the basic part of the buffer. To find the pH after adding the HCl, we need to: 1. Calculate moles of HCl that we added 2. Calculate the moles of TRIS base and TRISH+ after the reaction with HCl 3. Calculate the new concentrations of TRIS and TRISH+ in the 200.0 mL portion of the buffer 4. Use the Henderson-Hasselbalch equation with the new concentrations to find the pH of the buffer after adding HCl.

Key concepts

pH Calculation

Understanding the concept of pH is essential when working with buffer solutions like TRIS. The pH scale measures how acidic or basic a solution is, and it ranges from 0 to 14. A solution with a pH of 7 is neutral, while pH values below 7 indicate acidity and values above 7 indicate basicity. To calculate the pH of a solution, we use the formula:\[pH = -\log_{10}[H^+]\]The concentration of hydrogen ions \([H^+]\) determines the acidity of a solution. In the case of TRIS buffers, calculating pH involves understanding the balance between TRIS base and TRIS acid. This balance is critical for biochemical studies, where maintaining a stable pH is essential for reactions.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a formula used to relate the pH of a solution to the concentration of acid and its conjugate base, or base and its conjugate acid in the case of TRIS.For TRIS buffers, we use:\[pH = pK_a + \log_{10}\left(\frac{[TRIS]}{[TRIS H^+]}\right)\]This equation simplifies the process of calculating pH in buffer solutions by providing a straightforward relationship between the pH, the pKa, and the ratio of the concentrations of the buffer components. The Henderson-Hasselbalch equation is a powerful tool for buffer design, helping to maintain the desired pH even when small amounts of acids or bases are added.

Optimal pH

The optimal pH in a buffer solution is the pH at which the solution is most resistant to changes in acidity or basicity. For TRIS buffers, the optimal pH is closely related to the pKa of TRIS and occurs when the concentrations of the base and its conjugate acid are equal. This is when the buffering capacity is at its maximum.In practice, to find the optimal pH:

• Calculate the pKb from the known Kb value using \( pK_b = -\log_{10}(K_b) \).
• Convert pKb to pKa with the equation \( pK_a = 14 - pK_b \).
• The optimal pH is then approximately equal to pKa.

Finding the optimal pH is crucial for creating effective buffers in biochemical applications, ensuring they perform efficiently within their intended pH ranges.

Chemical Equilibrium

Chemical equilibrium is a state where the concentrations of reactants and products remain constant over time, indicating a balance in a reversible reaction. In buffer solutions like TRIS, equilibrium is pivotal in maintaining a stable pH.TRIS reacts with water to reach an equilibrium:\[\left(\mathrm{HOCH}_2\right)_3 \mathrm{CNH}_2 + \mathrm{H_2O} \rightleftharpoons \left(\mathrm{HOCH}_2\right)_3 \mathrm{CNH}_3^+ + \mathrm{OH}^-\]At equilibrium, the rate of formation of \((\mathrm{HOCH}_2)_3 \mathrm{CNH}_3^+\) and \(\mathrm{OH}^-\) matches their rate of consumption. The concept of chemical equilibrium helps us understand why buffers can effectively resist changes in pH when acids or bases are added, thus providing a stable environment for chemical reactions and processes.