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 [TRIS]/[TTRISH \(\left.^{+}\right]\) at \(\mathrm{pH}=7.00\) and at \(\mathrm{pH}=9.00\) 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 L. What is the pH of this buffer? What is the \(\mathrm{pH}\) after \(0.50 \mathrm{mL}\) of \(12 \mathrm{M}\) HCl is added to a 200.0-mL portion of the buffer?

Short answer

The optimal pH for TRIS buffers is 8.08. The ratio [TRIS]/[TRISH+] is about 0.083 at pH 7.00 and about 8.3 at pH 9.00. The pH of the prepared buffer is about 8.08, and after adding 0.50mL of 12M HCl to a 200.0-mL portion of the buffer, the pH is about 8.04.

Step-by-step solution

a. Calculate the optimal pH for TRIS buffers

The optimal pH occurs when the buffering capacity is the greatest, which is when the pH is equal to the pKb of the base. To find the pH, we need to convert Kb to pKb, and then to pOH, and finally to pH using the relation: pKb = -log(Kb) pOH = pKb pH = 14 - pOH Given Kb value is 1.19 x 10^-6. First, let's find pKb: pKb = -log(1.19 x 10^-6) ≈ 5.92 Now we can find the pOH, which is equal to pKb in this case: pOH = 5.92 Finally, we can find the optimal pH: pH = 14 - pOH = 14 - 5.92 ≈ 8.08 So the optimal pH for TRIS buffers is about 8.08.

b. Calculate [TRIS]/[TRISH+] at pH = 7.00 and pH = 9.00

To find the desired ratio, first, we convert pKb to Ka (acid dissociation constant) using the relation: Ka = Kw / Kb, where Kw = 1.0 x 10^-14 (ionic product of water) We proceed to find pKa through: pKa = -log(Ka) Then we use the Henderson-Hasselbalch equation to find the ratio [TRIS]/[TRISH+]: pH = pKa + log([TRIS]/[TRISH+]) Let's first find Ka and then pKa: Ka = Kw / Kb = (1.0 x 10^-14) / (1.19 x 10^-6) ≈ 8.4 x 10^-9 pKa = -log(Ka) ≈ 8.08 For pH = 7.00: 7.00 = 8.08 + log ([TRIS]/[TRISH+]) Now solve for the ratio: log ([TRIS]/[TRISH+]) = 7.00 - 8.08 = -1.08 [TRIS]/[TRISH+] ≈ 0.083 Now, let's do the same for pH = 9.00: 9.00 = 8.08 + log([TRIS]/[TRISH+]) log([TRIS]/[TRISH+]) = 9.00 - 8.08 = 0.92 [TRIS]/[TRISH+] ≈ 8.3 Therefore, at pH 7.00, the ratio [TRIS]/[TRISH+] is about 0.083, and at pH 9.00, the ratio is about 8.3.

c. Determine the pH of the prepared buffer and after adding HCl

First, we need to find the moles of TRIS base and TRISHCl, and then divide them by the total volume to get the concentrations. The formula of TRIS base is (HOCH2)3CNH2, and its molar mass is 121.14 g/mol. The formula of TRISHCl is (HOCH2)3CNH3+Cl-, and its molar mass is 157.60 g/mol. Moles of TRIS base = (50.0 g) / (121.14 g/mol) ≈ 0.4125 mol Moles of TRISHCl = (65.0 g) / (157.60 g/mol) ≈ 0.4123 mol Now we find the concentrations by dividing the moles by the total volume (2.0 L): [TRIS] = 0.4125 mol / 2.0 L ≈ 0.2063 M [TRISH+] = 0.4123 mol / 2.0 L ≈ 0.2062 M We can use the Henderson-Hasselbalch equation to find the pH: pH = pKa + log([TRIS]/[TRISH+]) Using the values we calculated above: pH ≈ 8.08 + log(0.2063/0.2062) ≈ 8.08 The pH of the prepared buffer is about 8.08. Now we will find the pH after adding 0.50 mL of 12 M HCl to a 200.0-mL portion of the buffer. First, calculate the moles of H+ ions added: Moles of H+ = (12 M) * (0.50 mL) * (1 L / 1000 mL) = 6 x 10^-3 mol Now, let's find the new concentrations of TRIS and TRISH+ in the 200 mL buffer: [TRIS] = (0.4125 mol - 6 x 10^-3 mol) / (2.0 L) ≈ 0.2033 M [TRISH+] = (0.4123 mol + 6 x 10^-3 mol) / (2.0 L) ≈ 0.2092 M Now we can use the Henderson-Hasselbalch equation again to find the new pH: pH ≈ 8.08 + log(0.2033 / 0.2092) ≈ 8.04 So, after adding 0.50 mL of 12 M HCl to a 200.0 mL portion of the buffer, the pH is about 8.04.

Key concepts

pH calculation

Calculating the pH of a solution helps us understand its acidity or basicity. The pH scale ranges from 0 to 14, where 7 is neutral. Values below 7 indicate acidity, while values above 7 indicate basicity. To perform pH calculations, it is essential to understand the relationship between hydrogen ion concentration \([H^+]\) and pH: \[ pH = -\log([H^+]) \].

In the context of TRIS buffer solutions, calculating pH involves converting the base dissociation constant (\(K_b\)) to its corresponding pH. This starts with finding the pK\(_b\), using: \[ pK_b = -\log(K_b), \] then calculating the pOH (which is equal to pK\(_b\) in a buffer solution), and finally transforming it to pH by using: \[ pH = 14 - pOH. \]

This method shows how the concentrations of the components in a buffer solution determine its ability to resist changes in pH, not only during initial preparation but also when small amounts of acids or bases are added.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a fundamental tool for calculating pH in buffer solutions. It relates the pH to the pKa (or pKb for bases) and the ratio of the concentrations of the buffer components. For TRIS buffers, the equation is:

\[ pH = pKa + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right) \]

In our case of TRIS buffer, the concentration of TRIS corresponds to the concentration of the base, and TRISH\(^+\) to the acid component. Given pKa derived from \(pKa = -\log(K_a)\), the equation helps predict how the buffer will respond to changes, such as the addition of more TRIS base or TRISH\(^+\) chloride.

This is invaluable for experiments that require tight pH control as it sets the stage for determining how much buffering agent is necessary to reach a specific pH target. By using this equation, we can achieve precise pH adjustments in biochemical experiments. The Henderson-Hasselbalch equation is a practical guide for adjusting the component concentrations to maintain the desired pH.

buffering capacity

Buffering capacity refers to the ability of a buffer solution to resist changes in pH upon addition of small amounts of acid or base. This is crucial in systems requiring stable pH environments, such as biochemical experiments involving proteins and enzymes, which are sensitive to pH changes.

TRIS buffer, for instance, has optimal buffering capacity in the pH range of 7 to 9. This is because, within this range, the amounts of TRIS and TRISH\(^+\) are equal or close in concentration. A buffer exhibits maximum capacity when the concentrations of its acidic and basic forms are the same, typically when pH equals pKa.

When a buffer solution, like a TRIS buffer, is used, it moderates pH changes by converting added acids or bases into water and additional buffer components that stabilize the pH level. Thus, it extends the buffer's useful lifetime and effectiveness. Understanding buffering capacity is essential for preparing laboratory conditions that are conducive to studying sensitive biochemical reactions. It guides us in creating buffers that maintain a stable pH, minimizing disturbance in experimental protocols.