Problem 51 - Analytical Chemistry

Question

Tris(hydroxymethyl)aminomethane \(\left[\left(\mathrm{HOCH}_{2}\right)_{3} \mathrm{CNH}_{2}-\right.\) Tris, or THAM \(]\) is a weak base frequently used to prepare buffers in biochemistry. Its \(K_{b}\) is \(1.2 \times 10^{-6}\) and \(\mathrm{p} K_{b}\) is \(5.92 .\) The corresponding \(\mathrm{p} K_{a}\) is \(8.08,\) which is near the \(\mathrm{pH}\) of the physiological buffers, and so it exhibits good buffering capacity at physiological \(\mathrm{pH}\). What weight of THAM must be taken with \(100 \mathrm{~mL}\) of \(0.50 M \mathrm{HCl}\) to prepare \(1 \mathrm{~L}\) of a \(\mathrm{pH} 7.40\) buffer?

Step-by-Step Solution

Answer

1.27 grams of THAM is needed to prepare the buffer.

Step 1: Understanding the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation, applicable to buffer solutions, is given by:\[\text{pH} = \text{p}K_a + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)\]For this particular exercise, the target pH is 7.40, \(\text{p}K_a\) is 8.08, and the acid concentration is given by the amount of \(HCl\), which is 0.05 moles in 1 L (0.5 M in 100 mL). THAM will be used as the base.

Step 2: Calculate Base to Acid Ratio

Rearranging the Henderson-Hasselbalch equation to find the ratio \([\text{Base}]/[\text{Acid}]\):\[7.40 = 8.08 + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)\]Subtract 8.08 from both sides to get:\[-0.68 = \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)\]Take the antilog to find:\[\frac{[\text{Base}]}{[\text{Acid}]} = 10^{-0.68} \approx 0.21\]

Step 3: Determine Base Amount Required

We have determined that the ratio \(\frac{[\text{Base}]}{[\text{Acid}]} = 0.21\). Given that we have 0.05 moles of \(HCl\) (acid), the moles of THAM (base) required is:\[[\text{Base}] = 0.21 \times [\text{Acid}] = 0.21 \times 0.05 \text{ moles} = 0.0105 \text{ moles}\]

Step 4: Calculate Mass of THAM

The molar mass of THAM is approximately 121.14 g/mol. Using the calculated moles of THAM:\[\text{Mass of THAM} = 0.0105 \text{ moles} \times 121.14 \text{ g/mol} = 1.27 \text{ grams}\]

Step 5: Final Verification

Verify the calculations and confirm that using 1.27 grams of THAM will indeed yield a buffer solution with a pH of 7.40 when mixed with 0.50 M HCl to make a total volume of 1 L. Ensure all conversions and assumptions match the initial parameters of the buffer preparation problem.

Key Concepts

Henderson-Hasselbalch equation

Buffer solutions are crucial in maintaining stable pH levels in various biochemical processes. The Henderson-Hasselbalch equation provides a simple way to estimate the pH of a buffer solution. This equation is expressed as: \[\text{pH} = \text{p}K_a + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)\]This equation helps calculate the pH of a solution by relating the concentration ratio of the base to its corresponding acid and the acid's dissociation constant (\(\text{p}K_a\)).

• **pH**: The measure of acidity or alkalinity.
• **pK_a**: The negative logarithm of the acid dissociation constant, which indicates acid strength in a solution.
• **Base/Acid Concentrations**: Refers to the molar concentrations of the base and corresponding acid in the buffer.

In this exercise, the Henderson-Hasselbalch equation is used to regulate the pH to 7.40 by adjusting the [Base]/[Acid] ratio using THAM as the base.

weak base

A weak base is a substance that partially accepts protons in a solution, leading to an incomplete dissociation. THAM, or Tris(hydroxymethyl)aminomethane, used in this problem, is a classic example of a weak base. Weak bases are crucial for buffer solutions as they react with strong acids to moderate pH changes. Here’s why THAM is suitable:

• **Partial Proton Acceptance**: Unlike strong bases, weak bases like THAM do not fully dissociate. Instead, they reach an equilibrium in solution, providing a gradual and stable buffer effect.
• **pK_b Characteristics**: With a \(K_b\) of \(1.2 \times 10^{-6}\) and a \(\text{p}K_b\) of 5.92, THAM is suited for physiological buffer ranges. A \(\text{p}K_a\) of 8.08 fits closely with physiological pH.
• **Buffering Capacity**: Its characteristics allow THAM to absorb excess H+ ions, preventing significant pH changes, which is essential for maintaining enzyme activity and biochemical reactions.