Question

Suppose you want to do a physiological experiment that calls for a pH 6.50 buffer. You find that the organism with which you are working is not sensitive to the weak acid \(\mathrm{H}_{2} \mathrm{A}\left(K_{a 1}=2 \times 10^{-2} ; K_{a 2}=5.0 \times 10^{-7}\right)\) or its sodium salts. You have available a 1.0 \(\mathrm{M}\) solution of this acid and a 1.0 \(\mathrm{M}\) solution of \(\mathrm{NaOH} .\) How much of the NaOH solution should be added to 1.0 \(\mathrm{L}\) of the acid to give a buffer at pH 6.50\(?\) (Ignore any volume change.)

Short answer

To create a buffer with the pH of 6.50, approximately 0.613 L of 1.0 M NaOH solution should be added to 1.0 L of the 1.0 M weak acid solution.

Step-by-step solution

(Step 1: Write down the Henderson-Hasselbalch equation)

The Henderson-Hasselbalch equation is given by: pH = pKa + log10(\(\dfrac{[A^{-}]}{[HA]}\)) where pH is the desired pH of the buffer, pKa is the logarithm of the acid dissociation constant (Ka), [A^-] is the concentration of the base (conjugate of the weak acid), and [HA] is the concentration of the weak acid.

(Step 2: Find the appropriate pKa value)

We are given two dissociation constants for the weak acid \(\mathrm{H}_{2} \mathrm{A}\). We need to choose the one that will be suitable for preparing the buffer with the desired pH. The choice should be made based on the difference between the pH and pKa values. The pKa value closest to the desired pH should be chosen. The first dissociation constant is \(K_{a_1} = 2 \times 10^{-2}\). We can find the pKa as follows: pKa1 = -log10(\(K_{a_1}\)) = -log10(2 * 10^-2) ≈ 1.7 The second dissociation constant is \(K_{a_2} = 5.0 \times 10^{-7}\). We can find the pKa as follows: pKa2 = -log10(\(K_{a_2}\)) = -log10(5 * 10^-7) ≈ 6.3 Since pKa2 (6.3) is closer to the desired pH (6.50) than pKa1 (1.7), we choose pKa2.

(Step 3: Set up and solve the equation)

Now that we have chosen the appropriate pKa value, we can use the Henderson-Hasselbalch equation to find the amount of NaOH needed: 6.50 = 6.3 + log10(\(\dfrac{[A^{-}]}{[HA]}\)) 0.2 = log10(\(\dfrac{[A^{-}]}{[HA]}\)) Now, we must find the ratio of the base to the acid: 10^0.2 = \(\dfrac{[A^{-}]}{[HA]}\) 1.585 = \(\dfrac{[A^{-}]}{[HA]}\) Since initially, we have 1 L of 1.0 M weak acid solution, the initial moles of [HA] can be calculated as follows: Initial moles of [HA] = 1.0 L * 1.0 M = 1.0 mol Let x be the moles of NaOH added to the acid solution. When x moles of NaOH are added, x moles of [HA] will be converted into [A^-]: [HA] = 1.0 - x [A^-] = x Now we can plug these values into our ratio equation: 1.585 = \(\dfrac{x}{1.0 - x}\) Now solve for x: x = 1.585 * (1.0 - x) x = 1.585 - 1.585 * x 2.585 * x = 1.585 x ≈ 0.613 mol

(Step 4: Calculate the volume of NaOH solution needed)

Since the concentration of NaOH solution is 1.0 M, we can use the formula: Moles of NaOH = Concentration of NaOH * Volume of NaOH 0.613 mol = 1.0 M * Volume of NaOH Volume of NaOH ≈ 0.613 L Therefore, to create a buffer with the pH of 6.50, we need to add approximately 0.613 L of 1.0 M NaOH solution to 1.0 L of the 1.0 M weak acid solution.

Key concepts

Buffer Solutions

A buffer solution is a special mixture that helps maintain a stable pH, even when small amounts of acid or base are added. This is particularly important in many biological and chemical processes where a consistent pH is crucial.
Buffers are usually made from a weak acid and its conjugate base, or a weak base and its conjugate acid. Here’s how they work:

• When an acid is added, the base component of the buffer neutralizes it, preventing significant pH change.
• When a base is added, the acid component of the buffer neutralizes it, keeping the pH steady.

By choosing the right components, a buffer can be tailored to maintain a specific pH range, making it versatile for various applications.

Acid Dissociation Constant

The acid dissociation constant, represented as \(K_a\), helps us understand how strong an acid is. It measures how easily an acid releases protons (\(H^+\)) in a solution. A smaller \(K_a\) value means a weaker acid, as it dissociates less in water.
Each weak acid has its unique \(K_a\) value. Here's why it matters:

• It determines how much of the acid is present in its ionized form (\(H^+\) and conjugate base) versus its non-ionized form (\(HA\)).
• A higher \(K_a\) value indicates a stronger weak acid because it dissociates more.

In buffers, knowing the \(K_a\) value helps us predict how well the buffer will maintain the pH, especially which pKa to use when making calculations. This plays a crucial part when using the Henderson-Hasselbalch equation.

pKa Calculation

The pKa is the negative logarithm of the acid dissociation constant, \(K_a\). It’s a more convenient way to express the acidity of a substance. The equation for calculating pKa is:

\[\text{pKa} = -\log_{10}(K_a)\]

Choosing the correct pKa is key when preparing buffer solutions. The smaller the difference between the pH and the pKa, the more effective the buffer will be:

• pKa gives insight into which form of the acid (ionized or non-ionized) is more prevalent at a given pH.
• A buffer's capacity to resist changes in pH is greatest at its pKa, because there are equal concentrations of the acid and its conjugate base.

Understanding and calculating pKa enables the effective use of the Henderson-Hasselbalch equation in buffer preparation.