## Q. 19.3

Preparing a Buffer

Problem An environmental chemist needs a carbonate buffer of pH 10.00 to study the effects of acid rain on limestone-rich soils. How many grams of $Na_2CO_3$ must she add to 1.5 L of 0.20 M $NaHCO_3$ to make the buffer ($K_a$ of $HCO_3^− = 4.7×10^{−11}$)?

## Verified Solution

Plan The conjugate pair is $HCO_3^−$ (acid) and $CO_3^{2−}$ (base), and we know the buffer volume (1.5 L) and the concentration (0.20 M) of $HCO_3^−$, so we need to find the buffer-component concentration ratio that gives a pH of 10.00 and the mass of $Na_2CO_3$ to dissolve. We convert $K_a$ to $pK_a$ and use Equation 19.1 to find the ratio $[CO_3^{2−}]/[HCO_3^−]$ that gives a pH of 10.00. Multiplying the given molarity of $HCO_3^−$ by the volume of solution gives the amount (mol) of $HCO_3^−$ and the ratio of $[CO_3^{2−}]/[HCO_3^−]$ gives the amount (mol) of $CO_3^{2−}$ needed, which we convert to mass (g) of $Na_2CO_3$.

$pH = pK_a + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right)$        (19.1)

Solution Calculating $pK_a$:
$pK_a = −\log K_a = −\log (4.7×10^{−11}) = 10.33$
Solving for $[CO_3^{2−}]/[HCO_3^−]$:

$pH = pK_a + \log \left(\frac{[CO_3^{2−}]}{[HCO_3^−]}\right) 10.00 = 10.33 + \log \left(\frac{[CO_3^{2−}]}{[HCO_3^−]}\right)$

$−0.33 = \log \left( \frac{[CO_3^{2−}] }{[HCO_3^−]}\right) \text{and} \left( \frac{[CO_3^{2−}]}{[HCO_3^−]}\right) = 10^{−0.33} = 0.47$

Calculating the amount (mol) of $CO_3^{2−}$ needed for the given volume of solution:

$\text{Amount (mol) of }HCO_3^− = 1.5 \text{L soln }× \frac{0.20\text{ mol }HCO_3^−}{1.0 \text{L soln}} = 0.30\text{ mol }HCO_3^−$
$\text{Amount (mol) of }CO_3^{2−} = 0.30\text{ mol }HCO_3^− × \frac{0.47\text{ mol }CO_3^{2−}}{1.0\text{ mol }HCO_3^−} = 0.14\text{ mol }CO_3^{2−}$
Calculating the mass (g) of $Na_2CO_3$ needed:

$\text{Mass (g) of }Na_2CO_3 = 0.14\text{ mol }Na_2CO_3 × \frac{105.99 g Na_2CO_3}{1\text{ mol }Na_2CO_3} = 15 g Na_2CO_3$

The buffer is prepared by dissolving 15 g of $Na_2CO_3$ into about 1.3 L of 0.20 M $NaHCO_3$ and adding more 0.20 M $NaHCO_3$ to make 1.5 L. Then a pH meter is used to adjust the pH to 10.00 by dropwise addition of concentrated strong acid or base.

Check For a useful buffer range, the concentration of the acidic component, $[HCO_3^−]$ in this case, must be within a factor of 10 of the concentration of the basic component, $[CO_3^{2−}]$. And we have 0.30 mol of $HCO_3^−$, and 0.14 mol of $CO_3^{2−}$; 0.30/0.14 = 2.1.
Make sure the relative amounts of components are reasonable: we want a pH below the $pK_a$ of $HCO_3^−$ (10.33), so we want more of the acidic than the basic species.