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.