To determine which weak acid would be best for preparing a buffer solution with a pH of 8.60, we need to consider the [tex]\(pK_a\)[/tex] values of the given acids. A buffer solution works best when the pH is close to the [tex]\(pK_a\)[/tex] of the acid. The [tex]\(pK_a\)[/tex] is related to the acid dissociation constant [tex]\(K_a\)[/tex] through the following formula:
[tex]\[
pK_a = -\log_{10}(K_a)
\][/tex]
Let's calculate the [tex]\(pK_a\)[/tex] values for each provided [tex]\(K_a\)[/tex]:
1. For [tex]\(K_a = 4.0 \times 10^{-7}\)[/tex]:
[tex]\[
pK_a = -\log_{10}(4.0 \times 10^{-7}) \approx 6.40
\][/tex]
2. For [tex]\(K_a = 2.6 \times 10^{-9}\)[/tex]:
[tex]\[
pK_a = -\log_{10}(2.6 \times 10^{-9}) \approx 8.59
\][/tex]
3. For [tex]\(K_a = 3.2 \times 10^{-5}\)[/tex]:
[tex]\[
pK_a = -\log_{10}(3.2 \times 10^{-5}) \approx 4.49
\][/tex]
4. For [tex]\(K_a = 5.0 \times 10^{-6}\)[/tex]:
[tex]\[
pK_a = -\log_{10}(5.0 \times 10^{-6}) \approx 5.30
\][/tex]
5. For [tex]\(K_a = 1.6 \times 10^{-10}\)[/tex]:
[tex]\[
pK_a = -\log_{10}(1.6 \times 10^{-10}) \approx 9.80
\][/tex]
6. For [tex]\(K_a = 0.00040\)[/tex]:
[tex]\[
pK_a = -\log_{10}(0.00040) \approx 3.40
\][/tex]
Next, we compare these [tex]\(pK_a\)[/tex] values to the target pH of 8.60. The best acid for the buffer solution will be the one with a [tex]\(pK_a\)[/tex] closest to 8.60.
Calculating the absolute differences:
[tex]\[
\text{Difference for } K_a = 4.0 \times 10^{-7}: |6.40 - 8.60| = 2.20
\][/tex]
[tex]\[
\text{Difference for } K_a = 2.6 \times 10^{-9}: |8.59 - 8.60| = 0.01
\][/tex]
[tex]\[
\text{Difference for } K_a = 3.2 \times 10^{-5}: |4.49 - 8.60| = 4.11
\][/tex]
[tex]\[
\text{Difference for } K_a = 5.0 \times 10^{-6}: |5.30 - 8.60| = 3.30
\][/tex]
[tex]\[
\text{Difference for } K_a = 1.6 \times 10^{-10}: |9.80 - 8.60| = 1.20
\][/tex]
[tex]\[
\text{Difference for } K_a = 0.00040: |3.40 - 8.60| = 5.20
\][/tex]
From these differences, [tex]\(K_a = 2.6 \times 10^{-9}\)[/tex] has the smallest difference of [tex]\(0.01\)[/tex], meaning its [tex]\(pK_a\)[/tex] (8.59) is closest to the desired pH of 8.60.
Therefore, the best weak acid to use when preparing a buffer solution with a pH of 8.60 is the acid with [tex]\(K_a = 2.6 \times 10^{-9}\)[/tex].
