To find the pH of a 0.415 M solution of disodium citrate (Na[tex]\(_2\)[/tex]C[tex]\(_6\)[/tex]H[tex]\(_6\)[/tex]O[tex]\(_7\)[/tex]), we can utilize the properties of citric acid (C[tex]\(_6\)[/tex]H[tex]\(_8\)[/tex]O[tex]\(_7\)[/tex]) and its dissociation constants. Citric acid has three dissociation constants, represented as [tex]\( K_{a1} \)[/tex], [tex]\( K_{a2} \)[/tex], and [tex]\( K_{a3} \)[/tex].
Given:
[tex]\( K_{a1} = 7.44 \times 10^{-4} \)[/tex]
[tex]\( K_{a2} = 1.73 \times 10^{-5} \)[/tex]
[tex]\( K_{a3} = 4.02 \times 10^{-7} \)[/tex]
Concentration of disodium citrate, [tex]\( [ \text{Na}_2\text{C}_6\text{H}_6\text{O}_7 ] = 0.415 \, M \)[/tex]
### Step-by-Step Solution:
1. Identify the Predominant Species:
Disodium citrate (Na[tex]\(_2\)[/tex]C[tex]\(_6\)[/tex]H[tex]\(_6\)[/tex]O[tex]\(_7\)[/tex]) will dissociate in water, primarily forming H[tex]\(_2\)[/tex]C[tex]\(_6\)[/tex]H[tex]\(_5\)[/tex]O[tex]\(_7^-\)[/tex] ions. The most significant [tex]$\text{H}^+$[/tex] ion contribution will come from the first dissociation constant ([tex]\( K_{a1} \)[/tex]).
2. Approximation of Dominant Equilibrium:
For this predominant species H[tex]\(_2\)[/tex]C[tex]\(_6\)[/tex]H[tex]\(_5\)[/tex]O[tex]\(_7^-\)[/tex], we can use the approximation for the dominant equilibrium involving [tex]\( K_{a1} \)[/tex]:
[tex]\[
\text{H}_2\text{C}_6\text{H}_5\text{O}_7^- \leftrightarrow \text{H}^+ + \text{HC}_6\text{H}_5\text{O}_7^{2-}
\][/tex]
Using the formula for the concentration of hydrogen ions [tex]\( [\text{H}^+] \)[/tex]:
[tex]\[
[\text{H}^+] = \sqrt{K_{a1} \cdot [\text{Na}_2\text{C}_6\text{H}_6\text{O}_7 ]}
\][/tex]
Substituting the given values:
[tex]\[
[\text{H}^+] = \sqrt{7.44 \times 10^{-4} \cdot 0.415}
\][/tex]
Computing the result:
[tex]\[
[\text{H}^+] \approx 0.01757 \, M
\][/tex]
3. Calculate the pH:
The pH of the solution can be calculated using the formula:
[tex]\[
\text{pH} = -\log_{10}[\text{H}^+]
\][/tex]
Substituting the concentration of hydrogen ions:
[tex]\[
\text{pH} = -\log_{10}(0.01757)
\][/tex]
Computing the result:
[tex]\[
\text{pH} \approx 1.755
\][/tex]
### Final Results:
- The concentration of disodium citrate: [tex]\(0.415 \, M\)[/tex]
- The concentration of hydrogen ions: [tex]\(0.01757 \, M\)[/tex]
- The pH of the solution: [tex]\(1.755\)[/tex]
Thus, the pH of the 0.415 M solution of disodium citrate is approximately [tex]\(1.755\)[/tex].
