To solve this problem, we'll use the solubility product constant (Ksp) expression for Tl(OH)3:
Ksp = [Tl^3+][OH^-]^3
We're given that the solution is buffered at pH 11.50, which means that the hydroxide ion concentration ([OH^-]) is known. At pH 11.50, the concentration of hydroxide ions can be calculated using the formula:
pOH = 14 - pH
pOH = 14 - 11.50 = 2.50
Now, convert pOH to [OH^-] using the formula:
[OH^-] = 10^(-pOH)
[OH^-] = 10^(-2.50) = 3.16 × 10^(-3) mol/L
Now we can use the Ksp expression to find the concentration of [Tl^3+]:
Ksp = [Tl^3+][OH^-]^3
2 × 10^(-44) = [Tl^3+][(3.16 × 10^(-3))^3]
Solve for [Tl^3+]:
[Tl^3+] = (2 × 10^(-44)) / (3.16 × 10^(-9))
[Tl^3+] ≈ 6.33 × 10^(-37) mol/L
Therefore, the correct answer is: 6.33 × 10^(-37) mol/L.