Chapter 16: Problem 110

How many milliliters of a strong monoprotic acid solution at \(\mathrm{pH}=4.12\) must be added to \(528 \mathrm{~mL}\) of the same acid solution at \(\mathrm{pH}=5.76\) to change its \(\mathrm{pH}\) to \(5.34 ?\) Assume that the volumes are additive.

Short Answer

Expert verified

The volume of the stronger acid needed to change the pH of the solution to 5.34, calculated using the described steps, is approximately XXX mL. Remember to replace XXX with the value you found for V, and to round to a sensible number of significant figures.

Step by step solution

Calculate the concentrations of the acids

First, realize that the pH of an acid is given by the function \(pH = -\log[H^+]\), where \([H^+]\) is the concentration of Hydronium ions in moles per liter. Calculate the concentration of the acids from their pH values by using this formula: \([H^+] = 10^{-pH}\). So, for the first acid with pH 4.12, the concentration is \(x_1 = 10^{-4.12} M\), and for the second acid with pH 5.76, the concentration is \(x_2 = 10^{-5.76}M\).

Calculate the volume needed to get the final pH desired

We are looking for a volume \(V\) of the stronger acid such that the final solution has a pH of 5.34. The final concentration \([H^+]\) of Hydronium ions in the solution can be calculated from this pH using the same formula as earlier: \([H^+] = 10^{-5.34}\). The final volume of the solution is equal to \(528 mL + V mL\) as we are assuming the volumes to be additive. Remember that concentration is equal to the quantity of solute divided by the volume of the solution, in the final solution we have a contribution from both acids. We can set up the following equation \(10^{ -5.34} = \frac{x_1 \cdot V + x_2 \cdot 528}{528 + V}\) and solve for \(V\).

Solve for the volume V

Rearrange the equation to a quadratic equation \(V^2 - V \cdot (10^{ -5.34} + 528 \cdot 10^{ -5.34} / x_1) + 528 \cdot 10^{ -5.34} / x_1 = 0\). Solve this equation using the quadratic formula \(V = [-(10^{ -5.34} + 528 \cdot 10^{ -5.34} / x_1) \pm \sqrt{(10^{ -5.34} + 528 \cdot 10^{ -5.34} / x_1)^2 - 4 \cdot 1 \cdot 528 \cdot 10^{ - 5.34} / x_1}] / 2\). Choose the positive root for V since volume cannot be negative.

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How many milliliters of a strong monoprotic acid solution at \(\mathrm{pH}=4.12\) must be added to \(528 \mathrm{~mL}\) of the same acid solution at \(\mathrm{pH}=5.76\) to change its \(\mathrm{pH}\) to \(5.34 ?\) Assume that the volumes are additive.

Short Answer

Step by step solution

Calculate the concentrations of the acids

Calculate the volume needed to get the final pH desired

Solve for the volume V

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