K_sp ; increasing pH with given [OH]

[lizzyb]

1. State the Problem

A water initially contains 40 mg/L of Mg2+. The pH of the water is increased until the concentration of hydroxide ions (OH-) is 0.001000 M. What is the concentration of magnesium ion in this water at this pH? Give you answer in milligrams per liter. Assume that the temperature of the solution is 25oC.

2. Relevant Material

Mg(OH)2 (s) <---> Mg2+ + 2 OH- pKs = 11.25
Answer (given in book): 0.4423 mg / L

3. Work Done So Far

The initial molarity is 1.646e-3 M.
The given answer molarity is 18.198e-6 M.

         -11.25               2
K_sp = 10       = [Mg2+] [OH-]     (1)

We're given [OH-] = 0.001 M ; plugging this into (1) gives [Mg2+] = 5.623e-6 M.

let [Mg2+] = s so [OH-] = 2s

Then

         -11.25           2      3
K_sp = 10       = s * (2s)  = 4 s  (2)

so s = 1.12e-4

Finally, I tried setting up a similar equation as (2):

let [Mg2+] = 1.646e-3 - s, [OH-] = s, then

         -11.25                       2 
K_sp = 10      = (1.646e-3 - s) * (2s)     (3)

so s = 2.949e-5 (the final [Mg2+] = 1.646e-3 - s = 1.6165e-3 M

What am I not doing correctly?

 

[Borek]

We're given [OH-] = 0.001 M ; plugging this into (1) gives [Mg2+] = 5.623e-6 M.

At this moment you have a correct answer almost ready. You know the final concentration of Mg in the solution, that's enough to calculate number of moles and mass dissolved in 1L. Whatever you did later didn't make any sense to me.

It will not be identical with the key, no idea why.

Note that concentration of 18.198e-6M is below the precipitation limit:

-log(18.198e-6 x 1e-3²) = 10.74, while you have listed pK_sp of 11.25.

 

[Blueshift5]

4. Response

Based on your calculations, it seems that you have correctly used the given information and equations to solve for the concentration of magnesium ion at the given pH. However, it is possible that there may be a small error in your calculations or rounding, leading to a slightly different answer than the one provided in the book. It is also important to consider the precision of the given values and the accuracy of the equations used. I would suggest double checking your calculations and using more precise values if available. Additionally, it may be helpful to consult with a colleague or professor to compare your approach and calculations to ensure accuracy.