### Calculate the K_{sp} of a Saturated Solution When Given the pH

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Please be aware that this problem does not require a concentration to be given in the problem. We know the pH and we know the solution is saturated. This information will be sufficient.

Warning: you need to know about K_{sp} AND acid base ideas to do this problem type. If you lack one or the other of these skills, just be aware you just might struggle a little in your understanding of this problem type.

To solve the problem, we must first calculate the [OH¯]. To do this, we will use the pH and acid base concepts. Then, we will use the K_{sp} expression to calculate the K_{sp}

Final Note: K_{sp} are almost always given at 25.0 °C in reference sources. All problems in this tutorial are taken to be at 25.0 °C. If you were to see a problem where the specified temperature was different, it's probably just that the reference source gave a K_{sp} that, for whatever reason, was not at 25.0 °C.

**Problem #1:** A saturated solution of Mg(OH)_{2} is prepared. The pH of the solution is 10.17. What is the K_{sp} for this compound?

**Solution:**

1) The chemical equation:

Mg(OH)_{2} <===> Mg^{2+} + 2OH¯

2) The K_{sp} expression:

K_{sp} = [Mg^{2+}] [OH¯]^{2}

3) Use the pH to get the pOH:

14.00 - 10.17 = 3.83

4) Use the pOH to get the [OH¯]:

[OH¯] = 10¯^{pOH} = 10¯^{3.83} = 1.479 x 10¯^{4} M

5) From the chemical equation, we note that the [Mg^{2+}] is half the value of the [OH¯], therefore:

[Mg^{2+}] = 1.479 x 10¯^{4} M divided by 2 = 7.395 x 10¯^{5} M

6) We now have the necessary values to put into the K_{sp} expression:

K_{sp} = (7.395 x 10¯^{5}) (1.479 x 10¯^{4})^{2}
K_{sp} = 1.62 x 10¯^{12}

Note: the book value for the K_{sp} of Mg(OH)_{2} is 5.61 x 10¯^{12}. The reason for the difference is that teachers like to change the values slightly so as to not allow a student to simply look up the correct answer and claim they did all their work on the calculator.

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