# What is the maximum number of moles of #"Mg"("OH")_2#, that can remain dissolved in a 5.0 L solution whose pH is 9.70?

!! CORRECT SOLUTION !!

The value listed here is

Start by calculating the concentration of hydroxide anions. You know that for aqueous solutions at room temperature, you have

In your case, you will have

and

Plug in your value to find

Now, according to the expression you have for the solution's solubility product constant, the equilibrium concentration of magnesium cations will be equal to

Plug in your values to find

Use the volume of the solution to find the number of moles of magnesium cations

In your case, you will have

The answer is rounded to two sig figs.

!! INCORRECT SOLUTION !!

Start by calculating the concentration of hydroxide anions. You know that for aqueous solutions at room temperature, you have

In your case, you will have

and

Plug in your value to find

Magnesium hydroxide's dissociation equilibrium looks like this

Notice that every mole of magnesium hydroxide that dissociates in aqueous solution produces

This tells you that the equilibrium concentration of hydroxide anions will be twice as high as the equilibrium concentration of magnesium cations.

You can thus say that this solution will contain

Use the volume of the solution to calculate how many moles of magnesium cations would be present

In your case, you will have

The answer is rounded to two sig figs.

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To determine the maximum number of moles of Mg(OH)2 that can remain dissolved in a solution, we need to consider the solubility equilibrium of Mg(OH)2 and its dissociation into Mg2+ and OH- ions. The solubility equilibrium constant expression for Mg(OH)2 is Ksp = [Mg2+][OH-]^2. Given that the solution has a pH of 9.70, we can use the pH to find the concentration of hydroxide ions ([OH-]) in the solution using the relationship pH = -log([H+]) and the fact that [H+][OH-] = 1.0 x 10^-14 at 25°C. Once we have the concentration of hydroxide ions, we can use the Ksp expression to find the maximum concentration of Mg2+ ions, which corresponds to the maximum number of moles of Mg(OH)2 that can remain dissolved. Finally, we can convert the concentration of Mg2+ ions to moles using the volume of the solution.

However, it's important to note that the solubility of Mg(OH)2 can also depend on factors such as temperature and the presence of other ions that can form complexes with Mg2+ or OH-. Without specific information about these factors, the calculation may provide only an estimate of the maximum solubility.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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