PH of mixture; dilute solution

[Qube]

Homework Statement

You mix the below two solutions. What is the resulting pH?

200 mL solution with $H_{3}O^{+} = 1 * 10^{-4}$
100 mL solution with $H_{3}O^{+} = 2 * 10^{-11}$

Homework Equations

$pH = -log[H_{3}O^{+}]$

$N_{H_{3}O^{+}} = M * L$

The Attempt at a Solution

I first found the number of moles of hydronium ion on each of the two solutions by multiplying volume in liters by the molarity of the solution. For the first solution, the number of moles of hydronium ion is $2 * 10^{-5}$. The second solution has $2 * 10^{-12}$ moles of hydronium ion.

I added the number of moles of hydronium ion and divided by the volume of the mixture - 300 mL or 0.3 L. This yields the new hydronium ion molarity.

I then took the negative logarithm of the hydronium ion molarity and found the pH to be approximately 4.

However, the key says the pH is instead approximately 10, which I noticed to be 14 - 4 ($pK_{w} = 14 = pH + pHO$).

Why is the solution basic instead of acidic?

 

[epenguin]

Homework Statement

You mix the below two solutions. What is the resulting pH?

200 mL solution with $H_{3}O^{+} = 1 * 10^{-4}$
100 mL solution with $H_{3}O^{+} = 2 * 10^{-11}$

Homework Equations

$pH = -log[H_{3}O^{+}]$

$N_{H_{3}O^{+}} = M * L$

The Attempt at a Solution

I first found the number of moles of hydronium ion on each of the two solutions by multiplying volume in liters by the molarity of the solution. For the first solution, the number of moles of hydronium ion is $2 * 10^{-5}$. The second solution has $2 * 10^{-12}$ moles of hydronium ion.

I added the number of moles of hydronium ion and divided by the volume of the mixture - 300 mL or 0.3 L. This yields the new hydronium ion molarity.

I then took the negative logarithm of the hydronium ion molarity and found the pH to be approximately 4.

However, the key says the pH is instead approximately 10, which I noticed to be 14 - 4 ($pK_{w} = 14 = pH + pHO$).

Why is the solution basic instead of acidic?

You were drawn maybe by the wording into thinking oh that's just water with this and other water with that H₃O⁺ concentration. But water is water for which [H₃O⁺] is 10^-7. So what you've got in those solutions is dilute acid and dilute alkali. First think how acid one and how alkaline the other is and you will see which of the answers, yours or the official one, is reasonable.

Suppose the one is HCl and the other is NaOH and work out the pH. I get 10.

 

[Qube]

Hmm, so I forgot the auto ionization of water. But wouldn't that be included in the hydronium ion concentrations given? Or should I add 10^-7 for each solution?

 

[Borek]

Actually this problem is faulty and has no reasonable answer as worded. It doesn't say anything about the other compounds in the solution, which is a crucial thing.

I can construct for you example with solutions that will give after mixing ANY pH between 4 and 10.7. Name pH and I will tell you how to do it. Seriously.

 

[epenguin]

That's true, I just made the simplest assumption about what the solutions were, which turns out to give the stated right answer.
It should also have been stated they were 10^-4 M etc.

Quite often on this forum we have to guess what the questions really were.