### Video Transcript

What is the pH value of 0.5 liters
of 0.2 molar KOH? A) Zero, B) between zero and seven,
C) seven, D) between seven and 10, or E) greater than 10.

pH is defined as the negative log
of the concentration of hydrogen ions in a solution. And we want to find the pH of a
solution containing KOH, which is potassium hydroxide. When potassium hydroxide dissolves
in water, it dissociates into K⁺ or potassium ions and OH⁻ or hydroxide ions. Since potassium hydroxide produces
OH⁻ ions and water, that makes it a base. With the information that’s given
in the problem, we would be able to find the amount of potassium hydroxide and
therefore the amount of hydroxide in the solution. But to find the pH, we need the
concentration of hydrogen ions, not the concentration of hydroxide ions.

In any aqueous solution, water
reacts with itself in a process known as autoionization or self-ionization to form
hydrogen ions and hydroxide ions. This is an equilibrium reaction
with the equilibrium expression equal to the concentration of hydrogen ions times
the concentration of hydroxide ions. The hydroxide ions from the
potassium hydroxide solution will participate in this equilibrium reaction. So we’ll be able to use this to
find the concentration of hydrogen ions and therefore find the pH of our
solution.

The equilibrium constant for this
expression Kw is equal to one times 10 to the minus 14 at 25 degrees Celsius. So the only thing we need to find
in order to solve for the concentration of hydrogen ions is the concentration of
hydroxide ions. Concentration is, of course,
defined as the amount of a substance in moles per liter of solution. According to our balanced chemical
equation, every one mole of potassium hydroxide dissociates to form one mole of
hydroxide ions.

Since the amount of potassium
hydroxide and the amount of hydroxide ions must be equal to each other and they’re
both dissolved in the same volume of solution, the concentration of potassium
hydroxide and the concentration of hydroxide ions must be equal to each other. So the concentration of hydroxide
ions in our solution is also 0.2 molar.

Now, we can solve for the
concentration of hydrogen ions in our solution by dividing both sides of the
equilibrium expression for the autoionization of water by the concentration of
hydroxide ions. So we can plug everything in. To make the math a little bit
easier to solve, let’s convert the concentration of hydroxide ions into scientific
notation, which would be two times 10 to the minus one.

Whenever you’re dividing exponents
that are the same base. In this case, both of our exponents
are a power of 10. You can make the math easier by
subtracting the value in the exponents. We can separate out the math of the
exponents and the other numbers. So one divided by two is one-half
or 0.5 and then 14 minus a minus one gives us negative 13. So the concentration of hydrogen
ions in the solution must be 0.5 times 10 to the minus 13.

Now, we can find the pH by taking
the negative log of the concentration that we just found. This math gets a little bit tricky
to do in our heads. But luckily, most of the answer
choices are ranges of pHs. So we might be able to approximate
here. So instead of finding the pH for a
concentration of 0.5 times 10 to the minus 13, let’s just find the pH for a
concentration of one times 10 to the minus 13 or just 10 to the minus 13.

If you see a log whose base is
unspecified, you can assume that it’s log base 10. Since logs have the ability to undo
exponents, the log base 10 of something raised to a power of 10 will just be the
value in the exponent. So the log of 10 to the minus 13 is
just minus 13. So our pH is approximately equal to
13. Since this pH is greater than 10,
this matches answer choice E.

So the pH of our potassium
hydroxide solution is greater than 10.