### Video Transcript

A solution at 25 degrees Celsius
has a hydroxide ion concentration of 1.26 times 10 to the negative nine moles per
cubic decimeter. What is the pH of the solution to
one decimal place?

The pH is defined as the negative
log of the concentration of hydronium ions. But we’ve been given a
concentration of hydroxide ions, not hydronium ions. However, these concentrations are
related to each other, as both hydroxide and hydronium participate in an equilibrium
reaction with water. This reaction is called the
autoionization of water. This is the equilibrium expression
for this reaction. The equilibrium constant for this
reaction 𝐾 𝑤 has a value of one times 10 to the negative 14 at 25 degrees
Celsius.

If we divide both sides of this
expression by the concentration of hydroxide ions, we’ll end up with an expression
that we can use to solve for the concentration of hydronium ions. We can plug in the value of 𝐾 𝑤
and the concentration of hydroxide ions. We leave off the units of the
hydroxide ion concentration when we plug it into this expression because 𝐾 𝑤 is a
unitless quantity. Solving gives us 7.937 times 10 to
the negative six for the concentration of hydronium, which is in units of moles per
cubic decimeter.

Now we can solve for the pH by
plugging the concentration of hydronium into the pH equation. pH is unitless, so we
still don’t need to worry about the units. Taking the negative log gives us
5.100, which the problem tells us to round to one decimal place. So a solution that has a hydroxide
ion concentration of 1.26 times 10 to the negative nine moles per cubic decimeter
has a pH of 5.1.

But there’s another way we can work
this problem. Let’s take another look at our
expression for 𝐾 𝑤. We can take the log of both sides
of this expression. Using the properties of logs, we
can separate the right side into two terms. This quantity is equal to the pH
times negative one. This quantity looks very similar to
the pH, except there’s hydroxide ions instead of hydronium ions. This quantity is called the
pOH. We can also plug in the value for
𝐾 𝑤 into our equation. Taking the log of one times 10 to
the negative 14 gives us negative 14. If we multiplied both sides of our
equation by negative one, we’ll arrive at a convenient equation that relates the pH
and the pOH.

We can subtract the pOH from both
sides of our equation to create an expression that we can use to solve for the pH,
which we can flip so that the pH is on the left-hand side of the equation. To find the pH, we first need to
solve for the pOH, which is equal to the negative log of the hydroxide ion
concentration. Plugging in the concentration given
in the problem gives us a pOH of about 8.9996. Now we can plug in the pOH, which
gives us a pH of about 5.1, which we can again round to one decimal place.

So either way we solve this
problem, the pH of a solution with a hydroxide ion concentration of 1.26 times 10 to
the negative nine moles per cubic decimeter is 5.1.