# What is difference between a pH of 8 and a pH of 12 in terms of H+ concentration?

Jul 19, 2016

Here's what I got.

#### Explanation:

The pH of a solution is simply a measure of the concentration of hydrogen ions, ${\text{H}}^{+}$, which you'll often see referred to as hydronium cations, ${\text{H"_3"O}}^{+}$.

More specifically, the pH of the solution is calculated using the negative log base $10$ of the concentration of the hydronium cations.

color(blue)(|bar(ul(color(white)(a/a)"pH" = - log(["H"_3"O"^(+)])color(white)(a/a)|)))

Now, we use the negative log base $10$ because the concentration of hydronium cations is usually significantly smaller than $1$.

As you know, every increase in the value of a log function corresponds to one order of magnitude. For example, you have

$\log \left(10\right) = 1$

$\log \left(10 \cdot 10\right) = \log \left(10\right) + \log \left(10\right) = 1 + 1 = 2$

$\log \left(10 \cdot {10}^{2}\right) = \log \left(10\right) + \log \left({10}^{2}\right) = 1 + 2 = 3$

and so on. In your case, the difference between a pH of $8$ and a pH of $12$ corresponds to a difference of four units of magnitude between the concentration of hydronium cations in the two solutions.

Keep in mind, however, that because you're dealing with numbers that are smaller than $1$, and thus with negative logs, that the solution with a higher pH will actually have a lower concentration of hydronium cations.

More specifically, you have

"pH"_1 = - log(["H"_3"O"^(+)]_1) = 8

This is equivalent to

["H"_3"O"^(+)]_1 = 10^(-"pH"_1) = 10^(-8)"M"

Similarly, you have

"pH"_2 = - log(["H"_3"O"^(+)]_2) = 12

This is equivalent to

["H"_3"O"^(+)]_2 = 10^(-"pH"_2) = 10^(-12)"M"

As you can see, the first solution has a concentration of hydronium cations that is

$\left({10}^{- 8} \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{M"))))/(10^(-12)color(red)(cancel(color(black)("M}}}}\right) = {10}^{\textcolor{red}{4}}$

times higher than the concentration of hydronium cations of the second solution. This corresponds to the fact that you have

${\text{pH"_2 - "pH}}_{1} = 12 - 8 = \textcolor{red}{4}$

Simply put, a solution that has a pH that is $\textcolor{red}{4}$ units lower than the pH of a second solution will have a concentration of hydronium cations that ${10}^{\textcolor{red}{4}}$ times higher than that of the second solution. 