# A solution with a pH of 10 is 100 times more basic than a solution with pH 8. Why?

Nov 26, 2016

Because the $p H$ scale is logarithmic.

#### Explanation:

$p H = - {\log}_{10} \left[{H}_{3} {O}^{+}\right]$ by definition.

And thus, if $p H = 10$, $\left[{H}_{3} {O}^{+}\right] = {10}^{- 10} \cdot m o l \cdot {L}^{-} 1$, (i.e. $p H = - {\log}_{10} \left({10}^{- 10}\right) = - \left(- 10\right) = 10$

And if $p H = 8$, $\left[{H}_{3} {O}^{+}\right] = {10}^{- 8} \cdot m o l \cdot {L}^{-} 1$, which is one hundred times more concentrated than the first instance, as required. In other words, if the $\Delta p H = 2$, there is a ${10}^{2}$, i.e. one hundredfold difference in $\left[{H}_{3} {O}^{+}\right]$.

Do not be intimidated by the $\log$ function. When we write ${\log}_{a} b = c$, we ask to what power we raise the base $a$ to get $c$. Here, ${a}^{c} = b$. And thus ${\log}_{10} 10 = 1 ,$, ${\log}_{10} 100 = 2 ,$${\log}_{10} {10}^{- 1} = - 1$. And ${\log}_{10} 1 = 0$.

I acknowledge that I have hit you with a lot of facts. But back in the day A level students routinely used log tables before the advent of electronic calculators. If you can get your head round the logarithmic function you will get it.