# $pH$ and $pOH$

## Definitions

Molarity of hydrogen ions $H_3O^+$, is generally expressed with powers of 10 and does not hold intuitively. That is why scientists introduced a definition that can easily be manipulated:

$pH$ $=$ $-log[H_3O^+]$ $pOH=-log[OH^-]$ where log is the decimal logarithm ($x$ $=$ $log\; y$ $\Leftrightarrow$ $y$ $=$ $10^x$ )

The ionic product of water gives: $[OH^-][H_3O^+]$ $=$ $10^{-14}$ $log([OH^-][H_3O^+])$ $=$ $-14$ (former definition) $log[OH^-]$ $+$ $log[H_3O^+]$ $=$ $-14$ $(log(xy)$ $=$ $log\;x$ $+$ $log\;y)$ $-log[OH^-]$ $+$ $(-log[H_3O^+])$ $=$ $14$

$pOH$ $+$ $pH$ $=$ $14$ In acid environment: $pH$ $\lt$ $7$, therefore $pOH$ $\gt$ $7$ In neutral environment: $pH=7$ and $pOH=7$ In basic environment: $pH\gt7$, therefore $pOH\lt7$

## Example

Coca-Cola has $pH=2,6$, therefore $[H_3O^+]$ $=$ $10^{-2,6}$ $=$ $2,5\cdot10^{-3}\frac{mol}{L}$ and $pOH$ $=$ $14-2,6$ $=$ $11,4$ therefore $[OH^-]$ $=$ $10^{-11,4}$ $=$ $4,0\cdot10^{-12}\frac{mol}{L}$ This is a highly acidic drink.