Gary L. Bertrand, Professor Emeritus
Department of Chemistry
Missouri University of Science and Technology
Discussion of Logarithms






Common logarithms are very useful in representing very big numbers or very small numbers.  In Chemistry and in Biological Sciences, a major application is in representing very low concentrations of hydrogen ion (H+) or hydroxide ion (OH-) as pH or pOH.

The precise definition of pH is "the negative common logarithm of the activity of hydrogen ion in solution".  For practical purposes, the activity is approximated as concentration in moles/L:
pH = - log 10 ([H+]) .
The lower case letter p before  upper case letters (X), such as H, OH, or K stands for
 "- log 10 (X)".
pOH = - log 10 ([OH-]) ,
pKa = - log 10 (Ka) ,
pMg = - log 10 ([Mg2+]) .

The concentrations of hydrogen ion and hydroxide ion are related through the ion-product of water (Kw) .  This results in a fixed relationship between pH and pOH.

Kw = [H+][OH-]
- log 10 (Kw) = - log 10 ([H+])  - log 10 ([OH-])
pKw = pH + pOH  .

A solution is neutral when the concentrations of hydrogen ion and hydroxide ion are the same, acidic when the concentration of hydfrogen ion is greater than that of hydroxide ion (pH is less than pOH), and basic when the concentration of hydroxide ion is greater than that of hydrogen ion (pH is greater than pOH).    At 24 oC, the ion product of water is 1.00 x 10-14 (pKw = 14.000), and a neutral solution has pH = 7.00 and pOH = 7.00.  This relationship changes slightly with temperature, as is shown in the table below.

 t oC  
pKw neutral pH
0 1.14 x 10-15 14.943 7.47
10 2.92 x 10-15 14.535 7.27
20 6.81 x 10-15 14.167 7.08
25 1.01 x 10-14 13.996 7.00
30 1.47 x 10-14 13.833 6.92
40 2.92 x 10-14 13.535 6.77
50 5.47 x 10-14 13.262 6.63
60 9.62 x 10-14 13.017 6.51
As the table shows, the pH of a neutral aqueous solution depends slightly on the temperature because the autoprotolysis constant of water (Kw) increases with temperature.  The pH may also depend on materials dissolved in water.  In popular (as opposed to technical) useage, however, pH = 7 is generally taken as an indication of neutrality in any aqueous solution at any temperature.