MathBench > Measurement

Logs and pH

Measuring Acidity: the pH scale

lemon slicesAny time that a measurement can vary over many orders of magnitude, that's a candidate for using a log scale.

For example, the proper function of biological systems depends on the correct concentration of hydrogen ions (H+, or more correctly, H3O+) in an aqueous solution. This concentration can be as high as 1 M (extremely acidic), or as low as 10-14 M (extremely basic). Instead of counting out the zeros every time, we use a log scale, invented by Soren Sorenson:

extremely acidic : 1M = 100 M, so.... pH = 0

extremely basic : 10-14 M, so.... pH = 14

Notice that Sorenson chose to drop the minus sign, so our extreme base has a pH of 14, not -14. This is nice because we avoid negative numbers, but a bit tricky, since it means that the BIGGER the pH, the LOWER the concentration of H+.

The end result of all of this is that neutral solutions have a pH of 7, while a pH below 7 is an acidic solution and a pH above 7 is a basic solution.*

*Fine Print: Although pH technically measures just H+, what's really going on is an equilibrium between H2O, H+ and OH-. This is because water is a weak acid that dissociates into H+ and OH-. The equilibrium constant (K) for this dissociation is:

K = [H+] [OH-] / [H2O] = 1.8 × 10-16 M at 25°C

We can simplify this by noting that there is a large excess concentration of water with a constant molarity of 55.56, so

K [H2O] = [H+] [OH-] = 1.8 × 10-16 × 55.56 = 1 × 10-14 M2,

or simply

K= [H+] [OH-] = 10-14 M2.

When a solution is neutral, [H+] = [OH-], and therefore both [H+] and [OH-] must be 10-7 M, and pH must be 7.

 

How does the concentration of H+ change as the pH changes? Let's think of pH as a series of steps:

if pH = 7, then [H+] = 10-7M = 0.0000001 M

if pH = 6, then [H+] = 10-6M = 0.000001 M

if pH = 5, then [H+] = 10-5M = 0.00001 M

if pH = 4, then [H+] = 10-4M = 0.0001 M

if pH = 3, then [H+] = 10-3M = 0.001 M

if pH = 2, then [H+] = 10-2M = 0.01 M

if pH = 1, then [H+] = 10-1M = 0.1 M

if pH = 0, then [H+] = 100M = 1 M

Every time we go one "step" on the pH scale, we increase (or decrease) the concentration of hydrogen ions by a factor of 10. This is an important characteristic of using logs:

adding 1 to the log, or subtracting 1 from the log, changes the original unlogged number by a factor of 10 (multiplies or divides by 10).

If you increased the pH of a solution from 4.5 to 5.5, what would happen to the concentration of H+?

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I think I have the answer: 10 times LESS

If you decreased the pH of a solution from 4.5 to 2.3, what would happen to the concentration of H+?

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I think I have the answer: 10^2.1 = 125.9 times MORE