Stop this qualitative technobabble, it's time for quantitative calculations! pH is mathematically simple, if the activity coefficients are about 1, i.e. assuming an ideal-dilute solution. And yes, pH can be negative or over 14. The 'p' is shorthand for "power of", to denote very small concentrations; pH 2 is ten times diluted from pH 1, and so on. That is, pH is -1 times tenth base logarithm of the ionic activity of hydrogen ions: pH = -log10 aH+.
The ionic activity is the "effective concentration" of the protons, which is different from the stoichiometric concentration ("how much we added") since ions tend to cluster in solutions. This actually requires a fair bit of calculations for pH measurement in concentrated or salt-laden solutions, such as volcanic waters. In dilute solutions, we can assume that the activity is the same as the stoichiometric concentration. (In 0.1 to about 0.01 M we have to calculate the activity; see Debye-Hückel.)
So, we're left with this: pH = -log10 aH+. (The square brackets mean "the concentration of" in this case, in mol/l or moles per litre as always; this is essentially the count of ions per litre.)
Water dissociates even on when nothing else is present, although the extent of ionization is very small. Water dissociation is an equilibrium reaction, so we can measure an equilibrium constant, which is called water dissociation constant KW.
The reaction: H2O <-> H+ + OH-
K = (aH+ aOH-) / aH2O
The water is a solvent, so its activity
is 1, aH2O
= 1. One way of looking at this is that we could say that the water is of a different phase
- it's not aqueous ions like H+
but liquid - and when a reaction has one reactant in a different phase, this reactant is left out of the function for the position of the equilibrium. The environment in an dilute aqueous solution is completely saturated with water, which that means we can assume
O] is constant
. Dissolving small amounts of substances to water
will not affect the volume
considerably or change the number of water molecules
. Thus, [H2
O] is left out, and the constant is approximated such that activities are assumed to be the same as concentrations:
KW = [H+][OH-]
It can be measured that the constant has the value KW
at 298 K
(room temperature). This is an actual experimental value, not a definition. An implication is that pH is not a "scale" in the same way centigrade
is, for example; it is a dimensionless number
based on real physical quantities
Neutral pure water has a pH of 7, as you will learn in a chemistry class. Here is the calculation:
In pure water, [H+] = [OH-], so
[H+][OH-] = [H+]2 = KW
==> [H+] = (10-14)0.5 = 10-7
By definition, pH = -log10 10-7
==> pH = 7.
What happens to hydrochloric acid
an activity of 1 mol HCl to 1 dm3 water
HCl -> H+ + Cl- (dissociation)
1 mol H+ ==> pH = -log 1
pH = 0.
In this case, we ignore the original dissociation of water, because the acid gives ten million times more H+
ions than the water does. This assumption is valid for (not too dilute) solutions with only acid and water. Of course, when concentrations of ions from the acid and from the water are closer, i.e. at 10-7
of acid, you have to take the original dissociation into account. As you can see, if we put more than 1 mol
of active protons, we get a negative pH
. The result is not accurate for concentration, though; using the Debye-Hückel
formula gives the result that an activity of 1 mol/l corresponds to 2 mol/l stoichiometric concentration, although Debye-Hückel probably doesn't work at this high a concentration.
When we remove or add hydrogen ions to the solution, the reaction shifts, because (surprise surprise!) the dissociation constant stays constant. (See Le Chatelier's principle.) This way we can understand how the pH can be over 7. When we increase the [OH-] part in the reaction, the [H+] part has to decrease.
So when we put, for instance, NaOH into the solution, it dissociates into OH-, which in turn decreases the number of hydrogen ions in the water. As in adding H+, we can neglect the small amount of OH- from the water in the calculations and assume that all OH- comes from the base.
KW = [H+][OH-]
[H+] = KW/[OH-]
2 mol dm-3 of NaOH
[H+] = 10-14/(2 mol dm-3)
pH = 14.3
There you see how the pH is not always from 0 to 14. As you may have noticed, a handy shortcut
for the pH is to take the logarithm
of the whole KW expression
. Because the numerical value of KW
, its tenth-base logarithm is -14.
KW = [H+][OH-] (-log10 both sides)
14 = -log10 [H+] - log10 [OH-] (There's the pH!)
pH = 14 - log10 [OH-]
pH is simply fourteen minus the logarithm of the concentration
of the strong acid
. In this case, log 2 is about -0.3, so pH = 14 - (-0.3) = 14.3. (Notice that in this concentrated solutions, this analysis breaks down, because activity
no longer the same as the concentration. Use the Debye-Hückel theory
in these cases.) If you want to apply this to weak base
s, calculate the concentration of hydroxide
ions first. In general:
pH = 14 - pOH
Not cut-n-paste. No source would have come up with this level of clarity and possibly some highly creative errors.