I found this fascinating because it looks to me as if the 'simple' explanation is more complicated than the usual explanation. In thinking about it it seems that this might be because we define pH in the wrong way. I'll give a more appropriate definition and I think an explanation of buffering and buffering capacity (alkalinity) fall right out. Maybe I'm wrong and I'm just making it more complicated but if that's so just ignore this. Warning: there is math involved. The explanation for buffering is no more complicated than the properties of ratios of numbers. When dealing with ratios it is convenient to use logarithms as they represent fractional changes rather than absolute. For example a doubling (be it increase from 0.1 to 0.2 or 1 to 2 or 1000 to 2000) is always represented by 0.3 log units.
Consider a base, such as the bicarbonate ion: HCO3-. If we add a proton to it we get carbonic acid: HCO3- + H+ --> H2CO3.
Definition of pH:
pH is a constant plus the logarithm of the ratio of the concentration of unprotonated base to the concentration of protonated base in a solution:
pH = pK + log([HCO3-]/[H2CO3])
The constant depends on the base. For bicarbonate ion it is 6.38
If there are a fixed number of C containing molecules in a liter of solution and we add n H+ ions then the number of H2CO3 molecules will increase by n and the number of HCO3- ions will decrease by the same number. It is simple math that the change in the log of a ratio will be biggest when the original ratio is large or small and least when it is near 1.
To see this consider a state in which there are 9999 Republicans for every Democrat (feel free to reverse the names if the rest of the analogy offends). The ratio is 1/9999 and the log of that is -4. Suppose we have a pill which if given to a Republican causes him to lose his mind and join the Democratic party. We would then have a ratio of 2/9998. The ratio has almost doubled and thus the log will increase by 0.3 units to -3.7. Suppose we pass out 10 pills and get a ratio of 10/9990. The log of that is -3. Ten pills have changed the 'pH' by 1 unit (whatever the constant may be). Now suppose that we have passed out 5000 pills so that the ratio is 5000/5000 =1 (log(1) = 0) but we still need more votes. We got a whole unit change in pH from 10 pills at the outset. How about now? Giving out 10 more pills would give a ratio 5010/4990 and the log of that is 0.0017. Our last 10 pills only made a 'pH' change of 0.002 units. Thus it takes a lot more pills to effect a given log ratio change when the ratio is close to 1 than it does when it is large or small.
The math is exactly the same for the chemical situation and our 'pills' are hydrogen ions. If you plot log(n/N-n) vs. n where N is some large number and n is the number of hydrogen ions (pills) you will get a curve with the same shape as Kai's Fig 2 i.e. a titration curve.
Now lets look at N - the total number of people in the state or the total number of C containing molecules in the water. Let's divide the numerator and denominator in the ratio by this number thus the ratio is (n/N)/(1 - n/N). Thus the ratio and hence the pH does not depend on the number of ions (or people) but only on the fraction of them that are given a pill or are protonated or not. But the number of pills you must pass out to achieve a given fraction clearly does depend on the number of people/molecules. In a state with 10,000 people you need to pass out 5,000 pills to reach pH 0, in a state with 100,000 you need to pass out 50,000. In chemistry this is the distinction between pH and alkalinity. pH depends only on the ratio and the alkalinity (which is the number of hydrogen ions that must be added to reach a particular pH) depends on the number of C molecules.
[Edit] Looking at pH - pK = log([HCO3-]/[H2CO3]) it is clear that the ratio of bicarbonate to carbonic is 1 (and the resistance to change in pH when H+ is added maximized) when pH = pK (as log(1) = 0) and this is why a buffering system has maximum buffering capacity when the pH of the solution is close to the pK.
Simpler? I leave that to the reader to decide.
Let me finish up here (and I put this at the end on purpose) by justifying my definition of pH. It is a concept from more advanced chemistry than what we typically encounter in high school or freshman college courses (I offended someone by saying something similar in another post but I don't remember this from either high school or college) and that is the concept of an Acidity Function. Hammet's Acidity function is defined as
Ho = pKbh + log(/[BH+])
where B is the unprotonated form of a base and BH+ the protonated form. The reaction BH+ --> B + H+ is governed the equilibrium equation Gb**Gh*[H+]/Gbh*[BH+] = Kbh. The G's are activity coefficients (more on them in a second). Taking minus the log of both sides gives -log(Gb/Gh) - log(/[BH]) - log(Gh*[H+]) = -log(Kbh).
By definition
pKbh = - log(KbH)
and, by the IUPAC definition,
pH = -log(Gh*[H+]) = -log(activity of hydrogen ion)
so subsituting pKbh into the definition of the acidity function we get
Ho = pKbh + log(/[BH+]) = -log(Gb/Gh) + pH
For the dilute aqueous solutions we are interested in Gb and Gh, the activity coefficents, are both very close to 1 so that, as log(1) = 0
Ho = pH.
