Acids and Bases, and What to Make of Them

Students are often confused when starting to study acids and bases, by the multiple definitions and duplicate definitions of acidity (K(a)) and basicity (K(b)). Let me attempt to shed some light.
The element hydrogen is frequently found in substances, but never as an atom. It's always bonded, either to itself (as H2), or to some other element, either in a binary covalent (e.g. H2S), more complex molecule (e.g. C6H12O6).
Such molecules are often quite stable, but sometimes and under the right conditions the hydrogen can come off as a proton, immediately donated to whatever solvent molecule(s) surround the dissociating molecule. We call the original molecule an acid, and we say the molecule has dissociated. This process is rapidly reversible; that hydrogen ion or any other can go back onto the remaining molecule fragment, which we call a conjugate base. 
The reaction for an acid dissociation is therefore:
Reactant1(initial molecule = conjugate acid) + H2O = H3O(+) + Product2(conjugate base, lacks the proton)
The proportion of time that the proton in question spends on its original molecule, vs. the time spent on a solvent molecule, determines whether we call the original molecule a *strong* acid, or a *weak* acid. Note that these are not the everyday understanding of strong and weak, referring to amount of substance in a solution; to avoid confusion, in chemistry we refer to these instead as "concentrated" and "dilute" respectively.
For a strong acid, the proton spends more time on the solvent; for a weak acid, it spends more time on the original molecule. At any one time, some proportion of the protons are on the acid, vs. on the solvent; we speak of the respective molecular species (whether electrically neutral or charged) as having concentrations (usually, moles/liter). Then we can write a mathematical expression for the equilibrium that results in the solution:
                                    K(a) = -------------------------
where product1 = H3O(+) (you can also write it as H(+), makes no difference), product2 = the conjugate base entity, and reactant1 = the original acid molecule.
Note that in this expression, by convention the solvent (usually water) does *not* appear, even though it is definitely a reactant. Its concentration is always already pre-figured into the value of the acidity constant K(a), because for most purposes the concentration of water (~55 molar) does not vary much. And when it does, such as when you've loaded H2SO4 into water at a high level, and such that all the available H2O has been transformed into H3O(+), it doesn't matter anyway for chemical purposes (although it may matter to you: the mixing of H2SO4 with water is highly exothermic, and adding water to the pure acid will boil the water off and spatter you with destructive, hot, oxidizing liquids!) other than affecting amounts (by dilution) for your end use.
There are a handful of common strong acids, you should learn the list. Also, for related molecules which incorporate molecular groupings like those of the strong acids, the acidity can be quite high. For example, acetic acid is a weak acid (pK(a) ~ 5), monochloroacetic acid is stronger, dichloroacetic acid still stronger, and trichloroacetic acid is actually a strong acid. The incorporation of the electron-attracting element chlorine (present in the strong acid HCl), 2 atoms away from the oxygen holding the proton, affects the oxygen that holds the proton, that much.
Now, the original "molecule" holding the proton which may come off, doesn't have to be a neutral molecule, though it frequently is. It can be positively charged (NH4(+)), neutral (HCl), or negative (HSO4(-)). It may carry multiple charges, or even some minus and some plus charges, on the same molecule (this is often the case with protein molecules in solution). Whatever it is, we can still consider the loss of the proton to be a well-defined event, with an equilibrium constant as above (even if we don't know where on the molecule the proton is exiting -- though if we do have this information, we can assign pK(a) values to each site where a proton may be lost).
So, acids (under the Bronsted-Lowry and Arrhenius definitions) are easy species to understand -- they have protons, and they come off when tickled appropriately.
Now for bases. Bases are substances which cause the appearance of OH(-) ions (hydroxide) when placed in water. They may possess hydroxide in their formula (e.g., NaOH) *or* they may have this effect by the related reaction in which they ACCEPT A PROTON from water. Many organic substances, particularly those with NH2, C-NH-C or C-N-C groupings, are apt to do this. Just because of the history of chemistry, an analogous mathematical expression was developed to express the basic dissociation equilibrium:
K(b)    =  -------------------
where OH- is product1, product2 = the conjugate acid formed, and reactant1 = the original base molecule species.
Whenever it's convenient, you can always convert from a given K(b) to the K(a) for the reaction proceeding in the opposite direction, by using the identity:
 K(a) * K(b) = K(w) = 10^(-14).
If you like to keep just one set of values in mind for acidities of chemical species, keep the K(a)'s, and convert all the K(b)'s to their equivalent K(a)'s. (pK(a) = 14 - pK(b).)
Although lots of chemistry takes place in water, with an effective pH range of ~-1.7 to ~ 15.7, this is far from the end of it. In other solvents, which have different abilities to accept or lose protons than does water, things which would all be strong acids or strong bases in water, may not be. For example, H2SO4 has pK(a) = -3, and so is a strong acid in water, but it is not in glacial acetic acid, which doesn't much like an added proton. However, HCLO4, with pKa = -8, IS a strong acid in glacial acetic acid (we can tell these sorts of things by colligative properties of the solutions: electrical conductivity, freezing point depression, boiling point elevation, osmotic pressure, and so on).
This has an important effect: if you're trying to perform a reaction requiring (temporary or permanent) addition or subtraction of a proton somewhere on a molecule, you might not be able to do it in water (because the strongest proton available is that of H3O(+), and the strongest stripper of protons is OH(-)). However, using an inherently stronger acid or base in the appropriate solvent may work.
Consider the pH ranges available for K(a)'s:
water = -1.7 to +15.7  (H(+) = -1.7, OH(-) = 15.7 -- the limitation to this range is the "leveling effect of H2O"
HI (hydroiodic acid) pK(a) = -9.3
Triflic acid (CF3SO3H) pKa ~ -14.6
Carborane acid H(CHB11Cl11) pKa < - 18
NH3 = 38 (to make NH2(-); note that this is much different than that for NH4(+), the usual species in water!)
CH4 = 51 (i.e., this means CH3(-) methide ion is such a strong base that it will rip protons from almost any other species having them!)
This range: -18 to +51,  (or 10^69) might not seem large, but for comparison, that's a difference in free proton levels comparable to one proton vs. an entire galaxy (1 mole H= 1 g = 6 x 10^23 atoms; 1 sun = 2 x 10^30 kg; galaxy = 6 x 10^11 sun masses, or about 10^69).
Now a few words on effects of acids and bases. Biologically, acids and bases are important because many of the molecules in our body won't function or even will fall apart if exposed to high levels of acid or base. The level of acidity (or alkalinity) generated by placing an acid or base in water depends on primarily three factors: (1) its concentration, (2) its strength (pK(a) or pK(b)), and (3) what the starting pH of the solution was (caused by other chemicals present). In fact, one can roll (2) and (3) together, and state as (2') the pK(a) or pK(b) relative to the buffered pH of the starting solution.
Now, you can calculate proton levels and accomplish reactions based on the acidity of a solution. However,

Strong Is Different from Corrosive!

The carborane acids are incredible proton donors, yet they are not highly corrosive. Corrosiveness is related to the negatively-charged part of the acid. Hydrofluoric acid (HF), for example, is so corrosive it dissolves glass. The fluoride ion attacks the silicon atom in silica glass while the proton is interacting with oxygen. Even though it is highly corrosive, hydrofluoric acid is not considered to be a strong acid because it does not completely dissociate in water. In a similar vein (pun intended), phosphoric acid, though not strong, will also cause massive tissue damage if you spill some on your skin.
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