I am a confused (and somewhat anal retentive) non-chemist who is trying to understand what pH is. I have come to understand it is a measure of acidity, but I have not yet figured out what that means to a precision that I find satisfying.
Please excuse me if these are dumb questions, but they will help me understand the definition:
Does everything have a pH? What is the scale that pH works upon? Does a proton have a pH? A quark? Does the sun have a pH? Does the Universe?
Did you not carry out some research?
Well, you can find information about it in Wikipedia, or even dictionaries like Merriam-Webster or Dictionary Online.
You will see that pH is calculated through:
$$pH = -\log(a_{\ce{H+}})$$
Or which can most of the time be stated as:
$$pH = -\log(\ce{[H+]})$$
$a_{\ce{H+}}$ is the hydrogen ion activity, the 'effective concentration' of hydrogen ions. Without active hydrogen ions, you don't have any pH.
You might find that the pH scale is from 0 to 14, but you can also have circumstances where you have negative pH and pH above 14. It depends on the extent of dissociation of an acid or base to produce active hydrogen ions, or hydroxide ions in the case of bases.
Since I mentioned hydroxide ions, it might also be worth nothing that:
$$14 = pH + pOH$$
And that basically, a pH of 1 is equivalent to a pOH of 13.
pH scales are just a special case of solvonium/solvate scales, which are a way to keep track of the concentration of solvonium/solvate ions. They can be defined for any (usually protic) liquid solvent in macroscopic amounts (the limit of many molecules). Particles don't have pH, and I'm not sure it's possible to define such scales in a useful manner for solids, gasses, plasmas or mixtures of solvents.
Let's look at some cases. Water is capable of self-ionization, according to the equation:
$$\ce{2H_2O_{(l)} -> H_3O^+_{(aq)} + OH^{-}_{(aq})} \ \ \ \ \ K=\frac{a_{H_3O^+}a_{OH^{-}}}{a_{H_2O}^2}\approx \frac{[H_3O^+][OH^-]}{[H_2O]^2}=3.2\times10^{-18}$$
, where the equilibrium constant $K$ is given at 25°C (it varies with temperature). If we consider relatively dilute solutions, then $[H_2O]\approx55.5\ mol.L^{-1}$ and one can obtain:
$$[H_3O^+][OH^-]=K_w^{25°C}=1\times10^{-14}$$
$K_w$ is called the ionic product of water. Taking the logarithm of both sides and multiplying everything by $-1$, we find
$$-log[H_3O^+]-log[OH^-]=-log\ K_w^{25°C}\rightarrow pH+pOH=pK_w^{25°C}=14$$
(At the end I switched $\ce{H_3O^+}$ for $\ce{H^+}$ since they are essentially the same thing. $\ce{H_3O^+}$ is nothing but $\ce{H^+}$ solvated in water)
The $p$ in $pH$ is nothing special; it is merely shorthand for the operator $-log()$, and it can be applied to any measure, though it's not very common to do so. Though at 25°C, the sum of pH and pOH is constant (equal to 14), there is nothing immediately forbidding either value being negative. They just can't be very negative because the approximations in the first two equations breaks down as such solutions are no longer dilute. Also, the sum of pH and pOH is only 14 at 1 bar for a temperature of 25°C. At 0°C, $K_w^{0°C}=1.1\times10^{-15}$, while at 100°C, $K_w^{100°C}=5.1\times10^{-13}$.
Now for other solvents. Water isn't the only substance to self-ionize; another example is is liquid ammonia ($\ce{NH_3}$). Here, we have:
$$\ce{2NH_3\ _{(l)} -> NH_4^+\ _{(am)} + NH_2^{-}\ _{(am})}$$
An ionic product can also be defined for ammonia:
$$[NH_4^+][NH_2^-]=K_{am}^{-34°C}\approx 10^{-30}$$
From the ionic product, we see that ammonia has a smaller tendency to self-ionize compared to water. This is due to several factors, such as the smaller electronegativity of the nitrogen atom and the lower dielectric constant of ammonia, which destabilize the ions formed. In this solvent, the neutral point is around pH 15.
After all this, what's the point of pH? As was mentioned at the start, it's a way to keep track of the concentration (or activity rather) of $\ce{H^+}$ in solvents, and it turns out this species is particularly important to chemists, not only because it's so common and a part of water. Bonding/dissociation/charge transfer happens due to the interaction of electrostatic potentials in molecules which push and pull electrons. Usually, a handful of fundamental units of charge are distributed in volume of a few atoms. However, if you stop to think about it, $\ce{H^+}$ has no electron cloud and is actually a lone proton, characterized by a charge of $+e$ concentrated in the volume of a nucleus. If you recall, an atomic nucleus has about a trillionth the volume of an atom's electron cloud. This means $\ce{H^+}$ has a phenomenally high charge density capable of generating a massive electric potential, and is therefore extremely reactive. Though helium and neon are the most inert substances we know, $\ce{H^+}$ will easily react exothermically with even them to produce $\ce{HeH^+}$ and $\ce{NeH^+}$. In fact, $\ce{H^+}$ cannot exist as a true lone ion in a condensed phase, and it will demand some electron density from any substance. A very large amount of acid-base theory was created solely to understand more about the reactivity of this unique species.
pH stands for "potential hydrogen", originally "power of hydrogen". In simplest terms, it is a logarithmic scale (based on powers of 10, similar to the Richter scale for earthquakes, the decibel scale for sound and signal levels, and others that seek to measure very large differences with smaller numbers) of how many free protons there are in an aqueous solution of a substance, which is in turn a measure of that solution's acidity or basicity.
The number is the absolute value of the exponent of this measurement, so if, for instance, a substance has a pH of 7, that means there's one free proton per 10 million water molecules (1/10,000,000 = 10-7). A pH of one, or less, means there's at least one proton for every 10 water molecules, while a pH of 14 means there is one proton or fewer per 100 trillion water molecules.
Protons, or H+ ions, are attracted to the negative charge region of a water molecule. Water's bent molecular structure, with the two hydrogens closer to one side and oxygen's extra lone pairs on the other, effectively makes each molecule a tiny magnet, which is the foundation for many important characteristics of water, like why ice floats and why some things dissolve in water while others don't. Anyway, the hydrogen loosely "binds" to one of those lone pairs, forming hydronium (H3O+).
There is a natural equilibrium of one hydronium ion per 10 million molecules. At this point, pure water (again being a polar solvent) will actually dissolve itself to a small degree, producing one hydronium and one hydroxide per 10 million. At this pH of 7, the opposite measure, pOH (power of hydroxide) is also 7, balancing the reactivity of the water.
Below a pH of 7, the positively-charged hydronium atoms come to dominate this equilibrium, decreasing the number of hydroxide ions and looking for a substance to which to donate its protons to reach a balance again. This solution, which is now a "proton donor", is one of the classical definitions of an acid (the Lowry-Bronsted definition). Ideally, it will donate its proton to a substance that needs one as badly as the acid needs to get rid of it, but if the acid has enough hydronium, it will stick those hydrogens wherever they'll fit into any other substance's molecules.
Above pH 7, the negatively-charged hydroxide ions reduce the hydronium and dominate equilibrium. The solution is now a "base", a "proton acceptor", and is looking for something from which to take protons. The ideal substance to do so is a proton donor - an acid - but similar to an acid, a base, if strong enough, will take hydrogen atoms from anywhere they're not nailed down.
Now, not all substances which are acidic have a free proton to donate (and so not all of them increase the amount of hydronium ions in water), and not all substances which are basic have a hydroxide anion, nor do they accept free protons into their structure. Therefore, there is another definition, the Lewis definition, stating that an acid is a substance that accepts a "lone pair" of electrons from another substance (a pair of electrons not forming a bond between atoms of the molecule) to equalize its own molecular charge, and a base is a substance that readily donates an electron pair. Regardless of the definition or the exact mechanism, these acids and bases have similar observed properties, and so even if explicit exchange of hydronium atoms isn't involved, the substances still indicate a pH using various testing methods. This prompted the change in name from "power of hydrogen" to "potential hydrogen", in effect making the measurement an "equivalent", applying to both Lewis acids and bases and Bronsted-Lowry acids and bases.
