|These paper lanterns - also known as Kongming lanterns - are taking off due to some of the physical laws that Avogadro was investigating during the late 1700s and early 1800s.|
What is Avogadro's number?
Well, one way to look at it is the actual number, which is:
602,214,150,000,000,000,000,000 +/- 1,000,000,000
Or, in shorthand, (6.0221415 ± 0.0000010) × 10^23.
It's actually formally defined as "the number of carbon-12 atoms in 12 grams of unbound carbon-12 in its rest-energy electronic state.
But why do we care how many atoms are in 12 grams of a particular isotope of carbon?
Why Avogadro's number is useful
Long before physicists and chemists had an actual number for Avogadro's number, they were using Avogadro's number to construct theories and physical laws.
In a lot of nature, certain numbers are important because they help define the relative size of different physical phenomenon.
One great example is the constant c in E = mc^2. In Einstein's famous relationship, c is the speed of light. We know from this equation that energy is related to mass by a proportional relationship to the speed of light squared. Even when we really didn't have a good idea of the precise value for the speed of light, we knew this relationship held.
Another great example is Planck's constant, or h. In an electromagnetic wave, such as light or microwaves, there is a relationship between the energy of the wave (v) and the energy of the wave (E) that is defined as E = hv.
These relationships - such as E = mc^2 and E = hv - are known as "scientific laws" or "physical laws."
In the case of Avogadro's number, one law where the number becomes important is Avogadro's Law:
Avogadro's Law: Under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules.
So, knowing the number of atoms or molecules in one gas at a given temperature and pressure will give us a proportion where we can figure out the number of atoms or molecules in all gases.
"pee vee equals en ar tee..."
And that's where "pee vee equals en ar tee..." comes in.
PV = nRT is what's known as the "ideal gas law," the major consequence of Avogadro's Law. Most gases follow the ideal gas law almost perfectly. In PV = nRT, P is "pressure," V is "volume," and T is "temperature," R is the universal gas constant and n is the number of moles of molecules or atoms of the substance - counting units in Avagadro's number.
American Scientist article on the history of Avogadro's number
How Stuff Works article on gases
Wolfram Research on Avogadro's number
[Note: correction Aug 4 2012 from "And what's that n? That' Avogadro's number, who, along with another important constant, Boltzmann's constant R, gives the proportional relationship between most gases." to "R is the universal gas constant and n is the number of moles of molecules or atoms of the substance - counting units in Avagadro's number."]