Avogadro's number ( NA ) is a fundamental physical constant used to convert moles of a substance to a quantity of particles which make up that substance. The closest known approximation of its value is 6.022 141 5x1023 mol-1 (with an uncertainty of 0.000 001 0x1023).
Its most common application is in finding the number of atoms in n grams of an element having n atomic mass units.
grams
Number of atoms of an element = --------------- x Avogadro's Number
atomic weight
Number of atoms of an element = moles x Avogadro's Number
Of course, its use is not limited to elements. To find the number of molecules of a substance, use the relative formula mass (add up all the atomic masses of the elements that make up the molecule). It is number of molecules (or atoms) of a substance contained in an amount of a given substance whose mass (in grams) is numerically equal to the atomic weight of the substance.† More generally, it is the number of particles in one mole of anything. A mole of pennies, for example, is enough money to buy everyone in the world 50 million cars.
Formal Definition
Specifically, Avogadro's number is defined as the number of 12carbon atoms which make up 0.012 kilograms of carbon. If this sounds like a circuitous way of arriving at a value, it is. The more direct quantity of 1hydrogen atoms which make up one gram of hydrogen was deemed inappropriate because of the relatively large amounts of deuterium ( 2hydrogen ) contaminating hydrogen samples. Carbon has a more constant isotopic composition, and very pure samples of 12carbon are relatively easy to produce. 12Carbon is the standard by which all other atomic masses are measured, being defined as exactly 12 amu.
It should be noted, however, that a mole of 12carbon weighs less than six moles of protons plus six moles of neutrons. For an explanation of why, see mass defect.
Avogadro's Law
Count Lorenzo Romano Amedeo Carlo Avogadro di Quaregna e Cerreto (1776–1856) actually had little to do directly with the number which bears his name. However, the work that he did in the field of chemistry was important enough in calculating it that it was named in his honor. It was first calculated after Avogadro's death, in 1865, by Johann Josef Loschmidt using kinetic gas theory, which is based on Avogadro's Law: "equal volumes of all [ideal] gases at the same temperature and pressure contain the same number of molecules".
Calculating Avogadro's Number
The process of calculating Avogadro's number is complex. Fortunately many of these steps have already been done for us, such as building the Periodic Table of the Elements.
Find the volumes of elemental gasses necessary to combine to form various known compounds. For example, N2O is made from 2 parts N to 1 part O by volume. NO is made from 1 part N to 1 part O by volume. Joseph Louis Gay-Lussac (1778–1850) has done this step for us. No, I don't know how he knew it was N2O before they built the periodic table.
Realize that this may mean that equal volumes of gases at the same temperature and pressure contain equal numbers of particles. This was Avogadro's Hypothesis. It was later confirmed for ideal gasses and is now known as Avogadro's Law.
Use Avogadro's Law to determine the atomic mass ratio of two equal volume samples of elemental gasses at STP. For example, 1L of hydrogen masses 0.08g and 1L of nitrogen masses 1.12g. Therefore, because Avogadro's Law states that 1L of hydrogen and 1L of nitrogen contain an equal number of atoms at STP, we know that an atom of nitrogen is 14 times heavier (1.12g/0.08g) than an atom of hydrogen.
Since hydrogen is the lightest substance known, assign it a value of 1 and all other elements a value equal to its mass relative to hydrogen. We now have our periodic table of the elements.
Define a "mole" as the quantity of 12carbon necessary to obtain 0.012kg. Define Avogadro's number as the number of atoms of 12carbon this is.
Use X-ray crystallography to calculate the number of titanium atoms in one mole (kids, get your parents to help you with this step).
Titanium's crystalline structure is made of body-centered cubic unit cells having an edge length of 330.6 pm. The density of titanium is 4.401 g/cm3. There are 47.88 grams of titanium in one mole. Therefore:
2 atoms Ti 1 unit cell 47.88g Ti 1cm3
------------ x ------------------ x ----------- x ----------- = 6.02x1023 atoms of Ti/mole
unit cell (3.306x10-8)3cm3 mole 4.401g Ti
The National Institute of Standards and Technology (NIST) used this procedure with silicon crystals to obtain the current standard value of Avogadro's number used today.
Sources:
http://gemini.tntech.edu/~tfurtsch/scihist/avogadro.htm
Herron, Kukla, Schrader, Erickson, and DiSpezio. Chemistry. Lexington, Massachusetts: Heath and Company, 1987.
† Original definition by datagirl.