The Mole
Simply put, the mole represents a number. Just as the term dozen refers to the number twelve, the mole represents the number 6.02 x 10^{23}.
Now that's a big number! While a dozen eggs will make a nice omelet, a mole of eggs will fill all of the oceans on earth more than 30 million times over.
The mole is used when we're talking about numbers of atoms and molecules. Atoms and molecules are very tiny things. A drop of water the size of the period at the end of this sentence would contain 10 trillion water molecules. Instead of talking about trillions and quadrillions of molecules (and more), it's much simpler to use the mole.
History of the Mole
The number of objects in one mole, that is, 6.02 x 10^{23}, is commonly referred to as Avogadro´s number. Amadeo Avogadro was an Italian physics professor who proposed in 1811 that equal volumes of different gases at the same temperature contain equal numbers of molecules About fifty years later, an Italian scientist named Stanislao Cannizzaro used Avogadro's hypothesis to develop a set of atomic weights for the known elements by comparing the masses of equal volumes of gas. Building on this work, an Austrian high school teacher named
Johann Josef Loschmidt calculated the size of a molecule of air in 1865, and thus developed an estimate for the number of molecules in a given volume of air. While these early estimates have since been refined, they led to the concept of the mole  that is, the theory that in a defined mass of an element (its atomic weight) there is a precise number of atoms: Avogadro's number.
Johann Josef Loschmidt calculated the size of a molecule of air in 1865, and thus developed an estimate for the number of molecules in a given volume of air. While these early estimates have since been refined, they led to the concept of the mole  that is, the theory that in a defined mass of an element (its atomic weight) there is a precise number of atoms: Avogadro's number.
Molar Mass
A sample of any element with a mass equal to that element's atomic weight (in grams) will contain precisely one mole of atoms (6.02 x 10^{23} atoms). For example, helium has an atomic weight of 4.00. Therefore, 4.00 grams of helium will contain one mole of helium atoms. You can also work with fractions (or multiples) of moles:
Mole/Weight Relationship Examples Using Helium
 
Moles Helium

Helium Atoms

Grams Helium

1/4

1.505 x 10^{23}
 
1/2

3.01 x 10^{23}
 
1

6.02 x 10^{23}
 
2

1.204 x 10^{24}
 
10

6.02 x 10^{24}

Other atomic weights are listed on the periodic table. For each listed, measuring out a quantity of the element equal to its atomic weight in grams will yield 6.02 x 10^{23} atoms of that element.
The atomic weight of an element identifies both the mass of one mole of that element and the total number of protons and neutrons in an atom of that element. How can that be? Let's look at hydrogen. One mole of hydrogen atoms will weigh 1.01 grams .
Each hydrogen atom consists of one proton surrounded by one electron. But remember, the electron weighs so little that it does not contribute much to an atom's weight. Ignoring the weight of hydrogen's electrons, we can say that one mole of protons (H nuclei) weighs approximately one gram. Since protons and neutrons have about the same mass, a mole of either of these particles will weigh about one gram. For example, in one mole of helium, there are two moles of protons and two moles of neutrons  four grams of particles.
Molecular Weight
If you stand on a scale with a friend, the scale will register the combined weight of both you and your friend. When atoms form molecules, the atoms bond together, and the molecule's weight is the combined weight of all of its parts.
For example, every water molecule (H_{2}O) has two atoms of hydrogen and one atom of oxygen. One mole of water molecules will contain two moles of hydrogen and one mole of oxygen.
A bottle filled with exactly 18.02 g water will contain 6.02 x 10^{23} water molecules. The concept of fractions and multiples described above also applies to molecules: 9.01 g of water would contain 1/2 mole, or 3.01 x 10^{23} molecules. You can calculate the molecular weight of any compound simply by summing the weights of atoms that make up that compound.
Molecular Weight Calculator
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