The mole (abbreviation "mol") is the SI base unit that measures an amount of a substance. One mole of a substance is a quantity of substance that contains Avogadro's number of entities, which is approximately 6.022×1023 entities. A mole is much like a “dozen” in that both units can be used to quantify any set of objects, but the mole is generally used for measuring the numbers of atoms, molecules, and subatomic particles in a given amount of substance.
One mole is defined as the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilograms of carbon-12, where the carbon-12 atoms are unbound, at rest, and in their ground rate. The number of atoms in 0.012 kilogram of carbon-12 is known as Avogadro's number and is determined empirically. The currently accepted value is 6.0221415(10)×1023 mol-1 (2002 publication of Committee on Data for Science and Technology).
According to the SI, the mole is not dimensionless, but has its very own dimensions, namely "amount of substance", comparable to other dimensions such as mass and luminous intensity. The SI additionally defines Avogadro's number as having the unit reciprocal mole, as it is the ratio of a dimensionless quantity and a quantity with the unit mole. 
The relationship of the atomic mass unit to Avogadro's number means that a mole can also be defined as: That quantity of a substance whose mass in grams is the same as its formula weight. For example, iron has an atomic weight of 55.845, so a mole of iron weighs 55.845 grams. This notation is commonly used by chemists and physicists.
The mass (in grams) of one mole of a chemical element or compound is called its molar mass. It is useful as a conversion factor between the number of grams of a pure substance (which can be measured directly) and the number of moles of that substance.
Most chemical engineers as well as many other engineers and scientists differentiate between gram moles and kilogram moles (kgmol or kmol): 55.845 grams in a gram mole of iron and 55.845 kilograms in a kilogram mole of iron. Similarly, engineers and scientists in the United States use the pound mole (lbmol). For example, there are 55.845 pounds in a lbmol of iron. In addition to kgmol, kmol, or lbmol, ton moles are also used. For instance carbon monoxide (CO) has a molecular weight of 28, and one mol of CO therefore contains 28 g, one lbmol of CO contains 28 lb, and one tonmol of CO contains 28 tons. It should be noted, however, that only the "gram mole" is endorsed by the SI – none of these derivates are official units. Properly, the gram mole is called simply the mole, and Avogadro's number is directly connected to this mole– its relation to the lbmole or other variants requires a conversion factor.
When the mole is used to specify the amount of a substance, the kind of elementary entities (particles) in the substance must be identified. The particles can be atoms, molecules, ions, formula units, electrons, or other particles. For example, one mole of water is equivalent to about 18 grams of water and contains one mole of H2O molecules, but three moles of atoms (two moles H and one mole O).
When the substance of interest is a gas, the particles are usually molecules. However, the noble gases (He, Ar, Ne, Kr, Xe, Rn) are all monoatomic, meaning each particle of gas is a single atom. All gases have the same molar volume of 22.4 liters per mole at standard temperature and pressure (STP).
A mole of atoms or molecules is also called a "gram atom" or "gram molecule".
The name mole (German Mol) is attributed to Wilhelm Ostwald who introduced the concept in the year 1902. It is an abbreviation for molecule (German Molekül), which is in turn derived from Latin moles, meaning "mass, massive structure". He used it to express the gram molecular weight of a substance. So, for example, one mole of hydrochloric acid (HCl) has a mass of 36.5 grams (atomic weights Cl: 35.5 u, H: 1.0 u).
Prior to 1959, both the IUPAP and IUPAC used oxygen to define the mole: the chemists defining the mole as the number of atoms of oxygen which had mass 16 g, the physicists using a similar definition but with the oxygen-16 isotope only. The two organizations agreed in 1959/1960 to define the mole as such:
This was adopted by the CIPM (International Committee for Weights and Measures) in 1967, and in 1971 it was adopted by the 14th CGPM (General Conference on Weights and Measures).
In 1980, the CIPM clarified the above definition, defining that the carbon-12 atoms are unbound and in their ground state.
As with other SI base units, there have been proposals to redefine the kilogram in such a way as to define some presently measured physical constants to fixed values. One proposed definition  of the kilogram is:
This would have the effect of defining Avogadro's number to be precisely NA = 6.0221415×1023 elementary entities per mole, and, consequently, the mole would become merely a unit of counting, like the dozen.
The mole is useful in chemistry because it allows different substances to be measured in a comparable way. Using the same number of moles of two substances, both amounts have the same number of molecules or atoms. The mole makes it easier to interpret chemical equations in practical terms. Thus the equation:
can be understood as "two moles of hydrogen plus one mole of oxygen yields two moles of water."
Moles are useful in chemical calculations, because they enable the calculation of yields and other values when dealing with particles of different mass.
Number of particles is a more useful unit in chemistry than mass or weight, because reactions take place between atoms (for example, two hydrogen atoms and one oxygen atom make one molecule of water) that have very different weights (one oxygen atom weighs almost 16 times as much as a hydrogen atom). However, the raw numbers of atoms in a reaction are not convenient, because they are very large; for example, just one milliliter of water contains over 3×1022 (or 30,000,000,000,000,000,000,000) molecules.
The first thing is to figure out how many molecules of ethane were burnt. We know that it was just enough to make one g, so we now need the molecular mass of ethane. This can be calculated: the mass in grams of one mole of a substance is by definition its atomic or molecular mass; The atomic mass of hydrogen is one, and the atomic mass of carbon is 12, so the molecular mass of C2H6 is (2 × 12) + (6 × 1) = 30. One mole of ethane is 30 g. So one g of ethane is 1/30th of a mole; the amount burnt was 1/30th of a mole (remember that it is a number, quite like "half a dozen").
Now we can calculate the number of molecules of CO2 given off. Since for two molecules of ethane we obtain four molecules of CO2, we have two molecules of CO2 for each molecule of ethane. So, for 1/30th of a mole of ethane, 2 × 1/30th = 1/15th of a mole of CO2 were produced.
Next, we need the molecular mass of CO2. The atomic mass of carbon is 12 and that of oxygen is 16, so one mole of carbon dioxide is 12 + (2 × 16) = 44 g/mol.
Finally, the mass of CO2 is 1/15 mol × 44 g/mol = 2.93 g of carbon dioxide.
Notice that the number of moles does not need to balance on either side of the equation. This is because a mole does not count mass or the number of atoms involved, but the number of particles involved (each of them composed of a variable number of atoms). However, we could likewise calculate the mass of oxygen consumed, and the mass of water produced, and observe that the mass of products (carbon dioxide and water) is equal to the mass of dioxygen plus ethane:
(Note: actually, according to the mass-energy relationship, there is a very slim difference between the mass of carbon, hydrogen and oxygen separated, on one side, and on the other side the mass of the molecules made of them– this has not been accounted for here.)
Note: all of the following are accurate to approximately one significant figure.
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