Mole (unit)

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The mole (symbol: mol) is the SI base unit that measures an amount of substance. One mole contains Avogadro's number (approximately 6.022×10²³) entities. A mole is much like "a dozen" in that both units can describe any set of elementary objects, although the mole's use is usually limited to measurement of subatomic, atomic, and molecular structures. This is all due to the fact that a mole represents an exceedingly large number of entities.

1 Definition
2 Elementary entities
3 History
- 3.1 Proposed future definition
4 Utility of moles
5 Example calculation
6 Moles of everyday entities
7 See also
8 References

[edit] Definition

A mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram (or 12 grams) of carbon-12, where the carbon-12 atoms are unbound, at rest and in their ground state.^[1] The number of atoms in 0.012 kilogram of carbon-12 is known as Avogadro constant, and is determined empirically. The currently accepted value is 6.02214179(30)×10²³ mol^-1 (2007 CODATA).

According to the SI, the mole is not dimensionless, but has its very own dimension, namely "amount of substance", comparable to other dimensions such as mass and luminous intensity.^[2] (By contrast, the SI specifically defines the radian and the steradian as special names for the dimensionless unit one.)^[3] The SI additionally defines Avogadro constant as having the unit reciprocal mole, as it is the ratio of a dimensionless quantity and a quantity with the unit mole.^[3] However, if in the future the kilogram is redefined in terms of a specific number of carbon-12 atoms (see below), then the value of Avogadro's number will be defined rather than measured, and the mole will cease to be a unit of physical significance.^[4]

The relationship of the atomic mass unit (u^[5]) 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 relative atomic mass of 55.845 u, so a mole of iron has a mass of 55.845 grams. This notation is very commonly used by chemists and physicists.

Scientists (chemical engineers in particular) oftentimes use mole units other than the 'mole.' They are still based on amount of substance, just a different quantity. Because of the variety of units, these professions sometimes refer to moles as 'gram-moles,' in that a mole of 'substance with some mass in u' will of have a mass of the same number, only in grams (see above). This is the only mole unit which uses Avogadro's number; all others require a conversion factor.

The simplest non-standard mole unit is the kg-mol, or kmol. The same ideology behind the mole applies; thus, a kmol of 'substance with some mass in u' will have a mass of the same number in kilograms. There are 1000 grams in a kg, so there are 1000 moles in a kmol.

The same goes for lb-mol. There are 453.59 grams in a pound, so there are 453.59 mol in a lb-mol. A lb-mol of 'substance with some mass in u' will have a mass of the same number in pounds. Similarly a ton-mol is based on the ton.

Only the 'gram-mole' is endorsed by the SI; the only officially-allowed derivatives are those formed by the usual metric prefixes, such as the millimole (mmol) and kilomole (kmol).

[edit] Elementary entities

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, that is each particle of gas is a single atom. All gases have the same molar volume of 22.4 litres per mole at STP (see Avogadro's Law).

A mole of atoms or molecules is also called a "gram atom" or "gram molecule," respectively.

[edit] History

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 "mass, massive structure". He used it to express the gram molecular weight of a substance. So, for example, 1 mole of hydrochloric acid (HCl) has a mass of 36.5 grams (atomic masses 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:

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12; its symbol is "mol."

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.

[edit] Proposed future definition

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:

The kilogram is the mass of exactly (6.0221415×10²³/0.012) unbound carbon-12 atoms at rest and in their ground state.

This would have the effect of defining Avogadro's number to be precisely N_A = 6.0221415×10²³ elementary entities per mole, and, consequently, the mole would become merely a unit of counting, like the dozen.

Another proposed definition of N_A is:

N_A = 602214141070409084099072 = 84446888³

This has the nice properties of being a perfect cube, and being well within the current experimental bounds of measurement.

[edit] Utility of moles

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:

2H₂ + O₂ → 2H₂O

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 mL of water contains over 3×10²² (or 30,000,000,000,000,000,000,000) molecules.

[edit] Example calculation

In this example, moles are used to calculate the mass of CO₂ given off when 1 g of ethane is burned. The equation for this chemical reaction is:

7 O₂ + 2 C₂H₆ → 4 CO₂ + 6 H₂O

that is,

7 molecules of oxygen react with 2 molecules of ethane to give 4 molecules of carbon dioxide and 6 molecules of water.

The first thing is to figure out how many molecules of ethane were burnt. We know that it was just enough to make 1 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 1, 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 1 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 2 molecules of ethane we obtain 4 molecules of CO₂, we have 2 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:

(7/2)(1/30th mol of dioxygen) (2 × 16 g/mol) = 7×16/30 g = 3.73 g
(6/2)(1/30th mol of water)(2×1 + 16 g/mol) = 1.8 g
3.73 g + 1 g = 2.93 + 1.8 g

(Note: According to the mass-energy relationship, there is a very small 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.)

[edit] Moles of everyday entities

Note: all of the following are accurate to approximately one significant figure.

Given that the volume of a grain of sand is approximately 10^-12 m³^[6], and given that the area of the United States is about 10¹³ m²^[7], it therefore follows that a mole of sand grains would cover the United States in approximately one centimeter of sand.
A human body contains very roughly one hundred trillion cells^[8]; there are roughly six billion people on Earth; so the total number of human cells on the planet is approximately 100×10¹²*6×10⁹=6×10²³, which is very close to one mole.
Since the Earth has a radius of about 6400 km^[9], its volume is approximately 10²¹ m³. Since about 500 large grapefruit will fit in one cubic meter^[10], it therefore follows that a mole of grapefruit would have approximately the same volume as the Earth.