The Mole

The mole is the unit of amount in chemistry.  It provides a bridge between the atom and the macroscopic amounts of material that we work with in the laboratory.  It allows the chemist to weigh out amounts of two substances, say iron and sulfur, such that equal numbers of atoms of iron and sulfur are obtained.  A mole of a substance is defined as:

The mass of substance containing the same number of fundamental units as there are atoms in exactly 12.000 g of 12C.

Fundamental units may be atoms, molecules, or formula units, depending on the substance concerned. At present, our best estimate of the number of atoms in 12.000 g of 12C is 6.022 x 1023, a huge number of atoms.  This is obviously a very important quantity.  For historical reasons, it is called Avogadro's Number, and is given the symbol NA.

Unfortunately, the clumsy definition of the mole obscures its utility.  It is nearly analogous to defining a dozen as the mass of a substance that contains the same number of fundamental units as are contained in 733 g of Grade A large eggs.  This definition completely obscures the utility of the dozen: that it is 12 things!  Similarly, a mole is NA things. The mole is the same kind of unit as the dozen -- a certain number of things.  But it differs from the dozen in a couple of ways.  First, the number of things in a mole is so huge that we cannot identify with it in the way that we can identify with 12 things.  Second, 12 is an important number in the English system of weights and measures, so the definition of a dozen as 12 things makes sense. However, the choice of the unusual number, 6.022 x 1023, as the number of things in a mole seems odd. Why is this number chosen?  Would it not make more sense to define a mole as 1.0 x 1023 things, a nice (albeit large) integer that everyone can easily remember? To understand why the particular number, 6.022 x 1023 is used, it is necessary to resurrect an older, in some ways more sensible and useful, definition of the mole, which is grounded in the atomic weight scale addressed above.

The atomic weight scale defines the masses of atoms relative to the mass of an atom of 12C, which is assigned a mass of exactly 12.000 atomic mass units (amu). The number 12 is chosen so that the least massive atom, hydrogen, has a mass of about 1 (actually 1.008) on the scale.  The atomic mass unit is a very tiny unit of mass appropriate to the scale of single atoms.  Originally, of course, chemists had no idea of its value in laboratory-sized units like the gram.  The early versions of the atomic weight scale were established by scientists who had no knowledge of the electron, proton, or neutron.  When these were discovered in the late 19th and early 20th centuries, it turned out that the mass of an atom on the atomic weight scale was very nearly the same as the number of protons in its nucleus.  This is a very useful correpondence, but it was discovered only after the weight scale had been in use for a long time.

In their desire to be able to count atoms by weighing, chemists gradually developed the concept of the "gram-atomic weight", which was defined in exact correspondence with the atomic weight scale:

     1 atom of 12C weighs 12.000 amu
     1 gram-atomic weight of 12C weighs 12.000 g

Thus the gram-atomic weight of an element was defined as the atomic weight of the element, expressed in grams.  Because the atomic weight scale is numerically preserved in the definition of gram atomic weights, the mass of 1 gram-atomic weight of any element could be immediately determined as the atomic weight in grams.  Thus 1 gram-atomic weight of sulfur weighs 32.06 g; 1 gram-atomic weight of hydrogen weighs 1.008 g, and so on.  Analogous terms, such as gram-molecular weight for the molecular weight of a compound expressed in grams, were similarly used.  However, having to use a different term depending on whether elements or compounds were being discussed was awkward and inconvenient. For this reason, the term "mole" was adopted to signify the atomic, molecular, or formula weight of a pure substance expressed in grams.

Alternative definition of the mole:

The atomic, molecular, or formula weight of the substance, expressed in grams.

Thus one mole of ethyl alcohol, C2H6O, weighs 46.069 g.  One mole of water weighs 18.015 g.  If we mix 46.069 g of ethyl alcohol with 18.015 g of water, we can be assured that the mixture contains 1 molecule of ethyl alcohol per molecule of water. Further, we will know that there are 2 atoms of C and 8 atoms of H per each 2 atoms of O.  Thus the mole allows us to weigh convenient amounts of material containing known numbers of atoms; i.e., it allows us to count atoms.

The mole enables us to count atoms in the laboratory.

The mole is useful whether or not we know how many atoms of carbon-12 there are in 12.000 g of carbon-12.  If we weigh one mole of iron and one mole of sulfur, we know that these two samples contain the same number of atoms. This is the important aspect of the mole.  How many atoms there are in a mole is of subsidiary importance.  Nonetheless, it has become possible to determine this number. It is, of course, 6.022 x 1023 atoms per mole. We thus see that this number is simply a consequence of the choice that 1 mole be the formula weight in grams.  It is very nice that we know it; but we do not need to know it for the mole to be useful.

I would even go so far as to say that the modern definition of the mole in terms of a certain number of atoms of 12C is unfortunate, in that it suggests that the number, 6.022 x 1023 things/mole, must be used in any and every calculation involving moles!  In practice, we seldom need to know how many atoms or molecules we are working with, so in mole calculations the number 6.022 x 1023 is rarely used. What is invariably used (except for sample calculations in chemistry textbooks; see below!) is the fact that 1 mole of substance is its formula weight in grams.

The dual definitions of the mole can be used to find the mass of 1 amu expressed in g.  Exactly 12 g of carbon contains NA atoms, each weighing exactly 12 amu. In equation form,

NA atoms/mole x 12 amu /atom = 12 g/mole

Simplifying, we obtain NA amu = 1 g.  It follows that 1 m = 1/NA g = 1.660 x 10-24 g. Avogadro's number is an experimentally measured quantity.  Although we are confident that we know its value quite well, some future experiment may cause us to make a small revision in the number. By necessity, the mass of 1 m in grams will change accordingly. This is not worrisome, because neither number is crucial to the utility of the mole.

The atomic weight plays a dual role in chemistry.  It is used to represent the mass of a single atom, molecule, or formula unit of a substance, in which case it has units amu/atom.  It is also used to represent the mass of a mole of substance, in which case it has units g/mole. Many chemists prefer to use the term molar mass for the mass of a mole of substance.  In this course, we will use the phrase Formula Weight for both situations.  The meaning will be evident from context.

The importance of the mole concept can be summed up as follows: any statement that can be made about the number of atoms of an element in a molecule or formula unit of a substance can also be made about the number of moles of an element in a mole of the substance.  This is true because 1 mole of substance contains NA atoms, molecules, or formula units of substance.  Based on the formula for glucose, C6H12O6, we can make the following statements:

     1 molecule of glucose contains 6 atoms of C, 12 atoms of H, and 6 atoms of O;
     1 mole of glucose contains 6 moles of C atoms, 12 moles of H atoms, and 6 moles of O atoms;
     10 molecule of glucose contains 60 atoms of C, 120 atoms of H, and 60 atoms of O;
     10 moles of glucose contains 60 moles of C, 120 moles of H, and 60 moles of O atoms;
     Any amount of glucose contains equal numbers of C and O atoms, and twice this number of H atoms;
     Any amount of glucose contains equal numbers of moles of C and O atoms, and twice this number of moles of H atoms.
     NA molecules of glucose contains 6 x NA atoms of C, 12 x NA atoms of H, and 6 x NA atoms of O.

Example:  How many moles of Fe are in 5.6 g Fe?  How many Fe atoms are contained in the sample?

Solution:  By definition, 1 mole of Fe is 56.0 g. 5.6 g Fe is therefore 0.1 mole of Fe. The number of Fe atoms in the sample is

0.1 mole x 6.022 x 1023 atoms/mole = 6.022 x 1022 atoms.

5.6 g of iron is not much iron. However, even this small amount contains a huge number of iron atoms.
 

Example:  How many sulfur atoms are in 1.56 g sulfur?

Solution:  We can calculate the number of moles of sulfur from the atomic weight and the given mass. Once we have this, the number of atoms is obtained from Avogadro's Number, NA.

moles S = 1.56 g x 1 mole/32.06 g  = 0.0487 mole

atoms S = 0.0487 moles S x 6.022 x 1023 atoms/mole = 2.93 x 1022 atoms
 

Example:  What is the mass in g of 1 atom of sodium?

Solution:  If we know the mass of 1 mole of Na, and how many atoms are in a mole, the mass of a single atom should be easy to obtain:

mass Na atom = 22.99 g Na/mole x 1 mole/6.023 x 1023 atoms = 3.817 x 10-23 g.
 

Example:  What mass of sulfur contains the same number of moles as are in 10.0 g Fe?

Solution:  Figure the number of moles of Fe.  This is the desired number of moles of S.  Convert moles of S to mass of S using the atomic weight.

moles Fe = 10.0 g Fe x 1 mole/55.85 g = 0.1791 moles
moles S = moles Fe = 0.1791
g S = 0.1791 moles S x 32.06 g S/mole = 5.71 g S

Example:   Hemoglobin is the oxygen-carrying protein of most mammals. Each molecule of hemoglobin contains 4 atoms of iron. The molecular weight of hemoglobin is about 64000 g/mole. How many moles of iron are contained in 0.50 moles of hemoglobin? Calculate the number of iron atoms in 0.128 g of hemoglobin.

Solution:  Based on its molecular weight, hemoglobin is clearly a large molecule containing many atoms.  We are not told what the atoms are, nor how many of each there are.  However, we are told that each molecule of hemoglobin contains 4 atoms of iron. We can write the formula for a molecule of hemoglobin as follows:

Fe4X

where X represents the collection of all other atoms present.  What we can say about molecules and atoms, we can say about moles.  Thus 1 mole of hemoglobin contains 4 moles of iron.  Similarly, 0.50 moles of hemoglobin contains 4 x 0.50 = 2.00 moles of iron.

To obtain the second required answer, we convert mass of hemoglobin to moles hemoglobin using the molecular weight:

moles Fe4X = 0.128 g Fe4X x 1 mole/64000 g = 2.00 x 10-6 moles

This contains 4 x 2.00 x 10-6 moles of iron. The number of iron atoms is obtained using Avogadro's number:

Number Fe atoms = 2.00 x 10-6 mole Fe4X  x  4 mole Fe/1 mole Fe4X x 6.022 x 1023 atoms Fe/mole Fe= 4.82 x 1018 Fe atoms

It is interesting that we do not even need to know what the other atoms are in the hemoglobin molecule, much less how many of them there are.  Knowledge of the number of iron atoms per molecule of hemoglobin is enough.
 

For oxygen, which exists in nature as diatomic molecules, O2, the statement "a mole of oxygen" is ambiguous.  Does it mean a mole of oxygen molecules or a mole of oxygen atoms?  These are different things. A mole of oxygen molecules contains 2 moles of oxygen atoms. For elements that exist as molecules, it is best to explicitly state whether molecules or atoms are meant. Thus "1 mole of oxygen molecules" means 6.022 x 1023 O2 molecules, or 2 x 6.022 x 1023 O atoms; "1 mole of oxygen atoms" means 6.022 x 1023 O atoms.