### MOLECULES, MOLES, AND ALL THAT

This site is intended to provide some basic definitions of quantities that are typically studied at great length, with many examples, in an introductory course in chemistry. The first points are about the evidence for the existence of atoms and molecules that emerged from quantitative studies of chemical reactions, beginning a little over 200 years ago.

[Note: The following discussion is not intended to give an accurate picture of experimental methods for obtaining data or of the historical development of our knowledge. It is intended only to explain the logic behind these developments.]

### The Law of Definite Proportions

When it became possible to measure accurately the mass of quantities undergoing a chemical reaction, early chemists discovered the law of definite proportions: If two substances combine to form a third, the ratio of the two combining masses is always the same, even if the actual masses vary. Here's an example: Carbon combines with oxygen to form carbon monoxide. Four grams of oxygen will combine with 3 grams of carbon (ratio = 1.33); 8 grams of oxygen will combine with 6 grams of carbon (same ratio); 14 grams of oxygen will combine with 10.5 grams of carbon (same ratio). If one puts together, in a closed container, 15 grams of oxygen with 10.5 grams of carbon, only 14 grams of oxygen will combine with the carbon; there will be 1 gram of uncombined oxygen left.

Similar definite ratios are found for more complicated reactions, such as when two substances react and the result is two or more other substances.

(You may notice that the ratio in our example, 1.33, is a ratio of two small integers, 4:3. This is true in some cases, but not always.)

A simple explanation for the law of definite proportions is that oxygen and carbon are made of atoms. All oxygen atoms are identical in mass, and all carbon atoms are identical in mass. But the mass of one oxygen atom is 1.33 times the mass of one carbon. So in a reaction, each oxygen atom combines with one carbon to form a carbon monoxide molecule, and whether there are 1000 pairs of atoms doing this, or 10,000, or any number, the ratio of the masses combining is always 1.33.

### The Law of Multiple Proportions

A second example: Iron combines with oxygen to form a compound called "ferrous oxide". The ratio of mass of oxygen to iron is 0.2865. It's not a ratio of two integers, but the ratio is always the same, regardless of how much iron and oxygen there is. Now iron and oxygen also form a different compound called "ferric oxide". In this reaction, the ratio of mass of oxygen to iron is 0.4297. How can this be reconciled with the hypothesis that there are atoms of iron and oxygen, and why are there two different compounds of iron and oxygen anyway?

The answer is suggested if we look at the ratio of the two numbers above:

0.4297/0.2865 = 1.500

This is a ratio of two small integers, 1.500 = 3:2.

Let's assume that in the molecule of ferrous oxide there is one oxygen atom and one iron atom, and the ratio of the mass of the oxygen atom to the mass of the iron atom is 0.2865. Then in ferric oxide, the mass ratio is larger by a factor of 1.5, because there are one and one-half oxygen atoms for each iron atom. Since atoms are not divisible into halves, we should instead say that in the ferric molecule there are 3 oxygen atoms for every 2 iron atoms. Thus, (Fe is the symbol for an iron atom, and O is the symbol for an oxygen atom) the ferrous oxide molecule is FeO, and the ferric oxide molecule is Fe2O3. The subscripts designate the number of atoms in the molecule. If there is no subscript, it is understood to be 1.

This, along with similar analyses of many other chemical reactions, leads to the law of multiple proportions. This law also provides strong evidence for the atom-molecule hypothesis.

### Molecular Weight

In the following I will use the word "molecule" to refer to either an atom or a molecule. On the basis of many experiments and measurements like those discussed above, scientists have been able to determine the relative masses of molecules, that is, the ratio of the mass of molecule A to molecule B. A more convenient way to talk about masses is to arbitrarily assign a particular number to the mass of one molecule, and then give the masses of other molecules relative to that one. The choice has been to define the molecular weight of the carbon atom to be exactly 12.00. [This choice makes the molecular weights of many of the lighter atoms come out close to integers -- but other choices could have been made.] Then the molecular weight of the oxygen atom is 1.33 X 12 = 16; the molecular weight of the hydrogen atom comes out close to 1. For the iron atom the calculation is as follows:

(Mol. Wt. of O)/(Mol Wt. of Fe) = 0.2865

16/(Mol. Wt. of Fe) = 0.2865

Mol Wt. of Fe = 16/0.2865 = 55.85

There is a detail that should be mentioned here: The exact number 12 is assigned to the common isotope of carbon that has 6 protons and 6 neutrons. Carbon found naturally on earth is a mixture of carbon-12 with a very small amount of the isotope carbon-13 (the latter having 6 protons and 7 neutrons). Hence the molecular weight of naturally found carbon is slightly more than 12, specifically 12.011. These molecular weights of naturally occurring elements are what you will find in the periodic table. In this course (or at least in this first semester) we will try to avoid this detail by not being so precise about molecular weights.

Molecular weight does not have a unit, since the "12" assigned to carbon is just a number, not a mass in kilograms or a weight in newtons. Note also that in discussing the quantities in chemical reactions, one is usually using grams or kilograms, so it would be better to think of molecular weight as a measure of mass rather than weight. In fact, some people use the term "molecular mass" rather than "molecular weight", but the latter is more commonly used.

A possible source of confusion is the fact that some common elements form a molecule of two (or more) atoms. Hydrogen, oxygen, and nitrogen come in the forms H2, O2, and N2. These are called diatomic molecules. Note that these gases are not always in the diatomic form. Sometimes they are "monatomic", in the form of single atoms. What form the gas takes depends on the environment it finds itself in -- primarily temperature and pressure. The point about diatomic gases doesn't change any of the discussion above. It's still true that 4 grams of oxygen combines with 3 grams of carbon, 14 grams of oxygen combines with 10.5 grams of carbon, etc. To be clear we should always use the modifiers "diatomic" or "monatomic". The molecular weight of monatomic oxygen is 16; the molecular weight of diatomic oxygen is 32.

Some examples:

 molecular weight monatomic hydrogen 1 diatomic hydrogen 2 carbon 12 monatomic nitrogen 14 diatomic nitrogen 28 monatomic oxygen 16 diatomic oxygen 32 water (H2O) 18 carbon monoxide (CO) 28 carbon dioxide (CO2) 44 ferrous oxide 71.85 (that is, 55.85 + 16) ferric oxide 159.7

The molecular weight of a molecule is just the sum of the molecular weights of the constituent atoms.

### Moles

Since experiments in chemistry and physics are done with real mass units, it is convenient to define a quantity called the mole (also called the "gram-molecular weight"). One mole of a substance is equal to its molecular weight in grams. i.e. 1 mole of carbon is 12 grams, one mole of oxygen is 16 grams, one mole of carbon monoxide is 28 grams, one mole of iron is 55.85 grams. [Note that the gram is used here, not the kilogram. This may tend to cause confusion, but you just have to deal with it.] If I have 36 grams of carbon, what I have is 3 moles. If I have 40 grams of monatomic oxygen, I have 2.5 moles. As an equation,

 number of moles = (mass in grams)/(molecular weight). (1)

If I have 1.2 kg of iron, the mass in grams is 1200, so the number of moles is 1200/55.85 = 21.5. The symbol n will be used for the number of moles. The unit of n is "moles", sometimes abbreviated "mol".

Since the molecular weight of a molecule is proportional to its actual mass, the number of moles is proportional to the number of molecules. To see this, think of the following analogy: Suppose you wanted to determine the number of nails in a barrel. There might be thousands or tens of thousands of nails, making it quite inconvenient to count them. An alternate procedure would be to find the total weight of all the nails in the barrel (for example by weighing a full barrel, weighing an empty barrel, and subtracting); then get a careful weight of one nail; then divide the total weight by the weight of one nail. The result will be the number of nails. If one nail is 8.2 grams (0.0082 kg), and if the total weight is 52 kg, the number of nails is 52/0.0082 = 6.3 X 103. Note, because the measurements are only good to two significant figures, the answer for N is given to two significant figures. You haven't counted the nails accurately to the integer.

Now, looking at Eq. (1), the "mass in grams" is the mass of all the molecules in the barrel, the "molecular weight" is not equal to but proportional to the mass of one molecule. So when you divide you get a number (not equal to but) proportional to the number of molecules. If n is the number of moles and N is the number of molecules, these two quantities can be related by the equation,

 N = N0n. (2)

N0 is a proportionality constant; it is called Avogadro's number. It is a universal constant, in the sense that this number, once it is known, it can be used to relate any quantity of any material to the number of molecules that make it up. The best value today for Avogadro's number is 6.02 X 1023. Note here too that if I use Eq. (2) to calculate N from an experimental measurement of n, the result will not give me a count of molecules to the nearest integer, but only to a certain number of significant figures.

Examples:

A. 10 grams of carbon: n = 10/12 = 0.833 moles, N = 6.02 X 1023 X 0.833 = 5.01 X 1023 molecules. There is not really a unit here. It is the number of molecules, but not given to the integer.

B. 150 grams of diatomic oxygen: n = 150/32 = 4.69 moles, N = 6.02 X 1023 X 4.69 = 2.82 X 1024 molecules. If somehow this 150 grams of oxygen got separated into the monatomic form (molecular weight = 16), you would have n twice as large and also N twice as large.

What if you wanted to know the mass of one molecule? Most likely you would want this mass in kilograms. We found above that 150 grams (or 0.15 kg) of diatomic oxygen consists of 2.82 X 1024 molecules. Therefore the mass in kg of one diatomic molecule is

mass = (0.15)/(2.82 X 1024) = 5.3 X 10-26 kg.

This is one way of calculating the masses of atoms and molecules. Note however that this depends on knowing N0. The chemical experiments that determine molecular weights do not tell you the value of this constant. They only tell you that such a constant exists.

Similarly, Newton's theory of gravitation, along with its explanation of the motions in the solar system, tell you that there is a universal constant, G, of gravitation. They don't tell you the value of G.