The Avogadro Constant

The Avogadro constant (NA or L), sometimes called Avogadro's number, tells us how many elementary entities (usually either molecules or atoms) are present in in one mole of a substance. It is named after Amedeo Carlo Avogadro, an Italian physicist born in Turin in 1776. Avogadro lived and worked in Turin until his death in 1856, with much of that time spent teaching at the University of Turin. His most notable contribution to science was in the field of molecular theory. In 1811, he put forward the hypothesis that the volume of a gas at a given temperature and pressure is proportional to the number of molecules present, regardless of the kind of gas involved. This principle is known as Avogadro's law. Avogadro, however, did not distinguish between atoms and molecules. As far as he was concerned, different substances were composed of different types of molecule, of which the atom was just one particular type (he called it the "elementary molecule").

We now know that the atom is the basic unit of matter from which molecules are composed. A single molecule of a substance consists of two or more atoms, chemically bonded together. We also know that some elemental substances (helium gas for example) are monatomic. This means that their constituent atoms do not bond together chemically to form molecules. Other elemental substances are defined by their molecular structure. A molecule of oxygen gas, for example, consists of two oxygen atoms. For this reason, the symbol for oxygen gas is O2. A molecule of ozone gas, on the other hand, is composed of three oxygen atoms, and has the symbol O3. In a chemical compound, the atoms will be of two or more different types. Water, for example, is a chemical compound. A water molecule consists of two hydrogen atoms chemically bonded with one oxygen atom, so water has the symbol H2O.

According to Avogadro's law, a given volume of ozone gas will contain exactly the same number of molecules as an identical volume of oxygen gas, despite the fact that an ozone gas molecule contains one more atom than an oxygen gas molecule. Furthermore, the same volume of helium gas will contain exactly the same number of atoms (helium is a monatomic gas). Avogadro's theories were largely ignored at the time of their publication, but they have far-reaching implications. If the number of molecules in a given mass of substance can be counted, for example, it should be possible to calculate the mass of a single molecule or (assuming the relative masses of the constituent atoms are known) the mass of an individual atom. Conversely, if the mass of a single atom or molecule is known, it should be possible to calculate the number of molecules or atoms in a given mass of a substance.

The problem, of course, is how to find the number of atoms or molecules in a given amount of substance in the first place, without first knowing the mass of a single atom or molecule. Almost fifty years after Avogadro published his theories, an Italian chemist, Stanislao Cannizzaro (1826 - 1910), recognised the significance of Avagadro's ideas. He carried out various experiments involving the measurement of gas vapour densities, from which he was able to determine the molecular weight of various substances. In 1858 Cannizzaro published his paper Sunto di un corso di Filosofia chimica (which roughly translates as Sketch of a course of chemical philosophy) in which he managed to dispel much of the confusion that existed over the distinction between atoms and molecules, and showed how atomic and molecular weights could be calculated from experimental data.

In 1865, a few years after the publication of Cannizzaro's paper, the Austrian scientist Johann Josef Loschmidt (1821-1895) became the first scientist to estimate the size of the molecules in air at standard temperature and pressure, based on existing theories about the dynamic behaviour of gas molecules. Loschmidt arrived at a value of slightly less than 10-7 centimetres for the diameter of the molecules in air, which we now know to be wrong by a factor of three (the correct value is closer to 0.3 × 10-7 centimetres). Nevertheless, Loschmidt's estimate was remarkably close to the true figure, given the number of approximations he had to make. In 1873, Scottish physicist James Clerk Maxwell used Loschmidt's findings to estimate the number of molecules in one cubic centimetre of gas at standard temperature and pressure as 1.9 × 1019. The modern estimate of this quantity (known as the Loschmidt constant) is approximately 2.69 × 1019.

The value of the Avogadro constant has been revised several times in its history, as more accurate means of determining both the mass and the size of atoms and molecules have been devised, and as more detailed and accurate knowledge of the fundamental nature of matter has been accumulated. It was the French physicist Jean Baptiste Perrin who in 1909 proposed the name Avogadro's number (N) for the number of molecules in exactly thirty-two grams (32 g) of oxygen. The mass of one mole of oxygen was at that time defined to be exactly thirty-two grams (this figure has since been revised to 31.9988 g/mol-1).

We should at this point pause to consider the nature of the unit used for the amount of substance, i.e. the mole. The unit first appeared in the early 1900s as the gram-molecule, or gram-molecular weight. Much earlier than this, in 1803 in fact, the English scientist John Dalton (1766 - 1844) had presented a paper to the Manchester Literary and Philosophical Society that included the first table of relative atomic weights. In Dalton's table, the hydrogen atom (with its single proton) formed the basis for comparison and was assigned the value one. Although the relative atomic weights of the various other elements and chemical compounds recorded in Dalton's table were not particularly accurate (carbon and oxygen being assigned relative atomic weights of 4.3 and 5.5 respectively, for example), he was probably the first scientist to attempt to quantify the difference in mass between the atoms of different elements.

In 1828, the Swedish chemist Jöns Jacob Berzelius (1779 - 1848) compiled his own table of relative atomic weights that included all of the known elements. This time, the element chosen as the standard against which the relative mass of other elements would be compared was oxygen, which (for reasons best known to himself) he assigned the value 100. Nevertheless, his work supported the atomic theory that had been proposed by John Dalton, namely that all inorganic chemical compounds are composed of combinations of whole numbers of atoms. His choice of oxygen as the basis for comparison of the atomic weights of different elements was in part due to his discovery that the relative atomic weights of most elements are not integer multiples of the atomic weight of hydrogen.

By the end of the nineteenth century, the relative atomic and molecular weights of known elements and chemical compounds had been determined with a great deal more accuracy. The gram-molecule was simply the number of grams of a substance that was equivalent to its relative atomic or molecular weight. The relative atomic weight of oxygen was defined at the time as being sixteen (16). This number reflects the fact that an atom of the most abundant isotope of oxygen has eight protons and eight neutrons. The mass of a proton was considered to be the same as that of a neutron, and the mass of an electron was considered to be negligible. An oxygen molecule consists of two oxygen atoms, so the relative molecular weight of oxygen was defined as thirty-two (32). A gram-molecule of oxygen was therefore thirty two grams of oxygen. The gram-molecule of any substance was by definition the amount of a substance that contained the same number of molecules as thirty-two grams of oxygen.

The use of oxygen to define the gram-molecule subsequently proved unsatisfactory because, while physicists were using the common isotope of oxygen (oxygen-16), chemists were using a somewhat heavier naturally-occurring mixture of isotopes (oxygen-16, oxygen-17 and oxygen-18). By 1960, chemists and physicists had agreed to adopt carbon-12 as the standard against which the relative atomic mass of each element would be determined (the word mass being used in preference to the word weight). Carbon-12 was chosen because it is a stable isotope whose mass can be measured very accurately, and because it can be separated relatively easily from other carbon isotopes.

The amount of substance or mole was added as the seventh base unit of the International System of Units in 1971 following the 14th General Conference on Weights and Measures. A mole is defined as the amount of a substance that contains the same number of elementary entities (usually molecules or atoms) as there are atoms in twelve grams of carbon-12. This number is, by definition, the Avogadro constant. It should be pointed out that the Avogadro constant is not a fundamental physical constant in its own right, as is sometimes supposed. It nevertheless provides a relatively straightforward means of determining the number of atoms or molecules in a known mass of a substance. It is particularly important to chemists, for example, for accurately determining the ratio of the different chemical substances required for a particular chemical reaction, since such reactions involve the forming or breaking of chemical bonds between individual atoms.

Carbon-12 has six protons, six neutrons and six electrons. By definition, its relative atomic mass is defined as twelve (12). The unified atomic mass unit, from which the relative atomic mass of all other elements is determined, is one-twelfth (1/12) of the mass of a single atom of carbon-12. To find out how much of a substance we need to make up one mole of that substance (i.e. its molar mass in grams) we just need to know its relative atomic mass. Where do we find that information? In any up-to date copy of the periodic table! A truly excellent interactive version of the periodic table can be found at www.ptable.com.

Despite the use of increasingly sophisticated modern techniques such as mass spectrometry and x-ray diffraction crystallography, determining exactly how many atoms there are in twelve grams of carbon-12 (or any other substance) is far from easy. Avogadro's constant can only be found experimentally, and efforts to find a more accurate value for it are ongoing. These efforts are coordinated by the International Avogadro Coordination (IAC) project. One of the stated aims of the project is to re-define the kilogram (the base unit of mass) in terms of a physical constant, rather than having to rely on the integrity of the International Prototype Kilogram and its official copies. The most recent determination of the Avogadro constant, made in 2015 according to the project's website, is NA = 6.022 140 76(12) × 1023 mol-1 (a relative uncertainty of 2.0 × 10-8). Note that since the adoption of the mole as a physical base unit, the Avogadro constant is no longer treated as a pure number, and has the unit reciprocal mole (mol-1).

Somewhat ironically, Avogadro himself did not attempt to quantify the number of molecules in a given volume of gas, or if he did he made no record of it. The use of the symbol L as an alternative to the symbol NA for the Avogadro constant is almost certainly due to the fact that a quantitative estimate (of the number of molecules in a cubic centimetre of gas) was made possible by the work of Loschmidt. It is worth noting at this point that physicists and chemists tend to have slightly different perspectives on the Avogadro constant, because of the different kinds of work these two groups of scientists are involved in. The nature of the chemical reactions that take place between different chemical elements and compounds depends upon the electrical configuration of those chemical elements and compounds. The electrical configuration of a substance depends on the number of electrons in the outer shells of its constituent atoms, which in turn depends on the number of protons in the nuclei of those atoms.

Since different isotopes of an element differ only in the number of neutrons in their nuclei, the presence of different isotopes of the same element in a chemical reaction does not usually have any bearing on its outcome. The number of atoms or molecules present in each amount of substance involved, on the other hand, is of the utmost importance to chemists, since it determines precisely how many chemical bonds are formed (or broken) in a chemical reaction. Chemists therefore tend to be more interested in the average relative atomic mass of different elements rather than the relative atomic mass of a particular isotope, because the substances they use in their experiments are unlikely to contain pure isotopes of their constituent elements.

Physicists, on the other hand, are often very concerned with the isotopic purity of substances. One well known example is the requirement for specific isotopes of uranium and plutonium in the production of nuclear weapons. The online periodic table linked to above provides a wealth of useful information in this respect. In it you can find the average relative atomic mass for each of the elements in the table, as well as the relative atomic mass of each of the element's isotopes. We have reproduced the figures for the isotopes of the element carbon below. From this you should be able to see that the vast majority of carbon atoms are carbon-12 atoms, and the mass of one mole of pure carbon-12 (its molar mass) will be twelve grams. Note that the average relative atomic mass of carbon is 12.0107, so you could reasonably expect a mole of substance taken from a random sample of carbon to have a mass 12.0107 grams.


Properties of Carbon Isotopes
IsotopeRelative Atomic MassAbundanceHalf-Life
Carbon-1212.098.93%Stable
Carbon-1313.003354837781.07%Stable
Carbon-1414.00324198870%5.7 × 103 years

The carbon-14 isotope is actually very useful, since it is created by the effects of cosmic radiation and occurs naturally in the Earth's atmosphere in trace amounts. Small amounts therefore occur in the tissue of all living organisms (since all life on earth is carbon-based). Carbon-14 has a half-life of 5.7 x 103 years, which means that half of a given quantity of carbon-14 will decay into another element over a period of 5,700 years. Once a living organism dies, it will not ingest any further carbon-14, and any existing carbon-14 in the remains will continue to decay. By measuring the amount of carbon-14 in organic remains, scientists can make a reasonable estimate of how long an organism has been dead, and thus work out at approximately how long ago it actually lived (you have probably heard of this process, which is called carbon-dating).