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In physical sciences, the amount of substance, n, of a sample can be defined informally as the number of some specified elementary entities (usually either atoms, or molecules, or ions, or electrons) present in the sample, but where this number is expressed in terms of some standard batch size.^[1] This is analogous to expressing the number of years in terms of centuries, or the number of doughnuts in terms of dozens;^[2] however, see below for a formal definition. What makes amount-of-substance a useful quantity—even though it may seem that it is a redundant one, given that it seems to contain the same information as does the total number of entities^[3]—is its particular convenience in real-world experimental settings, a convenience that largely stems from the specific choice that was made for the definition of the standard batch size.^[1] In physical sciences, the standard batch size (i.e. the unit of measurement for the amount of substance) is chosen to be equal to the total number of atoms in a 12 g sample composed entirely of carbon-12 atoms; the name of this batch size is the mole. For reasons why this is such a convenient definition, see below.

The number of atoms in 12 g of carbon-12 atoms, i.e. the number of entities in 1 mol, is informally called Avogadro's number. This number has been measured to the precision of about 50 parts per billion (i.e. [the uncertainty in the number]/[the number itself] = ±5×10⁻⁸), and its approximate numerical value is^[4] 6.02214179×10²³. Here we should note that, formally speaking, the term “Avogadro's number” is outdated and one should instead speak of Avogadro constant, which has the same numerical value but has units of mol⁻¹; see below.

It turns out (see below) that for a sample of identical molecules to contain exactly 1 mol of molecules, the mass of the sample should be as many grams as the relative molecular mass of an individual molecule. As an example of this recipe, suppose the molecules in question are water molecules, H₂O. The relative molecular mass of an H₂O molecule is about 2×1 + 1×16 = 18; it follows that to prepare a sample of 1 mol of water molecules, one needs to measure out 18 g of water. The recipe still works for “natural″ samples of molecules, in which not all of the molecules are completely identical because they contain different isotopes of the same elements; see below for an explanation and a sample computation.

1 Rationale for amount-of-substance being measured in terms of number of entities rather than in terms of mass or volume
2 Rationale for preferring amount-of-substance to absolute numbers
3 Analogy with electric charge
4 Analogy with "standard batch size"
5 Basic relations
6 Sample computations with isotopically pure as well as natural samples of molecules
7 Formal definition
8 Terminology
9 Derived quantities
10 History
11 See also
12 References

[edit] Rationale for amount-of-substance being measured in terms of number of entities rather than in terms of mass or volume

Why is it that in chemistry (and in parts of physics, such as in thermodynamics and statistical mechanics), the number of elementary entities (measured in whatever batch sizes) is a more appropriate measure of the “amount of substance” than are mass or volume? In chemistry, this is because in chemical reactions, the reagents react molecule-to-molecule, ion-to-ion, etc. For example, suppose that 16 g of oxygen and 16 g of hydrogen are allowed to react to produce water. Since the atomic weight of an oxygen atom is about 16, and of a hydrogen atom, about 1, it follows (see below) that the amount of substance of oxygen atoms is (approximately) 1 mol, and of hydrogen atoms, (approximately) 16 mol. But each water molecule consists of two hydrogen atoms and one oxygen atom; therefore, after all the oxygen atoms have each bonded to two hydrogen atoms to form water molecules, only 2 mol of hydrogen will have been spent, and 14 mol will remain unused. Here oxygen is said to be the limiting reagent; see the article on stoichiometry for more information.

In thermodynamics, the total number of particles (in absolute terms or in terms of moles) in a system is a basic thermodynamic variable. The difference in the energy of the system as one changes the number of particles by one (while adding or subtracting as much heat, and doing as much positive or negative work, so as to keep the entropy and the volume of the system constant) is a fundamental quantity called the chemical potential. A very standard result into which the number of particles enters directly (and which is taught even at the high school level) is the ideal gas law.

[edit] Rationale for preferring amount-of-substance to absolute numbers

Once the importance of numbers of entities has been established, two obvious questions suggest themselves: (i) why not simply speak in terms of actual numbers of particles^[3] (i.e. batch size=1), and (ii) if the batch size must be large, then why not simply specify it numerically (say, that it is equal to exactly 6022×10²⁰) rather than defining it through a standard-mass sample of a standard atomic species (i.e. a 12 g-sample of carbon-12 atoms).

The answer to both questions is that with the chosen definition of the standard batch size, the measurement of the amount of substance is a measurement of a ratio of total numbers of entities (number of entities in the sample of interest divided by the number of atoms in 12 g of carbon-12). This is very convenient because what chemists actually measure is masses and volumes of samples, from which they are able very precisely to compute the ratios of the numbers of entities in different samples (thanks to such chemical principles as the Law of definite proportions, the Law of multiple proportions, and the Law of conservation of mass). In particular, it is possible to measure the ratios of the numbers of entities in chemical samples more precisely than it is possible to count the absolute numbers of entities. This has the peculiar-sounding consequence that it is possible to know the amount of substance (in moles) in the sample more precisely than it is possible to know the absolute number of entities in the sample. To make this more intuitive, imagine that one had no idea how many carbon-12 atoms there are in 12 g of carbon-12. Nevertheless, it is possible that one could find out with great precision that the ratio of the number of atoms in some other sample to that of the number of atoms in 12 g of carbon-12 is, say 2.1213. Then one could confidently say that the first sample has 2.1213 moles of atoms, without having any idea how many that is in absolute terms.

This is somewhat similar to the situation that existed in Solar system astronomy for a time, where one did not know very well the absolute distances of the planets to the Sun, but one did know quite precisely the ratios of these distances to each other and, in particular, their ratios to the distance from the Earth to the Sun. The latter distance became known as the astronomical unit, and one way to describe the situation is to say that one knew quite precisely all the distances in terms of the astronomical unit, while the length of the astronomical unit itself was known quite poorly. (Today the situation in astronomy is in fact even more complicated, as distances on various astronomical scales—e.g. distances within a galaxy vs. distances between galaxies—are measured in terms of a multitude of standard lengths; see cosmic distance ladder for more information.)

Therefore, once one starts to pay attention to measurement uncertainty, one realizes that a measurement of the amount of substance is not equivalent to a measurement of the absolute number of entities. For while it is true that the mean measured value of one can be deduced from the mean measured value of the other, the situation is different for their associated uncertainties. Suppose that the (relative) uncertainty in Avogadro's number is indeed larger than that in the amount-of-substance measurement. As one converts from the amount-of-substance value to the absolute-number value, the uncertainties in the latter will be dominated by the uncertainty in Avogadro's number, and so will be greater than the uncertainty in the amount-of-substance value (see propagation of uncertainty). Thus, there is a loss of information in this conversion: if one only records the resulting absolute-number value (and its uncertainty), and then tries to reverse the computation and determine the amount-of-substance from the absolute number, the uncertainty in the resulting amount-of-substance value will still be dominated by the uncertainty in Avogadro's number. In particular, the uncertainty in the computed amount-of-substance will be larger than what was achieved in the original amount-of-substance measurement: information has been lost in the back-and-forth conversion.

Thus, one should record the amount-of-substance values and not the converted absolute-number values; subsequently, as the precision with which Avogadro's number is known grows, so will the precision with which one can determine the absolute-number from the recorded amount-of-substance measurements—at least until the precision of measurement of Avogadro's number starts to exceed the precision of measurement of the amount of substance.

Once that happens, which could be in a matter of years, then the kilogram will be likely be redefined as a standard number of carbon-12 atoms^[1] (see “Proposed future directions: Atom-counting approaches” in the article kilogram), instead of the mole being defined in terms of a standard mass of a standard sample.

To summarize, the definitions of the amount of substance and of its unit, the mole, have been chosen to “to allow measurements to be described in the most simple and adequate way.”^[1]

[edit] Analogy with electric charge

Amount of substance is not the only physical quantity which is measured differently at the microscopic and macroscopic scales. It has long been known that electric charge is quantised at the atomic or molecular scale: that is, the charge can only take certain values, which are integer multiples of the elementary charge e. However, e has a very small value, 1.602 176 487(40) × 10⁻¹⁹ C, too small to be useful for laboratory-scale measurements.

Consider the example of an 800-watt domestic microwave oven running for two minutes from a 230-volt mains power socket. The current is ⁸⁰⁰⁄₂₃₀ ≈ 3.48 amperes, and this current passes for 60 × 2 = 120 seconds. The total charge that passes through the oven is simply the current multiplied by the time, about 417 coulombs.

Knowing the value of the elementary charge, it is possible to calculate how many electrons passed through the oven: in this case, it is about 2.6 × 10²¹ electrons. However, the charge wasn't measured by counting out 2.6 × 10²¹ electrons: such a feat would be impossible. The charge was measured by multiplying a current by a time, which is also how a domestic electricity meter operates. The electric current is measured using Ampère's force law or similar methods.

Just as it is not necessary to know the value of the elementary charge to measure electric quantities, it is not necessary to know the value of the Avogrado constant to measure amount of substance. Electric charge is proportional to the number of elementary charges, just as amount of substance is proportional to the number of elementary entities. The constant of proportionality is determined experimentally in each case, but is unimportant for normal laboratory measurements.

[edit] Analogy with "standard batch size"

A common analogy for the concept of amount of substance is to claim that its unit (the mole) is analogous to a standard batch size in manufacturing industries. The analogy goes that it is usual to buy paper in reams or eggs in dozens, just as it is usual to measure chemical compounds in moles. The analogy is fallacious on an everyday scale, which is the only scale on which the mole is actually used as a unit.

Sheets of paper in a ream or eggs in a dozen are physically counted one-by-one, either mechanically or by hand: atoms and molecules cannot be counted in the numbers necessary to make up a mole. The IBM Roadrunner was the world's fastest computer in June 2008, capable of performing × 10¹⁵ floating point operations per second: if it could count one atom with every floating point operation, it would take nineteen years to count the number of carbon atoms in one mole of carbon (exactly 12 grams of carbon-12, slightly more if natural carbon is used). A laboratory technician, on the other hand, can measure out one mole of carbon in a few tens of seconds, depending on training and on the level of precision required.

A more accurate analogy would be the official gold reserves held by central banks. Each central bank knows the mass of gold that it holds in reserve, but the value of that gold in any given currency changes from day to day. When the central bank publishes its accounts, it must apply a conversion factor to convert the mass of gold into a monetary value so that it can incorporate it into the rest of its accounting scheme. Under this analogy, amount of substance is the "chemical value" of a given sample.

[edit] Basic relations

A basic consequence of the definitions of the mole and of the atomic weight (equivalently, of the relative molecular mass for molecules) is that for a sample of identical atoms to contain exactly 1 mol of atoms, the mass of the sample should be as many grams as the atomic weight of an individual atom. This practically convenient correspondence is the result of the fact that it is the mass of the same atomic species, carbon-12 atom, that is used in the definitions of both the mole and of the atomic weight.

To see this, let $X$ be an individual atom (or a molecule, or an ion, etc.). Let $m (X)$ be the mass of $X$ in grams. For example, $m (12 C)$ is the mass of a single carbon-12 atom.

Next we need the definition of the atomic mass unit u, which is given by

$1\;\text{u} = (1/12)\;m(^{12}C)\approx 1.66 \times 10^{-24}\;\text{g}.$

(More precisely,^[4] u = 1.660538782(83)×10⁻²⁴ g. Note that u is usually written with units of kg; however, since the definition of the mole involves 12 grams of carbon-12, it follows that when talking about moles, grams are more convenient.)

Atomic weight $A r$ of an atom $X$ is the number of times the mass of the atom $X$ is greater than 1 u. In symbols,

$A_{\text{r}}(X)=\frac{m(X)}{(1/12)\;m(^{12}C)}=\frac{m(X)}{1\;\text{u}}.$

Note that $A r$ is a dimensionless quantity, and that

$m(X)=A_{\text{r}}(X) \times (1\;u).$

Now since by definition of the mole, 1 mol of atoms of $12 C$ has a mass of 12 g, we have

$m(^{12}C)\times \tilde{N}_{\text{A}}=12\;g,$

where $\tilde{N}_{\text{A}}$ may be called Avogadro's number, a dimensionless quantity equal to about 6.022×10²³. (The reason for the tilde is that the tilde-less symbol, $N A$ , already has a subtly different meaning: it is the Avogadro constant, which, unlike $\tilde{N}_{\text{A}}$ , has units: mol⁻¹. The value of $N A$ is 6.022×10²³ mol⁻¹. While the insistence on the distinction between $\tilde{N}_{\text{A}}$ and $N A$ may seem like splitting hairs—after all, $N_{\text{A}}=\tilde{N}_{\text{A}}\,\text{mol}^{-1}$ —it is necessary if the equations are to be completely correct; see below for more information on this issue.)

Upon dividing both sides by $12,$ we get the fundamental relationship

$(1\;u)\times \tilde{N}_{\text{A}}=1\;g.$

It follows that the mass of 1 mol of atoms $X$ is given by

$m(X)\;\tilde{N}_{\text{A}}=[A_{\text{r}}(X)\;(1\;u)]\;\tilde{N}_{\text{A}}$

$=A_{\text{r}}(X)\;[(1\;u)\;\tilde{N}_{\text{A}}]$

$=A_{\text{r}}(X)\;(1\;g).$

This equality says precisely what we wanted to show: for a sample of $X$ -atoms to have exactly 1 mol of atoms, the mass of the sample should be as many grams as $A r (X)$ , the atomic weight of $X$ (recall that atomic weight is a dimensionless quantity). Example: since the atomic weight of ¹⁶O, oxygen-16, is about 16, it follows that in 16 g of ¹⁶O there will be exactly 1 mol of ¹⁶O atoms, i.e. about 6.022×10²³ ¹⁶O atoms.

A related equality gives the amount of substance (in moles), $n$ , contained in a sample of $y$ grams of atoms $X$ (where $y$ is just a number):

$n=\frac{y \times (1\;\text{g})}{m(X)\;\tilde{N}_{\text{A}}}=\frac{y\;(1\;\text{g})}{A_{\text{r}}(X)\;(1\;g)},$ or

$n=\frac{y}{A_{\text{r}}(X)}.$

[edit] Sample computations with isotopically pure as well as natural samples of molecules

[edit] Isotopically pure samples

Consider first a particular type of isotopically pure H₂O molecule, say one in which both hydrogen atoms are hydrogen-1 (atomic weight of 1.007825) and the oxygen atom is oxygen-16 (atomic weight of 15.9949146). The relative molecular mass of this molecule is 2×1.007825 + 15.9949146 = 18.0105646; therefore, a sample of 1 mol of (¹H)₂(¹⁶O) will have a mass of 18.0105646 g. In contrast, if all the hydrogen is actually deuterium (atomic weight of 2.01410178), then 1 mol of (²H)₂(¹⁶O) will have a mass of 2×2.01410178 + 15.9949146 = 20.02311816 g.

[edit] Natural samples

A natural sample of water will contain molecules with all combinations of all isotopes of hydrogen and oxygen. The recipe for computing the mass of 1 mol of natural water is simple: for atomic weights of hydrogen and oxygen, one should use standard atomic weights (see the next paragraph), which are precisely what is usually given in periodic tables and other chemistry references. In the example of water, standard atomic weights of hydrogen and oxygen atoms are, respectively, 1.00794 and 15.9994, and so a natural sample of 1 mol of water will have a mass of 2×1.00794 + 15.9994 = 18.01528 g.

[edit] A justification of the procedure in the case of natural samples

[edit] Atomic samples

A standard atomic weight of an element is the weighted average of atomic weights of individual stable isotopes, with averaging weights set to the natural abundance of each isotope; see this section for an example of how standard atomic weight is computed in the case of silicon. An important point stressed in that section is that natural abundances are necessarily estimates because different samples of an element generally contain a range of abundances.

To show that 1 mol of a natural atomic sample has the mass which is as many grams as the standard atomic weight of the atom, we proceed as follows. The number of atoms in the sample is $\tilde{N}_{\text{A}}$ , the Avogadro's number (see above for a discussion on the subtle—though in the present context irrelevant—distinction between the Avogadro's number and the Avogadro constant). For simplicity suppose there are 2 isotopes of the atom $X$ , labeled $α X$ and $β X$ , with abundances $w α$ and $w_{\beta}\,.$ We assume that the abundances are expressed and numbers rather than percentages; so that $w α + w β = 1.$ Then out of $\tilde{N}_{\text{A}}$ atoms in total, $w_{\alpha}\,\tilde{N}_{\text{A}}$ of them will be $α X$ , and $w_{\beta}\,\tilde{N}_{\text{A}}$ of them will be $β X .$ . Let the atomic weights of the two isotopes be $A r (α X)$ and $A r (β X),$ so that their masses are $A_{\text{r}}(^{\alpha}X)\,(1 \text{u})$ and $A_{\text{r}}(^{\beta}X)\,(1 \text{u}),$ respectively, where $u$ is the atomic mass unit (see above). Then the total mass of the sample is given by

$m_{\text{1 mol of X}}=(\text{number of atoms}\;\;^{\alpha}X)\times(\text{mass of 1 atom of}\;\;^{\alpha}X)$

$+(\text{number of atoms}\;\;^{\beta}X)\times(\text{mass of 1 atom of}\;\;^{\beta}X)$

$=\left[w_{\alpha}\,\tilde{N}_{\text{A}}\right]\times\left[A_{\text{r}}(^{\alpha}X)\,(1 \text{u})\right]+\left[w_{\beta}\,\tilde{N}_{\text{A}}\right]\times\left[A_{\text{r}}(^{\beta}X)\,(1 \text{u})\right]$

$=\tilde{N}_{\text{A}}\,\left[w_{\alpha}\,A_{\text{r}}(^{\alpha}X)+w_{\beta}\,A_{\text{r}}(^{\beta}X)\right]\,(1 \text{u}).$

But the quantity in the brackets is just the standard atomic weight of $X$ , $\bar{A}_{\text{r}}(X),$ and so

$m_{\text{1 mol of X}}=\tilde{N}_{\text{A}}\,(1 \text{u})\,\bar{A}_{\text{r}}(X).$

From the above, we know that $\tilde{N}_{\text{A}}\,(1 \text{u})=1 \text{g}$ , and so

$m_{\text{1 mol of X}}=(1 \text{g})\,\bar{A}_{\text{r}}(X),$

i.e., 1 mol of a natural atomic sample has the mass which is as many grams as the standard atomic weight of the atom. The computation is completely analogous if there are more than two isotopes.

[edit] Molecular samples

Analogously, one defines the standard relative molecular mass $\bar{M}_{\text{r}}$ as the weighted average over all isotopically pure molecules. The demonstration that 1 mol of a natural sample of the molecule then has the mass which is as many grams as the standard relative molecular mass is indistinct from the case of an atomic sample with many isotopes. What remains to be shown is that the standard relative molecular mass can be computed from the standard atomic weights using the recipe that is a mirror image of the recipe in the case of a molecule composed of particular isotopes. On the example of water, this recipe reads

$\bar{M}_{\text{r}}(\text{H}_{2}\text{O})=2\,\bar{A}_{\text{r}}(H)+\bar{A}_{\text{r}}(O).$

For definiteness, consider the HCl molecule. Hydrogen atom can be either ¹H or ²H, while the chlorine atom can be either ³⁵Cl or ³⁷Cl. If the rate of chemical reaction between hydrogen and chlorine does not depend on which isotopes enter into it (which is often approximately true, but is generally not strictly true because of effects such as kinetic fractionation, equilibrium fractionation, and mass-independent fractionation), then the probability that an ¹H atom reacts with an ³⁷Cl atom (rather than with a ³⁵Cl atom) is the same as that an ²H atom reacts with an ³⁷Cl atom—namely, it is equal to the natural abundance of ³⁷Cl in either case. But the hydrogen isotopes have their own abundances, and then it follows that the abundance of, say, (²H)(³⁷Cl) is just the product of the natural abundances of ²H and ³⁷Cl. Let $M r (X)$ be the relative molecular mass of isotopically pure molecule X, and $A r (X)$ , the atomic weight of the isotope X; let $w (Y)$ the natural abundance of the isotope (or isotopically pure molecule) Y, expressed not as a percentage but as a number, so that, e.g. $w (1 H) + w (2 H) = 1$ . It will follow that the standard relative molecular mass of HCl, $\bar{M}_{\text{r}}(\text{HCl}),$ is simply computed from the standard atomic weights of hydrogen and chlorine, $\bar{A}_{\text{r}}(\text{H})$ and $\bar{A}_{\text{r}}(\text{Cl}):$

$\bar{M}_{\text{r}}(\text{HCl})=\bar{A}_{\text{r}}(\text{H})+\bar{A}_{\text{r}}(\text{Cl}).$

To see this, we begin with the definition of $\bar{M}_{\text{r}}(\text{HCl}):$

$\bar{M}_{\text{r}}(\text{HCl})=w(^{1}\text{H}\,^{35}\text{Cl})\;M_{\text{r}}(^{1}\text{H}\,^{35}\text{Cl})+ w(^{1}\text{H}\,^{37}\text{Cl})\;M_{\text{r}}(^{1}\text{H}\,^{37}\text{Cl})$

$+w(^{2}\text{H}\,^{35}\text{Cl})\;M_{\text{r}}(^{2}\text{H}\,^{35}\text{Cl})+ w(^{2}\text{H}\,^{37}\text{Cl})\;M_{\text{r}}(^{2}\text{H}\,^{37}\text{Cl})$

To proceed, it is convenient to use summation notation, where $j$ ranges over 1 and 2, while $k$ ranges over 35 and 37. The definition above is then written as

$\bar{M}_{\text{r}}(\text{HCl})=\sum_{j}\,\sum_{k}\,w(^{j}\text{H}\,^{k}\text{Cl})\;M_{\text{r}}(^{j}\text{H}\,^{k}\text{Cl}).$

Now we use the assumption, discussed above, that the abundance of, say, (²H)(³⁷Cl) is just the product of the natural abundances of ²H and ³⁷Cl. This means that for all $j$ and $k,$ we have $w(^{j}\text{H}\,^{k}\text{Cl})=w(^{j}\text{H})\,w(^{k}\text{Cl}).$ We also use the fact that, e.g.,

$M_{\text{r}}(^{1}\text{H}\,^{37}\text{Cl})=A_{\text{r}}(^{1}\text{H})+A_{\text{r}}(^{37}\text{Cl}).$

We get

$\bar{M}_{\text{r}}(\text{HCl})=\sum_{j}\,\sum_{k}\,w(^{j}\text{H})\,w(^{k}\text{Cl})\;(A_{\text{r}}(^{j}\text{H})+A_{\text{r}}(^{k}\text{Cl}))$

$=\sum_{j}\,\sum_{k}\,w(^{j}\text{H})\,w(^{k}\text{Cl})\;A_{\text{r}}(^{j}\text{H})+\sum_{j}\,\sum_{k}\,w(^{j}\text{H})\,w(^{k}\text{Cl})\;A_{\text{r}}(^{k}\text{Cl})\;$

$=\left[\sum_{j}\,w(^{j}\text{H})\;A_{\text{r}}(^{j}\text{H})\right]\,\sum_{k}\,w(^{k}\text{Cl})+\left[\sum_{j}\,w(^{j}\text{H})\right]\,\sum_{k}\,w(^{k}\text{Cl})\;A_{\text{r}}(^{k}\text{Cl}),$

where the two equalities follow by applying the property of distributivity. Now we recall that

$\sum_{k}\,w(^{k}\text{Cl})=\sum_{j}\,w(^{j}\text{H})=1,$

as well as that

$\bar{A}_{\text{r}}(\text{H})=\sum_{j}\,w(^{j}\text{H})\;A_{\text{r}}(^{j}\text{H})$

and

$\bar{A}_{\text{r}}(\text{Cl})=\sum_{k}\,w(^{k}\text{Cl})\;A_{\text{r}}(^{k}\text{Cl}).$

It follows that

$\bar{M}_{\text{r}}(\text{HCl})=\bar{A}_{\text{r}}(\text{H})+\bar{A}_{\text{r}}(\text{Cl}),$

which is what we wanted to show.

[edit] Formal definition

The International System of Units (SI) defines the amount of substance to be a base physical quantity that is proportional to the number of elementary entities present. The constant of proportionality depends on the unit chosen for amount of substance; however, once this choice is made, the constant is the same for all possible kinds elementary entities.^[5] The identity of “elementary entities” depends on the context and it must be stated; usually these entities are one of the following: atoms, molecules, ions, or elementary particles such as electrons. Amount of substance is sometimes referred to as chemical amount.

The SI unit for amount of substance, which is one of SI base units (since it is a unit of a base quantity) is the mole (symbol: mol). The mole is defined as the amount of substance that has an equal number of elementary entities as there are atoms in 12 g of carbon-12.^[6] That number is equivalent to the Avogadro constant, N_A, which has the value^[4] of 6.02214179(30)×10²³ mol⁻¹. The precision works out to be about 50 part per billion and is limited by the uncertainty in the value of the Planck constant. Note that under the SI, Avogadro constant has units and it is thus improper to refer to it as “Avogadro's number,″ since a “number” is supposed to be dimensionless. With mole as the unit, the constant of proportionality between amount of substance and the number of elementary entities is 1/N_A.

There is no reason to expect that the mass of any integer number of carbon-12 atoms should add up to exactly 12 g, from which it follows that the exact Avogadro's number is not necessarily an integer. After all, the definition of a gram is that it is (1/1000) of a kilogram, and the definition of a kilogram is that it is the mass of the prototype kilogram, a particular solid cylinder, kept in a vault in France, that is made of a platinum-iridium alloy and that therefore bears no particular relation to carbon-12 atoms (see also kilogram).

Because one should distinguish between physical quantities and their units, it is improper to refer to amount of substance as “the number of moles” or the “mole number,” just as it is improper to refer to the physical quantity of length as “the meter number.”^[3]

The only other unit of amount of substance in current use is the pound mole (symbol: lb-mol.), which is sometimes used in chemical engineering in the United States.^[7]^[8]:1 lb-mol. ≡ 453.592 37 mol (this relation is exact, from the definition of the international avoirdupois pound).

[edit] Terminology

Look up amount in Wiktionary, the free dictionary.

When quoting an amount of substance, it is necessary to specify the entity involved (unless there is no risk of ambiguity). One mole of chlorine could refer either to chlorine atoms (as in 58.44 g of sodium chloride) or to chlorine molecules (as in 22.711 dm³ of chlorine gas at STP). The simplest way to avoid ambiguity is to replace the term "substance" by the name of the entity and/or to quote the empirical formula.^[5]^[9] For example:

amount of chloroform, CHCl₃

amount of sodium, Na

amount of hydrogen (atoms), H

n(C₂H₄)

This can be considered to be a technical definition of the word "amount", a usage which is also found in the names of certain derived quantities (see below).

[edit] Derived quantities

When amount of substance enters into a derived quantity, it is usually as the denominator: such quantities are known as "molar quantities".^[10] For example, the quantity which describes the volume occupied by a given amount of substance is called the molar volume, while the quantity which describes the mass of a given amount of substance is the molar mass. Molar quantities are sometimes denoted by a subscript Latin "m" in the symbol,^[10] e.g. C_p,m, molar heat capacity at constant pressure: the subscript may be omitted if there is no risk of ambiguity, as is often the case in pure chemistry.

The main derived quantity in which amount of substance enters into the numerator is amount of substance concentration, c. This name is often abbreviated to "amount concentration",^[11] except in clinical chemistry where "substance concentration" is the preferred term^[12] (to avoid any possible ambiguity with mass concentration). The name "molar concentration" is incorrect,^[13] if commonly used.

[edit] History

The alchemists, and especially the early metallurgists, probably had some notion of amount of substance, but there are no surviving records of their having generalised the idea beyond a set of "recipes". Lomonosov in 1758 questioned the idea that mass was the only measure of the quantity of matter,^[14] but only in relation to his theories on gravitation. The development of the concept of amount of substance was coincidental with, and vital to, the birth of modern chemistry.

1777: Wenzel publishes Lessons on Affinity, in which he demonstrates that the proportions of the "base component" and the "acid component" (cation and anion in modern terminology) remain the same during reactions between two neutral salts.^[15]
1789: Lavoisier publishes Treatise of Elementary Chemistry, introducing the concept of a chemical element and clarifying the Law of conservation of mass for chemical reactions.^[16]
1792: Richter publishes the first volume of Stoichiometry or the Art of Measuring the Chemical Elements (publication of subsequent volumes continues until 1802). The term "stoichiometry" used for the first time. The first tables of equivalent weights are published for acid–base reactions. Richter also notes that, for a given acid, the equivalent mass of the acid is proportional to the mass of oxygen in the base.^[15]
1794: Proust's Law of definite proportions generalises the concept of equivalent weights to all types of chemical reaction, not simply acid–base reactions.^[15]

With the concept of atoms came the notion of atomic weight. While many were sceptical about the reality of atoms, chemists quickly found atomic weights to be an invaluable tool in expressing stoichiometric relationships.

1805: Dalton publishes his first paper on modern atomic theory, including a "Table of the relative weights of the ultimate particles of gaseous and other bodies".^[17]
1808: Publication of Dalton's A New System of Chemical Philosophy, containing the first table of atomic weights (based on H = 1).^[18]
1809: Gay-Lussac's Law of combining volumes, stating an integer relationship between the volumes of reactants and products in the chemical reactions of gases.^[19]
1811: Avogadro hypothesizes that equal volumes of different gases contain equal numbers of particles, now known as Avogadro's law.^[20]
1813/1814: Berzelius publishes the first of several tables of atomic weights based on the scale of O = 100.^[15]^[21]^[22]
1815: Prout publishes his hypothesis that all atomic weights are integer multiple of the atomic weight of hydrogen.^[23] The hypothesis is later abandoned given the observed atomic weight of chlorine (approx. 35.5 relative to hydrogen).
1819: Dulong–Petit law relating the atomic weight of a solid element to its specific heat capacity.^[24]
1819: Mitscherlich's work on crystal isomorphism allows many chemical formulae to be clarified, resolving several ambiguities in the calculation of atomic weights.^[15]

The ideal gas law was the first to be discovered of many relationships between the number of atoms or molecules in a system and other physical properties of the system, apart from its mass. However this was not sufficient to convince all scientists that atoms and molecules had a physical reality, rather than simply being useful tools for calculation.

1834: Clapeyron states the ideal gas law.^[25]
1834: Faraday states his Laws of electrolysis, in particular that "the chemical decomposing action of a current is constant for a constant quantity of electricity".^[26]
1856: Krönig derives the ideal gas law from kinetic theory.^[27] Clausius publishes an independent derivation the following year.^[28]
1860: The Karlsruhe Congress debates the relation between "physical molecules", "chemical molecules" and atoms, without reaching consensus.^[29]
1865: Loschmidt makes the first estimate of the size of gas molecules and hence of number of molecules in a given volume of gas, now known as the Loschmidt constant.^[30]
1886: van't Hoff demonstrates the similarities in behaviour between dilute solutions and ideal gases.
1887: Arrhenius describes the dissociation of electrolyte in solution, resolving one of the problems in the study of colligative properties.^[31]
1893: First recorded use of the term "mole" to describe a unit of amount of substance, by Ostwald in a university textbook.^[32]
1897: First recorded use of the term "mole" in English.^[33]
1901: van't Hoff receives the very first Nobel Prize in Chemistry, partly for determining the laws of osmotic pressure.^[34]
1903: Arrhenius receives the Nobel Prize in Chemistry, in part for his work on the dissociation of electrolytes.^[35]

By the turn of the twentieth century, the supporters of atomic theory had more or less won the day, but many questions remained, not least the size of atoms and their number. The development of mass spectrometry of one of the techniques that revolutionized the way that physicists and chemists made connections between the microscopic world of atoms and molecules and the macroscopic observations of laboratory experiments.

1905: Einstein's paper on Brownian motion dispels any last doubts on the physical reality of atoms, and opens the way for an accurate determination of their mass.^[36]
1909: Perrin coins the name "Avogadro constant" and makes an estimate of its value.^[37]
1913: Discovery of isotopes of non-radioactive elements by Soddy^[38] and Thomson.^[39]
1914: Richards receives the Nobel Prize in Chemistry for "for his determinations of the atomic weight of a large number of elements".^[40]
1920: Aston proposes the whole number rule, an updated version of Prout's hypothesis.^[41]
1921: Soddy receives the Nobel Prize in Chemistry "for his work on the chemistry of radioactive substances and investigations into isotopes".^[42]
1922: Aston receives the Nobel Prize in Chemistry "for his discovery of isotopes in a large number of non-radioactive elements, and for his whole-number rule".^[43]
1926: Perrin receives the Nobel Prize in Physics, in part for his work in measuring the Avogadro constant.^[44]
1959/1960: Unified atomic weight scale based on ¹²C = 12 adopted by IUPAP and IUPAC.^[45]
1968: The mole recommended for inclusion in the International System of Units (SI) by the International Committee for Weights and Measures (CIPM).^[6]
1972: The mole approved as the SI base unit of amount of substance.^[6]

[edit] See also

[edit] References

^ ^a ^b ^c ^d De Bièvre, P. and P. D. P. Taylor (1997). "Traceability to the SI of amount-of-substance measurements: from ignoring to realizing, a chemist's view". Metrologia 34: 75. doi:10.1088/0026-1394/34/1/10.
^ "Chemical reaction". Encyclopedia of science and technology. Routledge. 2001. http://books.google.com/books?id=Cmyt74ury34C&pg=PT78&dq.
^ ^a ^b ^c McGlashan, M. L. (1977). "Amount of substance and the mole". Phys. Educ. 12: 278. doi:10.1088/0031-9120/12/5/001.
^ ^a ^b ^c Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. doi:10.1103/RevModPhys.80.633. http://physics.nist.gov/cuu/Constants/codata.pdf. Direct link to value.
^ ^a ^b International Union of Pure and Applied Chemistry. "amount of substance, n". Compendium of Chemical Terminology Internet edition.
^ ^a ^b ^c International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), pp. 114–15, ISBN 92-822-2213-6, http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf
^ Talty, John T. (1988). Industrial Hygiene Engineering: Recognition, Measurement, Evaluation, and Control. William Andrew. pp. 142. ISBN 0815511752.
^ Lee, C.C. (2005). Environmental Engineering Dictionary (4th ed.). Rowman & Littlefield. pp. 506. ISBN 086587848X.
^ International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 46. Electronic version.
^ ^a ^b International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 7. Electronic version.
^ International Union of Pure and Applied Chemistry. "amount-of-substance concentration". Compendium of Chemical Terminology Internet edition.
^ International Union of Pure and Applied Chemistry (1996). "Glossary of Terms in Quantities and Units in Clinical Chemistry." Pure Appl. Chem. 68:957–1000.
^ "Molar concentration" should refer to a concentration per mole, i.e. an amount fraction. The use of "molar" as a unit, equal to 1 mol/dm³, symbol M, is frequent, but not (as of May 2007) completely condoned by IUPAC: See International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 42 (n. 15). Electronic version.
^ Lomonosov, Mikhail (1758/1970). "Mikhail Vasil'evich Lomonosov on the Corpuscular Theory". in Leicester, Henry M.. Cambridge, MA: Harvard University Press. pp. 224–33. http://www.archive.org/details/mikhailvasilevic017733mbp.
^ ^a ^b ^c ^d ^e "Atome". Grand dictionnaire universel du XIXe siècle. 1. Paris: Pierre Larousse. 1866. pp. 868–73. . (French)
^ Lavoisier, Antoine (1789). Traité élémentaire de chimie, présenté dans un ordre nouveau et d'après les découvertes modernes. Paris: Chez Cuchet. http://gallica.bnf.fr/ark:/12148/bpt6k3930k.table. . (French)
^ Dalton, John (1805). "On the Absorption of Gases by Water and Other Liquids". Memoirs of the Literary and Philosophical Society of Manchester, 2nd Series 1: 271–87. http://web.lemoyne.edu/~giunta/dalton52.html.
^ Dalton, John (1808). A New System of Chemical Philosophy. Manchester. http://www.archive.org/details/newsystemofchemi01daltuoft.
^ Gay-Lussac, Joseph Louis (1809). "Memoire sur la combinaison des substances gazeuses, les unes avec les autres". Mémoires de la Société d'Arcueil 2: 207. English translation.
^ Avogadro, Amadeo (1811). "Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique 73: 58–76. English translation.
^ Excerpts from Berzelius' essay: Part II; Part III.
^ Berzelius' first atomic weight measurements were published in Swedish in 1810: Hisinger, W.; Berzelius, J.J. (1810). "Forsok rorande de bestamda proportioner, havari den oorganiska naturens bestandsdelar finnas forenada". Afh. Fys., Kemi Mineral. 3: 162.
^ Prout, William (1815). "On the relation between the specific gravities of bodies in their gaseous state and the weights of their atoms". Annals of Philosophy 6: 321–30. http://web.lemoyne.edu/~giunta/PROUT.HTML.
^ Petit, Alexis Thérèse; Dulong, Pierre-Louis (1819). "Recherches sur quelques points importants de la Théorie de la Chaleur". Annales de Chimie et de Physique 10: 395–413. English translation
^ Clapeyron, Émile (1834). "Puissance motrice de la chaleur". Journal de l'École Royale Polytechnique 14 (23): 153–90.
^ Faraday, Michael (1834). "On Electrical Decomposition". Philosophical Transactions of the Royal Society. http://chimie.scola.ac-paris.fr/sitedechimie/hist_chi/text_origin/faraday/Faraday-electrochem.htm.
^ Krönig, August (1856). "Grundzüge einer Theorie der Gase". Annalen der Physik 99: 315–22. doi:10.1002/andp.18561751008. http://gallica.bnf.fr/ark:/12148/bpt6k15184h/f327.table.
^ Clausius, Rudolf (1857). "Ueber die Art der Bewegung, welche wir Wärme nennen". Annalen der Physik 100: 353–79. doi:10.1002/andp.18571760302. http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.table.
^ Wurtz's Account of the Sessions of the International Congress of Chemists in Karlsruhe, on 3, 4, and 5 September 1860.
^ Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der kaiserlichen Akademie der Wissenschaften Wien 52 (2): 395–413. English translation.
^ Arrhenius, Svante (1887). Zeitschrift fur physikalische Chemie 1: 631. English translation.
^ Ostwald, Wilhelm (1893). Hand- und Hilfsbuch zur ausführung physiko-chemischer Messungen. Leipzig.
^ Helm, Georg; (Transl. Livingston, J.; Morgan, R.) (1897). The Principles of Mathematical Chemistry: The Energetics of Chemical Phenomena. New York: Wiley. pp. 6.
^ Odhner, C.T. (December 10, 1901). Presentation Speech for the 1901 Nobel Prize in Chemistry.
^ Törnebladh, D.R. (December 10, 1903). Presentation Speech for the 1903 Nobel Prize in Chemistry.
^ Einstein, Albert (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen". Annalen der Physik 17: 549–60. doi:10.1002/andp.19053220806. http://www3.interscience.wiley.com/homepages/5006612/549_560.pdf.
^ Perrin, Jean (1909). "Mouvement brownien et réalité moléculaire". Annales de Chimie et de Physique, 8^e Série 18: 1–114. Extract in English, translation by Frederick Soddy.
^ Soddy, Frederick (1913). "The Radio-elements and the Periodic Law". Chemical News 107: 97–99. http://web.lemoyne.edu/~giunta/soddycn.html.
^ Thomson, J.J. (1913). "Rays of positive electricity". Proceedings of the Royal Society A 89: 1–20. doi:10.1098/rspa.1913.0057. http://web.lemoyne.edu/~giunta/canal.html.
^ Söderbaum, H.G. (November 11, 1915). Statement regarding the 1914 Nobel Prize in Chemistry.
^ Aston, Francis W. (1920). "The constitution of atmospheric neon". Philosophical Magazine 39 (6): 449–55.
^ Söderbaum, H.G. (December 10, 1921). Presentation Speech for the 1921 Nobel Prize in Chemistry.
^ Söderbaum, H.G. (December 10, 1922). Presentation Speech for the 1922 Nobel Prize in Chemistry.
^ Oseen, C.W. (December 10, 1926). Presentation Speech for the 1926 Nobel Prize in Physics.
^ Holden, Norman E. (2004). "Atomic Weights and the International Committee—A Historical Review". Chemistry International 26 (1): 4–7. http://www.iupac.org/publications/ci/2004/2601/1_holden.html.

[De_Bievre-0] De Bièvre, P. and P. D. P. Taylor (1997). "Traceability to the SI of amount-of-substance measurements: from ignoring to realizing, a chemist's view". Metrologia 34: 75. doi:10.1088/0026-1394/34/1/10.

[1] "Chemical reaction". Encyclopedia of science and technology. Routledge. 2001. http://books.google.com/books?id=Cmyt74ury34C&pg=PT78&dq.

[McGlashan.2C_1977-2] McGlashan, M. L. (1977). "Amount of substance and the mole". Phys. Educ. 12: 278. doi:10.1088/0031-9120/12/5/001.

[CODATA-3] Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. doi:10.1103/RevModPhys.80.633. http://physics.nist.gov/cuu/Constants/codata.pdf. Direct link to value.

[GoldBook-4] International Union of Pure and Applied Chemistry. "amount of substance, n". Compendium of Chemical Terminology Internet edition.

[SI-5] International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), pp. 114–15, ISBN 92-822-2213-6, http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf

[6] Talty, John T. (1988). Industrial Hygiene Engineering: Recognition, Measurement, Evaluation, and Control. William Andrew. pp. 142. ISBN 0815511752.

[7] Lee, C.C. (2005). Environmental Engineering Dictionary (4th ed.). Rowman & Littlefield. pp. 506. ISBN 086587848X.

[8] International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 46. Electronic version.

[molar-9] International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 7. Electronic version.

[10] International Union of Pure and Applied Chemistry. "amount-of-substance concentration". Compendium of Chemical Terminology Internet edition.

[11] International Union of Pure and Applied Chemistry (1996). "Glossary of Terms in Quantities and Units in Clinical Chemistry." Pure Appl. Chem. 68:957–1000.

[12] "Molar concentration" should refer to a concentration per mole, i.e. an amount fraction. The use of "molar" as a unit, equal to 1 mol/dm³, symbol M, is frequent, but not (as of May 2007) completely condoned by IUPAC: See International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 42 (n. 15). Electronic version.

[13] Lomonosov, Mikhail (1758/1970). "Mikhail Vasil'evich Lomonosov on the Corpuscular Theory". in Leicester, Henry M.. Cambridge, MA: Harvard University Press. pp. 224–33. http://www.archive.org/details/mikhailvasilevic017733mbp.

[Larousse-14] "Atome". Grand dictionnaire universel du XIXe siècle. 1. Paris: Pierre Larousse. 1866. pp. 868–73. . (French)

[15] Lavoisier, Antoine (1789). Traité élémentaire de chimie, présenté dans un ordre nouveau et d'après les découvertes modernes. Paris: Chez Cuchet. http://gallica.bnf.fr/ark:/12148/bpt6k3930k.table. . (French)

[16] Dalton, John (1805). "On the Absorption of Gases by Water and Other Liquids". Memoirs of the Literary and Philosophical Society of Manchester, 2nd Series 1: 271–87. http://web.lemoyne.edu/~giunta/dalton52.html.

[17] Dalton, John (1808). A New System of Chemical Philosophy. Manchester. http://www.archive.org/details/newsystemofchemi01daltuoft.

[18] Gay-Lussac, Joseph Louis (1809). "Memoire sur la combinaison des substances gazeuses, les unes avec les autres". Mémoires de la Société d'Arcueil 2: 207. English translation.

[19] Avogadro, Amadeo (1811). "Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique 73: 58–76. English translation.

[20] Excerpts from Berzelius' essay: Part II; Part III.

[21] Berzelius' first atomic weight measurements were published in Swedish in 1810: Hisinger, W.; Berzelius, J.J. (1810). "Forsok rorande de bestamda proportioner, havari den oorganiska naturens bestandsdelar finnas forenada". Afh. Fys., Kemi Mineral. 3: 162.

[22] Prout, William (1815). "On the relation between the specific gravities of bodies in their gaseous state and the weights of their atoms". Annals of Philosophy 6: 321–30. http://web.lemoyne.edu/~giunta/PROUT.HTML.

[23] Petit, Alexis Thérèse; Dulong, Pierre-Louis (1819). "Recherches sur quelques points importants de la Théorie de la Chaleur". Annales de Chimie et de Physique 10: 395–413. English translation

[24] Clapeyron, Émile (1834). "Puissance motrice de la chaleur". Journal de l'École Royale Polytechnique 14 (23): 153–90.

[25] Faraday, Michael (1834). "On Electrical Decomposition". Philosophical Transactions of the Royal Society. http://chimie.scola.ac-paris.fr/sitedechimie/hist_chi/text_origin/faraday/Faraday-electrochem.htm.

[26] Krönig, August (1856). "Grundzüge einer Theorie der Gase". Annalen der Physik 99: 315–22. doi:10.1002/andp.18561751008. http://gallica.bnf.fr/ark:/12148/bpt6k15184h/f327.table.

[27] Clausius, Rudolf (1857). "Ueber die Art der Bewegung, welche wir Wärme nennen". Annalen der Physik 100: 353–79. doi:10.1002/andp.18571760302. http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.table.

[28] Wurtz's Account of the Sessions of the International Congress of Chemists in Karlsruhe, on 3, 4, and 5 September 1860.

[29] Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der kaiserlichen Akademie der Wissenschaften Wien 52 (2): 395–413. English translation.

[30] Arrhenius, Svante (1887). Zeitschrift fur physikalische Chemie 1: 631. English translation.

[31] Ostwald, Wilhelm (1893). Hand- und Hilfsbuch zur ausführung physiko-chemischer Messungen. Leipzig.

[32] Helm, Georg; (Transl. Livingston, J.; Morgan, R.) (1897). The Principles of Mathematical Chemistry: The Energetics of Chemical Phenomena. New York: Wiley. pp. 6.

[33] Odhner, C.T. (December 10, 1901). Presentation Speech for the 1901 Nobel Prize in Chemistry.

[34] Törnebladh, D.R. (December 10, 1903). Presentation Speech for the 1903 Nobel Prize in Chemistry.

[35] Einstein, Albert (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen". Annalen der Physik 17: 549–60. doi:10.1002/andp.19053220806. http://www3.interscience.wiley.com/homepages/5006612/549_560.pdf.

[36] Perrin, Jean (1909). "Mouvement brownien et réalité moléculaire". Annales de Chimie et de Physique, 8^e Série 18: 1–114. Extract in English, translation by Frederick Soddy.

[37] Soddy, Frederick (1913). "The Radio-elements and the Periodic Law". Chemical News 107: 97–99. http://web.lemoyne.edu/~giunta/soddycn.html.

[38] Thomson, J.J. (1913). "Rays of positive electricity". Proceedings of the Royal Society A 89: 1–20. doi:10.1098/rspa.1913.0057. http://web.lemoyne.edu/~giunta/canal.html.

[39] Söderbaum, H.G. (November 11, 1915). Statement regarding the 1914 Nobel Prize in Chemistry.

[40] Aston, Francis W. (1920). "The constitution of atmospheric neon". Philosophical Magazine 39 (6): 449–55.

[41] Söderbaum, H.G. (December 10, 1921). Presentation Speech for the 1921 Nobel Prize in Chemistry.

[42] Söderbaum, H.G. (December 10, 1922). Presentation Speech for the 1922 Nobel Prize in Chemistry.

[43] Oseen, C.W. (December 10, 1926). Presentation Speech for the 1926 Nobel Prize in Physics.

[Holden-44] Holden, Norman E. (2004). "Atomic Weights and the International Committee—A Historical Review". Chemistry International 26 (1): 4–7. http://www.iupac.org/publications/ci/2004/2601/1_holden.html.

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Contents