Figure 1: Nuclide, Abundance, Element, and Molar Mass
The notion of atom is absolutely central to chemistry, according to a modern (post-1900) understanding of what chemistry is. This is related to the older notion of chemical element. The best way to define things is to start at the bottom and work our way up, step by step. We start with three fundamental particles (proton, neutron, and electron), then define atom, and then define chemical element.
Each atom has a nucleus consisting of one or more protons and zero or more neutrons. Conventionally, the number of protons is denoted Z, while the number of neutrons is denoted N. Protons and neutrons are both classified as nucleons. (They are the only nucleons you are likely to encounter.)
Each proton carries one unit of positive electrical charge. In contrast, neutrons are electrically neutral. As a result, the nucleus carries exactly Z units of positive electrical charge.
The number Z is called the proton number or synonymously the atomic number.
We name atoms according to atomic number. For example, hydrogen has Z=1 by definition, helium has Z=2 by definition, et cetera.
The notion of chemical element is a macroscopic notion (in contrast to atoms, which are ultramicroscopic). Specifically, a chemical element is a collection of atoms, all with the same Z value.
The periodic table (e.g. reference 1) is arranged in order of atomic number. You can find the name of the element corresponding to a particular Z value, and vice versa, using the periodic table. See reference 2 for details on how to think about the periodic table.
If you are going to associate the Z-value with the symbol for an element, the convention is to write it as a subscript to the left of the symbol; for example 2He has Z=2 while 3Li has Z=3. This is of course redundant, since the element-name uniquely specifies the proton number and vice versa. But sometimes redundancy is good, especially if you don’t have the periodic table memorized or readily accessible.
We can define the nucleon number to be the number of protons plus the number of neutrons. It is denoted A, so we can write A := Z + N.
If you are going to associate the A-value with the symbol for an element, the convention is to write it as a superscript to the left of the symbol; for example 4He has A=4 while 7Li has A=7.
There is no corresponding notation for N. There is no need for it, since if you know Z and A you can instantly infer N, namely N = A − Z.
There are at least two different definitions for the word atom.
This definition is particularly convenient when speaking about a collection of atoms, some of which are ionized and some are not.
This definition is more-or-less obligatory when speaking about the atoms inside a molecule, where it is usually not possible to keep track of which electrons “belong” to which atom.
If we start with a carbon atom and strip away all its electrons, we are left with a bare carbon nucleus. In that case calling it a nucleus is preferable to calling it an atom, if only because nucleus is the more specific term. However, in cases where the state of charge is not known, the term “atom” can be stretched to cover all the possibilities, from zero electrons on up.
If we start with a neutral atom and add or subtract some number of electrons, we create an electrically-charged entity called an ion. The same word – ion – also applies to electrically charged molecules.
For most purposes, when people talk about the size and shape of an atom, they mean the size and shape of the atom’s distribution of electrons. This distribution is a somewhat fuzzy cloud. The details of this distribution are formalized in terms of wavefunctions (also sometimes called orbitals) as discussed in reference 3. For the atoms inside molecules, any notion of the size of an atom is necessarily imprecise, because (as previously mentioned) it is not generally possible to keep track of which electrons “belong” to a given atom.
The nucleus is tiny, roughly 100,000 times smaller than the overall size of the atom. The electrons form a cloud around the nucleus.
Summary:
In this section, we restrict our attention to ordinary operations involving macroscopic quantities. This means it will be completely impractical to measure quantities by keeping an accurate count of the atoms.
To measure the quantity of a macroscopic sample, often the most precise method is to measure its mass.
| Tangential remark: In Europe, kitchen recipes commonly call for ingredients such as flour, sugar, shortening, etc. to be measured by mass, specified in grams or kilograms. This is accurate and convenient, assuming you have a kitchen scale. | Alas, in the US, kitchen recipes virtually always call for such ingredients etc. to be measured by volume. This is often inaccurate and inconvenient. Switching to European-style recipes would not be easy, because most households in the US don’t even have a kitchen scale. |
For many purposes, we want to be able to prepare samples where the number of atoms is controlled with reasonable precision. For example, this includes setting up chemical reactions so that the stoichiometry comes out right.
In ordinary chemistry-lab situations, it is not convenient to count the atoms one by one. Instead it is appropriate and conventional to scale things by a factor of Avogadro’s constant, which has the value 6.0221415(10)×1023 particles per mole. (In much of Europe this is called Loschmidt’s number, for good reason, as discussed in section 5.4.) The actual definition is arranged so that one mole of isotopically-pure 12C has a mass of exactly 12 grams. As a consequence, a mole of protons has a mass of approximately 1 gram ... more precisely it’s 1.00727646688(13)g.
For all practical purposes, you can think of the concept of mole as follows: It is just a number. A mole is like a dozen, only much larger. We can speak of a dozen carbon atoms, or a mole of carbon atoms. (At the moment, this practical definition is not the official SI definition, as discussed in section 7.)
Note that the definition of mole speaks of particles, not atoms. A mole of oxygen atoms weighs about 16 grams, while a mole of oxygen molecules weighs about 32 grams, because the oxygen molecule (O2) contains two oxygen atoms in a single particle.
To an excellent approximation, the mass of a molecule is the sum of the masses of its constituent atoms. It is traditional in chemistry classes to take this as an exact equality for practical purposes.1
For each naturally-occurring element, the molar mass has been carefully measured. This information is normally printed on the periodic table (e.g. reference 1). For instance, the mass of a mole of chlorine atoms is 35.453(2) grams, while the mass of a mole of bromine atoms is 79.904(1) grams. For typical intro-level chem-lab work, these numbers may be rounded off to 35.5 and 80, respectively.
If you divide one gram by Avogadro’s constant, you get a unit known as the unified atomic mass unit or (u) for short. One u = 1.6605402(10)×10−27 kg. The term dalton (Da) is a widely-used synonym, and is particularly useful when you want to attach a metric prefix (example: kilodalton).
The term AMU (for atomic mass unit) had two different meanings before the units were “unified” in 1961. There was a physics AMU and a chemistry AMU, both of which were close to the modern “unified” AMU but not exactly equal. Therefore you need to be careful when reading the older literature. Nowadays AMU is shorthand for unified atomic mass unit, although this is frowned upon in some circles; some editors will insist that you say “unified” and not say AMU.
The term atomic mass is very widely used (but not recommended) as a synonym for the molar mass of a chemical element. Talking about “atomic mass” tends to cause confusion about what is average versus what is typical, as we see from the following:
| Suppose that on average, there are 2.3 children per family. | It is vanishingly unlikely that you will see an “average” family walking down the street with 2.3 children. |
| The molar mass of naturally-occurring Cl is 35.5 grams per mole, so on average that’s 35.5 dalton per Cl atom. (See section 4 for a discussion of natural abundances.) | You will never find a Cl atom with mass 35.5 dalton. |
| The molar mass of naturally-occurring Br is 80 grams per mole, so on average that’s 80 dalton per Br atom. | You will never find in nature a Br atom with mass 80 dalton. (What you actually find is roughly a 50/50 mixture of 79Br and 81Br.) |
Very rarely one hears the term relative atomic mass. Somehow this is supposed to convey the idea of “average”, but this certainly seems like an abuse of the word “relative”. My advice is simple: If you mean average, say “average”.
For most practical chem-lab applications, the term “molar mass” is appropriate and unambiguous.
Operationally, the molar mass is determined by measuring each isotope using a high resolution mass spectrometer, and then computing the weighted average (weighted by natural abundance). An example of this calculation is shown in section 4. Additional examples can be found in reference 4.
It is important to keep in mind that a physical quantity remains the same, no matter what units are used to measure it. For example, the speed of light is the speed of light, no matter whether it is measured in meters per second or furlongs per fortnight. Voltage is voltage, even if it is measured in kilovolts or microvolts. In accordance with this principle, we are not required to measure molar mass in units of grams per mole (although it is overwhelmingly common and conventional to do so).
Dimensionally speaking, the molar mass (regardless of units) has the same dimensions as the so-called atomic mass or average atomic mass ... but molar mass does a much better job of communicating the concept, especially since for most elements there’s no such thing as an “average” atom. Also the idea of molar mass applies just fine to things that cannot have an atomic mass because they are not atoms at all (e.g. neutrons and molecules).
Also note that molar mass fits into a logical pattern with other quantities such as molar volume.
In addition to the foregoing argument about dimensions, we can make a point about units: one gram per mole is numerically equal to one dalton per particle. So we have here not just equivalent dimensions, but equivalent units. This makes it particularly easy to switch from “average atomic mass” to molar mass. All you need to do is change the name of the table; you don’t need to change the numerical entries in the table.
The term “atomic weight” is even more deprecated than “atomic mass”. It appears to be slowly dying out, but only slowly, and is still encountered rather frequently. It has multiple drawbacks. It is (or was) supposed to be synonymous with atomic mass, which is already disfavored relative to molar mass. On top of that, mass is not the same as weight. Suppose you are in a laboratory in outer space, where everything is weightless. Describing the mass of things is much more helpful than describing their weight. Remember that the gram is a unit of mass, not weight, so you shouldn’t be measuring atomic weight in grams, grams per mole, daltons, or anything like that.
For molecules, it is common to see the term “formula weight”, but this has multiple drawbacks for all the same reasons. It is much better to stick to the preferred term: molar mass.
Students sometimes question whether it is worth knowing the value of Avogadro’s number or (equivalently) the size of atoms. This is a valid, nontrivial question, but the answer turns out to be yes, for reasons discussed in section 5.
This concludes the discussion of the properties of macroscopic collections of atoms.
This section deals with the properties of individual atoms and their nuclei.
Disclaimer: you shouldn’t need to deal with individual atoms in order to do intro-level chemistry. Almost all chemistry (especially intro-level chemical reaction work) deals with macroscopic samples, i.e. with samples where the atom-count is closer to 1023 than it is to 1. Therefore the molar mass, as given in the periodic table, is sufficient information to let you get on with your work, even though it may be less than sufficient to satisfy your curiosity about individual atoms, and/or about non-chemical processes such as radioactive decay.
Recall that an atomic nucleus is composed of Z protons and N neutrons.
A particular value of the pair (Z, N) defines our notion of nuclide. (This is analogous to the way a Z-value defined our notion of chemical element.)
The nucleon number (A = Z + N) can also be called the baryon number because the class of nucleons is a subset of the class of baryons. Protons and neutrons are not the only baryons, but they are the only ones you are likely to encounter in the chemistry lab, so for chemistry purposes nucleon number is always equal to baryon number. Baryon number is an important concept, because it is a conserved quantity, just as electrical charge is a conserved quantity. Nucleon number is not rigorously conserved, but nearly so, in the sense that ordinary chem-lab processes, including radioactive decay, conserve total nucleon number.There exist some deprecated terms more-or-less synonymous with nucleon number. These include mass number, and (even worse) atomic mass number. These terms are very, very commonly used ... but they practically beg to be misinterpreted by non-experts. The so-called mass number is not a mass; it is a number, i.e. the nucleon number. Similarly the so-called atomic mass number is not a mass; it is a number. And it is a nuclide property, not an atom property, since we are specifying the N-value as well as the Z-value. You can infer the approximate mass from the nucleon number, and sometimes vice versa, but that does not make them the same thing. Inference is not the same as equivalence. Mass is mass ... and number is number. You need to be able to parse the term “mass number” when other people use it, but you should avoid using it yourself. Say nucleon number or baryon number instead.
To denote a nuclide, the conventional notation is to write the A-value as a superscript to the left of the symbol corresponding to the Z-value. For example, 3He has one neutron and two protons, for a total of three nucleons. The neutron number (N) is rarely written down explicitly; you are expected to infer it from the nucleon number (A) and the proton number (Z).
Each nuclide (Z, N) is considered an isotope of the corresponding chemical element (Z). For example, 4He is the most-common isotope of He. By extension, two nuclides with the same Z value are called isotopes of each other. For example, 3He is called an isotope of 4He and vice versa.2
Protons and neutrons each have a molar mass close to 1 dalton, and other contributions to the mass of the nuclide are small in comparison. Therefore, for example, 4He will have a molar mass close to 4 grams per mole.
| If you know the nucleon number of a nuclide, you can infer its molar mass to a good approximation. Conversely, if you know the molar mass of a nuclide, you can infer its nucleon number exactly. | If you know the molar mass of a chemical element, do not try to infer “the” nucleon number or even the “typical” nucleon number. It cannot be done reliably. Counterexamples abound, as discussed below. |
| Nuclides, yes. | Elements, no. |
The situation for the first few elements is shown in figure 1.
For clarity, the numbers in the figure have been rounded off, keeping only enough precision to make the point about molar mass being a weighted average of the nuclide masses. With the exception of the boron abundances, the numbers are known to considerably more precision than this. For details, see the references (section 8).
Qualifiers such as “natural” or “naturally-occurring” appear over and over again in this document, because it is possible to obtain samples that don’t have the natural distribution of isotopes. It is easy to buy a mole of 3He, although it is more expensive than ordinary 4He. Similarly it is easy to buy a mole of heavy water (D2O), although it is more expensive than ordinary water. In seawater, there is roughly one deuterium atom for every 6500 hydrogen atoms. You can shift this quite a bit in either direction by fractional distillation or gaseous diffusion. This can even happen inadvertently.
The natural product is not always cheaper; natural uranium (which contains about half a percent of 235U) is much more valuable than depleted uranium (from which most of the 235U has been removed).
Therefore, if you want to speak clearly, you can’t simply talk about the molar mass of this-or-that element; you need to specify that you’re talking about the molar mass of the naturally occurring element.
But even that isn’t entirely sufficient. It is possible to find different natural sources with different distributions of isotopes. So if you want to be really precise, you need to specify the source: e.g. seawater (not just natural water), or atmospheric nitrogen (not just natural nitrogen).
Usually, if you just grab a stock-bottle from the stockroom, the molar mass will be very close to what you expect, close enough for most purposes ... but for precise work, you might want to double-check. In particular, commercially available lithium compounds are sometimes significantly depleted of 6Li, leading to an unnaturally high molar mass (reference 5).
The molar mass of a natural sample of a chemical element can be expressed as a weighted average of the isotopes of that element, weighted by their natural abundance. The molar mass of each isotope, along with the natural abundance, can be obtained from the table of nuclides (reference 6), although sometimes more accurate abundance data can be obtained from reference 7.
| molar mass | natural | ||||
| / dalton | abundance | ||||
| 35Cl | 34.96885 | × | 75.78% | = | 26.4994 |
| 37Cl | 36.96590 | × | 24.22% | = | 8.95314 |
| ——— | |||||
| average: | 35.453 |
| nuclide mass | natural | ||||
| / dalton | abundance | ||||
| 79Br | 78.9183376(20) | × | 50.686(26)% | = | 40.00055 |
| 81Br | 80.9162911(30) | × | 49.314(26)% | = | 39.90306 |
| ——— | |||||
| average: | 79.90361(52) |
The bromine molar mass computed here is reasonably consistent with the values given in the Los Alamos periodic table (reference 1), namely 79.904(1). Their error bars are twice as large. Perhaps they are using older data, or perhaps they are accounting for sources of uncertainty that I am overlooking.
It is worth emphasizing that the molar mass of natural Br is 80, even though the nuclide 80Br has zero natural abundance. It can be created artificially, but it is radioactive with a short half-life, as you can ascertain from the table of nuclides (reference 6).
To repeat:
| Yes, given the masses and abundances of the nuclides, you can compute the molar mass of the element, by a process of averaging. | No, given the molar mass of the element, you cannot reliably undo the averaging to obtain the nucleon numbers, nuclide masses, or abundances. |
For most elements, uncertainty as to the natural abundances is the dominant contribution to the overall uncertainty of the element’s molar mass (far greater than the contribution from the uncertainty in the mass of each isotope). However, there are 21 elements in the periodic table for which only one isotope is found in nature. The molar mass of these elements is known to dramatically greater precision, compared to other elements, because there is no uncertainty as to the natural abundance.
Sometimes people ask how we “really” know that atoms exist, how we “really” know the size of atoms, and why we should care. These are nontrivial questions, for reasons discussed in section 5.2. Many things that we see in our daily lives are sensitive to the size of atoms, but the dependence may not be immediately obvious.
Ask yourself: If atoms tomorrow were ten times bigger than today, would you notice? How would you notice? What would you look for?
Here are some possible answers:
I don’t want to get into a metaphysical debate over whether the bumps seen in figure 2 “are” atoms. It suffices to say that the bumps are in one-to-one correspondence with atoms, and that the spacing between bumps tells us the spacing between atoms.
Even when you are not clearly seeing individual electrons, you might see shot noise, which is sensitive to the size of electrons.
The e-over-m data can come from ions moving in a magnetic field, or from electrochemistry (electroplating, electrolysis, et cetera).
If you want to check these things yourself, it is certainly possible to do kitchen-table experiments to estimate the diffusion constant, thermal conductivity, and/or viscosity of gases. With a little more effort, you can do this as a function of pressure, which provides a powerful way of verifying that mean free path and cross section are behaving as advertised.
Observation of Brownian motion is well within the capabilities of high-school or even elementary-school students. Getting from the raw observation to a quantitative determination of the size of molecules requires more analysis than young kids can handle, but the raw data provides at least some glimmer of an appreciation for what is going on in the microscopic and submicroscopic realms.
The Millikan experiment can be done in a high-school setting by students of modest ability using modest equipment. It’s not easy, and it requires hours, not minutes ... but it can be done.
We define the weak atomic hypothesis to be the hypothesis that atoms exist, and have some smallish but nonzero size. That is, we hypothesize that Avogadro’s number “exists” in some vague sense, but we don’t know the value of the number.
There are many important chemistry and physics experiments that provide good evidence for the weak atomic hypothesis. They provide evidence that atoms exist, but do not tell us anything about the size of atoms. Examples include:
This law was important in the history of chemistry. It predates the periodic table, and helped lay the groundwork for it.
This doesn’t require any notion of stoichiometry, and doesn’t even require you to have a correct chemical formula. That’s important, because almost none of the solids you see around you have a definite formula, i.e. none of them uphold Dalton’s so-called law of multiple proportions. This includes glass, ceramics, most metal objects, most plastic objects, wood, animal hair and tissue, minerals such as feldspar, et cetera.
At low temperatures, the law of Dulong and Petit fails. Nowadays the solution is to use something like the Debye model, which works well at low temperatures and correctly reproduces the law of Dulong and Petit at high temperatures.
Both these models (Debye and Dulong/Petit) deal only solids and deal only with the part of the specific heat that is due to phonons, i.e. the “lattice” specific heat. They leave out contributions from spin degrees of freedom, electrons, crystallographic phase changes, etc., which are sometimes very significant.
This requires you to get the chemical formulas right. As an example of what can go wrong, if you think the natural oxide of X is XO when in fact it is XO2, you will get the molar mass of X wrong by a factor of two.
Suppose you have established the weak atomic hypothesis. Further suppose you have measured lots of stoichiometric ratios using the methods enumerated above.
Then, if you discover the size of one type of atom, that tells you the size of all the others. The size of one atom plus stoichiometry tells you the size of another, and then you iterate until you have a complete table.
By way of contrast to the strong atomic hypothesis (section 5.1) and even the weak atomic hypothesis (section 5.2), there are quite a few things that can be adequately explained in terms of a continuous fluid, without any need to mention atoms. Examples include
We use the term hydrodynamic limit or hydrodynamic approximation to refer to situations where the phenomena of interest are well described by macroscopic properties, such as those listed above. Such situations commonly arise when the length-scale of interest is huge compared to the size of particles and huge compared to the spacing between particles.
It may be possible to derive and explain the macroscopic properties in terms of atomic theory, but once we have done so (or even if we have not), we don’t to keep track of the atoms, because the macroscopic properties tell us what we need to know, in the hydrodynamic limit.
The hydrodynamic approximation works well in almost all everyday situations, which is not surprising given how small atoms are. For thousands of generations, people were able to live their lives without knowing anything about atoms.
Amedeo Avogadro died without ever knowing the value of Avogadro’s number. If you had guessed a number 100 times too big or 100 times to small, he would have been unable to refute your guess.
Johann Loschmidt is generally credited with the first scientific measurement of the size of atoms (reference 15). In much of Europe, the thing we call Avogadro’s number is called Loschmidt’s number, which makes a certain amount of sense.
Beware that in the US, the same term, Loschmidt’s number, is on rare occasions used with a different meaning, namely the number of particles in one cm3 of gas at standard temperature and pressure.
The terms “atomic mass”, “relative atomic mass”, and “atomic weight” are deprecated; use molar mass instead.
Any terms involving atomic “weight” are deprecated; use the corresponding notions of mass instead.
The terms “mass number” and “atomic mass number” are deprecated; use nucleon number (aka baryon number) instead.
The typical periodic table gives you the molar mass, which is sufficient for doing ordinary chem-lab experiments with macroscopic samples. If you are curious about isotopes, the ordinary periodic table does not suffice; you need a chart of the nuclides.
If you want the formal definition of mole, here it is:
1. 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".
That’s a direct quote from reference 17. They don’t explain what they mean by “substance” or “amount of substance”.
This definition is less useful, harder to understand, and no more precise, compared to the practical numerical definition given in section 2.1.
Also, this is a moving target. In the metrology community, there are serious efforts toward redefining the mole to be an exact pure number. (This is analogous to the process whereby the speed of light was redefined to have an exact value.)
The notion of “amount of substance” is a macroscopic 19th-century concept. It predates any knowledge of the numerical value of Avogadro’s number ... just as the macroscopic 19th-century notions of element and compound predate the modern microscopic notions of atom and molecule. The modern notions are simpler and in all ways better.