## The Nuclear Atom and Atomic Mass

As stated above, we define the elements by their number of protons. We define the atomic number, Z, as the number of protons in an atom. For a neutral atom (not a charged ion), the number of electrons must equal the number of protons. However, the number of neutrons can vary within atoms of a given element.

Atoms of the same element with different numbers of neutrons are called isotopes. Most elements have multiple isotopes. For a given isotope, we define the mass number, A, as the atomic number plus the number of protons. We can write this as A = Z + N where A is mass number, Z is atomic number, and N is number of neutrons.

Read this section, which explains how we write atomic symbols with atomic numbers and mass numbers.

When we see the mass of an element on the periodic table, we are seeing the weighted average of the masses of all isotopes of that element. Then read the section "Isotopic Mixtures and Abundances" near the bottom of the page. This section describes how to determine the average atomic mass of an element if we know the isotope masses and their relative abundance. Try the practice problems to do these types of calculations yourself.

### 1. The Nuclear Atom

The precise physical nature of atoms finally emerged from a series of elegant experiments carried out between 1895 and 1915. The most notable of these achievements was Ernest Rutherford's famous 1911 alpha-ray scattering experiment, which established that:

• Almost all of the mass of an atom is contained within a tiny (and therefore extremely dense) nucleus which carries a positive electric charge whose value identifies each element and is known as the atomic number of the element.
• Almost all of the volume of an atom consists of empty space in which electrons, the fundamental carriers of negative electric charge, reside.

The extremely small mass of the electron (1/1840 the mass of the hydrogen nucleus) causes it to behave as a quantum particle, which means that its location at any moment cannot be specified; the best we can do is describe its behavior in terms of the probability of its manifesting itself at any point in space. It is common (but somewhat misleading) to describe the volume of space in which the electrons of an atom have a significant probability of being found as the electron cloud. The latter has no definite outer boundary, so neither does the atom. The radius of an atom must be defined arbitrarily, such as the boundary in which the electron can be found with 95% probability. Atomic radii are typically 30-300 pm.

#### The Nucleus is Composed of Protons and Neutrons

The nucleus is itself composed of two kinds of particles. Protons are the carriers of positive electric charge in the nucleus; the proton charge is exactly the same as the electron charge, but of opposite sign. This means that in any [electrically neutral] atom, the number of protons in the nucleus (often referred to as the nuclear charge) is balanced by the same number of electrons outside the nucleus.

Because the electrons of an atom are in contact with the outside world, it is possible for one or more electrons to be lost, or some new ones to be added. The resulting electrically-charged atom is called an ion.

The other nuclear particle is the neutron. As its name implies, this particle carries no electrical charge. Its mass is almost the same as that of the proton. Most nuclei contain roughly equal numbers of neutrons and protons, so we can say that these two particles together account for almost all the mass of the atom.

#### The Atomic Number is the Nuclear Charge, and Thus the Number of Electrons in the Neutral Atom

What single parameter uniquely characterizes the atom of a given element? It is not the atom's relative mass, as we will see in the section on isotopes below. It is, rather, the number of protons in the nucleus, which we call the atomic number and denote by the symbol Z. Each proton carries an electric charge of +1, so the atomic number also specifies the electric charge of the nucleus. In the neutral atom, the Z protons within the nucleus are balanced by Z electrons outside it.

Moseley searched for a measurable property of each element that increases linearly with atomic number. He found this in a class of X-rays emitted by an element when it is bombarded with electrons. The frequencies of these X-rays are unique to each element, and they increase uniformly in successive elements. Moseley found that the square roots of these frequencies give a straight line when plotted against Z; this enabled him to sort the elements in order of increasing atomic number.

Atomic numbers were first worked out in 1913 by Henry Moseley, a young member of Rutherford's research group in Manchester.

You can think of the atomic number as a kind of serial number of an element, commencing at 1 for hydrogen and increasing by one for each successive element. The chemical name of the element and its symbol are uniquely tied to the atomic number; thus the symbol "Sr" stands for strontium, whose atoms all have Z = 38.

#### Mass Number

This is just the sum of the numbers of protons and neutrons in the nucleus. It is sometimes represented by the symbol A, so

A = Z + N

in which Z is the atomic number and N is the neutron number.

#### Nuclides and Their Symbols

The term nuclide simply refers to any particular kind of nucleus. For example, a nucleus of atomic number 7 is a nuclide of nitrogen. Any nuclide is characterized by the pair of numbers (Z ,A). The element symbol depends on Z alone, so the symbol 26Mg is used to specify the mass-26 nuclide of manganese, whose name implies Z=12. A more explicit way of denoting a particular kind of nucleus is to add the atomic number as a subscript. Of course, this is somewhat redundant, since the symbol Mg always implies Z=12.

#### Isotopes are Nuclides Having the Same Atomic Number

Two nuclides of the same element (and thus with identical atomic numbers) but different neutron numbers (and therefore different mass numbers) are known as isotopes. Most elements occur in nature as mixtures of isotopes, but twenty-three of them (including beryllium and fluorine, shown in the table) are monoisotopic. For example, there are three natural isotopes of magnesium: 24Mg (79% of all Mg atoms), 25Mg (10%), and 26Mg (11%); all three are present in all compounds of magnesium in about these same proportions.

Approximately 290 isotopes occur in nature.

The two heavy isotopes of hydrogen are especially important – so much so that they have names and symbols of their own:

Deuterium accounts for only about 15 out of every one million atoms of hydrogen. Tritium, which is radioactive, is even less abundant. All the tritium on the earth is a by-product of the decay of other radioactive elements.

### 2. Relative Atomic Masses: The Atomic Weight Scale

For historical reasons, the term atomic weight has a special meaning in Chemistry; it does not refer to the actual "weight" of an atom, which would be expressed in grams or kg.

Atomic weights, sometimes called relative weights, are more properly known as relative atomic masses, and being ratios, are dimensionless.

Please note that although the terms mass and weight have different meanings, the differences between their values are so small as to be insignificant for most practical purposes, so the terms atomic weight and atomic mass can be used interchangeably.

Atoms are of course far too small to be weighed directly; weight measurements can only be made on the massive (but unknown) numbers of atoms that are observed in chemical reactions. The early combining-weight experiments of Dalton and others established that hydrogen is the lightest of the atoms, but the crude nature of the measurements and uncertainties about the formulas of many compounds made it difficult to develop a reliable scale of the relative weights of atoms. Even the most exacting weight measurements we can make today are subject to experimental uncertainties that limit the precision to four significant figures at best.

#### Atomic Weights are Average Relative Masses

In the earlier discussion of relative weights of atoms, we explained how Dalton assigned a relative weight of unity to hydrogen, the lightest element, and used combining weights to estimate the relative weights of the others he studied. Later on, when it was recognized that more elements form simple compounds with oxygen, this element was used to define the atomic weight scale. Selecting O = 16 made it possible to retain values very close to those already assigned on the H=1 scale.

Finally, in 1961, carbon became the defining element of the atomic weight scale. But because, by this time, the existence of isotopes was known, it was decided to base the scale on one particular isotope of carbon, C-12, whose relative mass is defined as exactly 12.000. Because almost 99% of all carbon atoms on the earth consist of 6C12, atomic weights of elements on the current scale are almost identical to those on the older O=16 scale.

Most elements possess more than one stable isotope in proportions that are unique to each particular element. For this reason, atomic weights are really weighted averages of the relative masses of each that are found on earth.

Atomic weights are the ratios of the average mass of the atoms of an element to the mass of an identical number of 6 C12 atoms.

You can visualize the atomic weight scale as a long line of numbers that runs from 1 to around 280. The beginning of the scale looks like this:

#### Atomic Weights (Relative Atomic Masses) of the First Ten Elements

The red vertical lines beneath each element symbol indicate where that element is located on the atomic weight scale.

Of these ten elements, only two, beryllium and fluorine, have a single isotope. The other eight atomic weights are weighted averages of the relative masses of the multiple isotopes that these (and most) elements possess. This is especially noticeable in the case of boron, whose average relative mass falls between 10 and 11. (Historically, observations of cases such as these led to the very concept of isotopes.)

For many elements, one particular isotope so dominates the natural mixture that the others have little effect on the average mass. For example, 99.99 percent of hydrogen atoms consist of 1H1, whereas 1H2, the other stable isotope, amounts to only 0.01 percent. Similarly, oxygen is dominated by 8O16 (over 99.7 percent) to the near exclusion of its two other isotopes.

Showing varying precisions of atomic weights of the first ten elements

Atomic weights are listed in tables found in every chemistry textbook; you can't do much quantitative chemistry without them! The "standard" values are updated every few years as better data becomes available.

You will notice that the precisions of these atomic weights, as indicated by the number of significant figures, vary considerably.

• Atomic weights of the 26 elements having a single stable isotope (monoisotopic elements) are the most precisely known. Two of these, boron and fluorine, appear in the above table.

• Owing to geochemical isotopic fractionation (discussed farther on), there is always some uncertainty in averaging the atomic weights of elements with two or more stable isotopes.

• Industrial processes associated mainly with nuclear energy and weapons production require the isolation or concentration of particular isotopes. When the by-product elements or compounds from which these isotopes have been depleted eventually get distributed in the environment or sold into the commercial marketplace, their atomic weights can vary from "official" values. This has occurred, for example, with lithium, whose isotope Li-6 has been used to produce hydrogen bombs.

• Naturally-occurring radioactive elements (all elements heavier than 82Pb) all gradually decay into lighter elements, most of which are themselves subject to radioactive decay. These radioactive decay chains eventually terminate in a stable element, the most common of which is one of the three stable isotopes of lead. Subsequent geochemical processes can cause lead ore bodies from such sources to mix with "primeval" Pb (derived from the cosmic dust that formed the solar system), leading to a range of possible average atomic weights. For these reasons, lead, with a listed average atomic weight of 207.2, has the least-certain mass of any stable element.

#### Weighing Atoms: Mass Spectrometry

A major breakthrough in Chemistry occurred in 1913 when J.J. Thompson directed a beam of ionized neon atoms through both a magnetic field and an electrostatic field. Using a photographic plate as a detector, he found that the beam split into two parts, and suggested that these showed the existence of two isotopes of neon, now known to be Ne-20 and Ne-22.

This, combined with the finding made a year earlier by Wilhelm Wien that the degree of deflection of a particle in these fields is proportional to the ratio of its electric charge to its mass, opened the way to characterizing these otherwise invisible particles.

Thompson’s student F.W. Aston improved the apparatus, developing the first functional mass spectrometer, and he went on to identify 220 of the 287 isotopes found in nature; this won him a Nobel prize in 1921. His work revealed that the mass numbers of all isotopes are nearly integers (that is, integer multiples of the mass number 1 of the protons and neutrons that make up the nucleus.

Neutral atoms, having no charge, cannot be accelerated along a path so as to form a beam, nor can they be deflected. They can, however, be made to acquire electric charges by directing an electron beam at them, and this was one of the major innovations by Aston that made mass spectrometry practical.

Mass spectrometry begins with the injection of a vaporized sample into an ionization chamber where an electrical discharge causes it to become ionized. An accelerating voltage propels the ions through an electrostatic field that allows only those ions having a fixed velocity (that is, a given charge) to pass between the poles of a magnet. The magnetic field deflects the ions by an amount proportional to the charge-to-mass ratios. The separated ion beams are detected and their relative strengths are analyzed by a computer that displays the resulting mass spectrum. In modern devices, a computer also controls the accelerating voltage and electromagnet current so as to being successive ion beams into focus on the detector.

Schematic diagram of a mass spectrometer

Photo of a mass spectrometer

The mass spectrometer has become one of the most widely used laboratory instruments. Mass spectrometry is now mostly used to identify molecules. Ionization usually breaks a molecule up into fragments having different charge-to-mass ratios, each molecule resulting in a unique "fingerprint" of particles whose origin can be deduced by a jigsaw puzzle-like reconstruction. For many years, "mass-spec" had been limited to small molecules, but with the development of novel ways of creating ions from molecules, it has now become a major tool for analyzing materials and large biomolecules, including proteins.

The mass spectrum of magnesium shows that it consists of three isotopes of masses 24 through 26. The height of each peak shows the abundance of each isotope.

#### Isotopic Mixtures and Abundances

Only 26 of the elements that occur on the Earth exist as a single isotope; these are said to be monoisotopic. The remaining elements consist of mixtures of between two and ten isotopes. The total number of natural isotopes is 339; of these, 254 are stable, while the remainder are radioactive, meaning that they decay into stable isotopes.

Recalling that a given isotope (also known as a nuclide) is composed of protons and neutrons, each having a mass number of unity, it should be apparent that the mass number of a given nuclide will be an integer, as seen in the mass spectrum of magnesium above.

It also follows that the relative atomic masses (“atomic weights”) of monoisotopic elements will be very close to integers, while those of other elements, being weighted averages, can have any value.

#### Problem Example 4

Estimate the average atomic weight of magnesium from the isotopic abundance data shown in the above mass spectrometry plot.

Solution: We just take the weighted average of the mass numbers:

(0.7899 × 24) + (0.1000 × 25) + (0.1101 × 26) = 24.32

Note: The measured atomic weight of Mg (24.305) is slightly smaller than this because atomic masses of nuclear components are not strictly additive, as will be explained further below.

When there are only two significantly abundant isotopes, you can estimate the relative abundances from the mass numbers and the average atomic weight.

#### Problem Example 5

The average atomic weight of chlorine is 35.45 and the element has two stable isotopes 17Cl35 and 17Cl37. Estimate the relative abundances of these two isotopes.

Solution: Here you finally get to put your high-school algebra to work! If we let x represent the fraction of Cl35, then (1-x) gives the fraction of Cl37. The weighted average atomic weight is then

35x + 37(1-x) = 35.45

Solving for x gives 2x = 1.55, x = 0.775, so the abundances are 77.5% Cl35 and 22.5% Cl37.

[Problems of this kind almost always turn up in exams]

#### Problem Example 6

Elemental chlorine, Cl2, is made up of the two isotopes mentioned in the previous example. How many peaks would you expect to observe in the mass spectrum of Cl2?

Solution: The mass spectrometer will detect a peak for each possible combination of the two isotopes in dichlorine: 35Cl-35Cl, 35Cl-37Cl, and 37Cl-37Cl.

#### Isotope Effects: When Different Isotopes Exhibit Different Chemical Behavior

The chemical behavior of an element is governed by the number and arrangement of its electrons in relation to its nuclear charge (atomic number). Because these quantities are identical for all isotopes of a given element, they are generally considered to exhibit identical chemical properties.

However, it turns out that the mass differences between different isotopes can give rise to very slight differences in their physical behavior that can, in turn affect their chemical behavior as well. These isotope effects are most evident in the lighter elements, in which small differences in neutron number lead to greater differences in atomic mass.

Thus no element is more subject to isotope effects than hydrogen: an atom of "heavy hydrogen" 1H2 (also known as deuterium and often given the symbol D) has twice the mass of an atom of 1H2. When this isotope is combined with oxygen, the resulting "heavy water" D2O exhibits noticeably different physical and chemical properties: it melts at 3.8° C and boils at 101.4° C. D2O apparently interferes with cell division in organisms; mammals given only heavy water typically die in about a week.

When two or more elements whose atoms contain multiple isotopes are present in a molecule, numerous isotopic modifications become possible.

For example, the two stable isotopes of hydrogen and of oxygen (O16 and O18) give rise to combinations such as H2O18, HDO16, etc., all of which are readily identifiable in the infrared spectra of water vapor.

The amount of the rare isotopes of oxygen and hydrogen in water varies enough from place to place that it is now possible to determine the age and source of a particular water sample with some precision. These differences are reflected in the H and O isotopic profiles of organisms. Thus the isotopic analysis of human hair can be a useful tool for crime investigations and anthropology research.

#### Isotopic Fractionation

Isotope effects manifest themselves in both physical and chemical changes. In general,

• Chemical bonds involving lighter isotopes tend to be more readily broken, so reactions that depend on the rupture of such bonds will lead to products that are slightly enriched in the lighter isotopes.
• Phase changes such as vaporization similarly favor the lighter isotopes. Thus when water vapor condenses in clouds to form rain, the heavier water isotopes become slightly more concentrated in the liquid phase.

These two effects give rise to isotopic fractionation as chemical substances move through the environment — or on a much smaller scale, through the various metabolic processes that occur in organisms. Over time, this leads to changes in the isotopic signatures of elements in different realms of the world that can reveal information that would otherwise be hidden.

The extent of isotopic fractionation depends on the temperature. This fact has been put to practical use to estimate the global average temperature in past times by measuring the degree of enrichment of the heavier isotopes of oxygen in glacial ice cores and also in ancient sediments containing the shells of microorganisms.

#### Atomic Weights, Molecular Weights and Formula Weights

Molecules are composed of atoms, so a molecular weight is just the sum of the atomic weights of the elements it contains.

#### Problem Example 7

What is the molecular weight of sulfuric acid, H2SO4?

Solution: The atomic weights of hydrogen and of oxygen are 1.01 and 16.00, respectively (you should have these common values memorized.) From a table, you can find that the atomic weight of sulfur is 32.06. Adding everything up, we have

(2 x 1.01) + 32.06 + (4 x 16.00) = 98.08

Because some solids are not made up of discrete molecules (sodium chloride, NaCl, and silica, SiO2 are common examples), the term formula weight is often used in place of molecular weight. In general, the terms molecular weight and formula weight are interchangeable.

### 3. Atomic Masses and the Atomic Mass Unit

Here again is the beginning of the atomic weight scale that you saw above:

You should understand by now that atomic weights are relative weights, based on a scale defined by 6C12 = 12. But what is the absolute weight of an atom, expressed in grams or kilograms? In other words, what actual mass does each unit on the atomic weight scale represent?

The answer is 1.66053886 × 10–27 kg. This quantity (whose value you do not need to memorize) is known as the unified atomic mass unit, denoted by the abbreviation u. (Some older texts leave off the "unified" part, and call it the amu.

The unified atomic mass unit is defined as 1/12 of the mass of one atom of carbon-12.

Why such a hard-to-remember number? Well, that's just how Nature sometimes does things. Fortunately, you don't need to memorize this value, because you can easily calculate its value from Avogadro's number, NA, which you are expected to know:

1 u = 1/NA gram = 1 ÷ (1000 NA) Kg

#### Masses of the Subatomic Particles

Atoms are composed of protons, neutrons, and electrons, whose properties are shown below:

particle mass, g mass, u charge symbol
electron 9.1093897 × 10–28 5.48579903 × 10–4 1– –10e
proton 1.6726231 × 10–24 1.007276470 1+ 11H+, 11p
neutron 1.6749286 × 10–24 1.008664904 0 01n

Two very important points you should note from this table:

• The mass of the electron is negligible compared to that of the two nuclear particles;
• The proton and neutron have masses that are almost, but not exactly, identical.

#### Nuclear Masses and Mass Defect

As we mentioned in one of the problem examples above, the mass of a nucleus is always slightly different from the masses of the nucleons (protons and neutrons) of which it is composed. The difference, known as the mass defect, is related to the energy associated with the formation of the nucleus through Einstein's famous formula e = mc2. This is the one instance in chemistry in which conservation of mass-energy, rather than of mass alone, must be taken into account. But there is no need for you to be concerned with this in this part of the course.

For all practical purposes, until you come to the section of the course on nuclear chemistry, you can consider that the proton and neutron have masses of about 1 u, and that the mass of an atom (in u) is just the sum of the neutron and proton numbers.

#### What you should be able to do

Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially important that you know the precise meanings of all the highlighted terms in the context of this topic.

• Give a chemical definition of element, and comment on the distinction between the terms atom and element.

• You should know the names and symbols of the more common elements, including those whose symbols are derived from their Latin names.

• Describe, in your own words, the Laws of Chemical Change: mass conservation, constant composition, and multiple proportions.

• Explain how these laws follow from Dalton's atomic theory.

• Describe Rutherford's alpha-ray scattering experiment and how it led to the present model of the atom.

• Define atomic number and mass number, and explain the relation between them.

• Define isotope and nuclide, and write the symbol for a nuclide of a given element with a given number of neutrons.

• Explain the purpose of a mass spectrometer and its general principle of operation.

• Describe the atomic weight scale.

• Find the molecular weight or formula weight from a chemical formula.

• Define the unified atomic mass unit, and write out the mass numbers of the proton, neutron, and electron.