Why does the periodic table have exactly this shape?
Write electron configurations and orbital diagrams, and predict how size, ionization energy, electron affinity and electronegativity change across the table, and why.
ExperimentsThe evidence that shells are realPeel an atom one electron at a time and read its shells from the energies.
Experiment 1 · 1913 onward
Peeling an atom
If electrons are arranged in shells, what should it cost to remove them one by one?
By 1913 Niels Bohr had put electrons in shells, and Henry Moseley had ordered the elements by nuclear charge. But shells were still an idea on paper. Physicists were learning to measure the energy needed to pull electrons off atoms (the Franck–Hertz experiment of 1914 was one way). If shells are real, the energies should show them: removing an inner electron should cost far more than removing an outer one. Take one element, peel its electrons off one at a time, and see whether the shells appear in the numbers. (This is ionization energy, the cost of removing an electron. Electronegativity, how hard an atom pulls on shared electrons, comes later in the unit.)
Element:
Bar number = which electron is removed (1st, 2nd, …). Red labels show how many times bigger a bar is than the one before. Click in the gap between two bars to place or remove a divider.
Step 1 of 3 · Read the shells of sodiumEach bar is the energy to remove the next electron from a sodium atom: bar 1 is the first removed, the easiest, and they run left to right. Click between two bars to put a divider where the cost jumps. Sodium has three shells, so find the two big jumps.
Electrons removed11
Shells found1
Valence electrons (before the first jump)?
Biggest jump×9.2
The maths
Ionization energy: X(g) → X⁺(g) + e⁻ (the energy to remove one electron from a gaseous atom, per mole, in kJ/mol)
Electrons removed so far: 11. Step between bars k and k+1 is the ratio IEₖ₊₁ / IEₖ, shown above each pair of bars when it is above 2.
Questions you might have (4)Why does the cost jump when a shell runs out?
An electron in a lower shell is both closer to the nucleus and less shielded from it. In sodium, the 10th electron removed comes from the second shell (2s or 2p). It feels a pull of about 7+ from the nucleus's 11+ charge, instead of the roughly 2+ the outer electron felt. That is a far stronger hold, so it takes about five times the energy of the 9th electron.
Why do the numbers rise even within a shell?
Each electron you remove leaves the others with the same nuclear charge but one fewer electron to repel them. They are pulled in slightly harder, so the next one is a little more tightly held. And pulling a negative electron from an ion that is already positive takes more than pulling it from a neutral atom.
How are these energies measured?
By firing electrons at a gas, or by shining ultraviolet light on it, and finding the lowest energy at which atoms lose an electron (the first ionization energy), then doing the same for the ions that result. Spectroscopy of the ions gives the later values. The numbers here are the standard tabulated ones, in kJ/mol, as in the periodic-table data used throughout this course.
Why is the first jump not always the biggest?
The biggest jump of all is the last one: removing the very deepest electrons, the 1s pair, which feel almost the full nuclear charge. For aluminium, that step is bigger than the one that ends the valence electrons. For the chemistry you need, read the first big jump: the electrons before it are the valence electrons.
Run the experiment to see what it showed.
TheoryThe rules that place the electronsWhere each electron goes, and what forbids the alternatives.
Theory 1 · Pauli, 1925; Hund, 1927
Build the atom
Where does each electron go, and what stops it going somewhere else?
In 1925 Wolfgang Pauli proposed that no two electrons in an atom can have exactly the same set of quantum numbers, and in 1927 Friedrich Hund worked out how electrons spread out over orbitals of equal energy. Add the plain rule that electrons settle in the lowest place available, and you have everything that decides the electron arrangement of every atom. You will find the three rules the way their discoverers did: by being told no when you break one. The bench counts the electrons and rejects a wrong move.
Build:
1s2 max
2s2 max
2p6 max
3s2 max
3p6 max
4s2 max
3d10 max
4p6 max
↑ Energy rises up the ladder. Click a box to add the next electron to it.
Step 1 of 4 · CarbonCarbon has 6 electrons. Click a box to put the next electron in it. Start at the bottom, where the energy is lowest, and work up.
Electrons placed0 / 6
Unpaired electrons0
Outer subshellnone
The maths
Rules: (1) lowest energy first (2) two to a box, spins opposite (3) one in each box before pairing
Electrons placed: 0 of 6. Configuration so far: none
Questions you might have (4)Why would 4s be lower than 3d?
A 4s electron spends part of its time very close to the nucleus, inside the inner shells, where it feels a large pull. A 3d electron never gets that close. Even though 4s is in a bigger shell, that close approach more than makes up for it, so 4s ends up slightly lower for potassium and calcium. Once 3d starts filling the two are close, which is why the transition metals are so varied.
Why two to a box, and why opposite spins?
Spin is a property each electron has, with two possible settings, up or down. The Pauli exclusion principle says no two electrons in an atom can have exactly the same set of quantum numbers. A box fixes n, l and mₗ, so the only thing left to tell two electrons apart is spin, and there are only two choices.
Why spread out before pairing?
Two electrons in the same box are squeezed into the same region and repel each other strongly. In separate boxes of the same subshell they are in different regions, with the same energy. Parallel spins also keep them further apart on average. So the spread-out arrangement is lower in energy.
Are there exceptions?
A few, such as chromium (4s¹ 3d⁵) and copper (4s¹ 3d¹⁰), where a half-full or full 3d subshell makes up for moving an electron out of 4s. The rules here are right for every atom in this bench and for most of the table. A later section counts the exceptions.
Work through it to see what it shows.
ExperimentsA test of the boxesIf unpaired electrons are real, a magnet should be able to find them.
Experiment 2 · 1922
Are the unpaired electrons really there?
Fire a beam of atoms through a magnet. What happens to an atom with an unpaired electron, and to one without?
In 1922 Otto Stern and Walther Gerlach, in Frankfurt, sent a beam of silver atoms through a magnet that was stronger on one side. Atoms are tiny magnets, and classical physics said they could point any way, so the beam should smear into a band. Stern and Gerlach saw two sharp lines. Our last section drew boxes with paired and unpaired electrons. A magnet is the obvious tool to find out whether the unpaired ones are really there: an atom with no unpaired electron should feel nothing, and one with an unpaired electron should be pushed.
Atoms:
Predict the screen for Silver:
Step 1 of 3 · SilverSilver has one electron in its outer 5s box, unpaired. First predict what the screen will show, then fire the oven. A tiny magnet could point any way, so classical physics expects a smear.
Unpaired electrons1
Beams expected, 2S + 12
Atoms landed0
The maths
A spinning electron is a tiny magnet. Paired electrons point opposite ways and cancel. In a field that is stronger on one side, the net push is proportional to the unpaired spin.
Spin has S = 1/2, so the number of beams is 2S + 1 = 2.
Questions you might have (4)Why does the field have to be stronger on one side?
In a uniform field, a tiny magnet is only twisted, not pushed anywhere. It takes a field that changes from one place to another to give a net force. That is why one pole is a wedge and the other is flat.
Why only s-electron atoms?
Electrons in p, d and f orbitals also circle the nucleus, which adds a second magnet. Atoms such as oxygen or iron then give more complicated patterns. With only s electrons there is no orbital motion, so the spin is the whole story.
What did Stern and Gerlach expect?
In 1922 the accepted picture let a tiny magnet point in any direction, so the beam should have smeared into a band. They saw two sharp lines of silver on the plate. Spin as a property with only two settings came a few years later, to explain exactly this.
Is this the same as a magnet picking up a solid?
It is related. In a solid, unpaired electrons make the material weakly attracted to a magnet: that is called paramagnetism, and a material with all paired electrons is weakly repelled: diamagnetism. A single atom in a beam is a cleaner test, because each atom is on its own.
Run the experiment to see what it showed.
TheoryNow let the rules runFill every subshell in order, all the way up, and see what shape appears.
Theory 2 · Mendeleev, 1869; Bohr, 1922
The table builds itself
What happens to the periodic table if you just add electrons, one at a time, by the rules?
In 1869 Dmitri Mendeleev arranged the elements by mass and by chemical behaviour, and left gaps for elements nobody had found yet. In 1922 Niels Bohr argued that the table's shape was not an accident of chemistry but a consequence of how electrons fill shells. You now have the rules Bohr needed: lowest energy first, two to a box, spread before pairing. Run them from hydrogen upward, one electron at a time, and see whether a periodic table appears.
Step 1 of 3 · Let the table startBelow is a tape of every subshell in filling order, each as long as it holds electrons (1s holds 2, 2p holds 6...). Press Play, or step by one electron, and watch each element take its place.
Electrons added (Z)0
Elementnone yet
Last electron goes innone yet
Blocknone yet
The maths
Each subshell has (2 × l + 1) boxes, and each box holds 2 electrons. s: 1 box = 2 p: 3 boxes = 6 d: 5 boxes = 10 f: 7 boxes = 14
Questions you might have (4)Why do rows have different lengths?
Each row starts when a new s subshell starts to fill and ends when a p subshell is full. The row length is the sum of the subshells filled in between: 2 (1s), 8 (2s + 2p), 8 (3s + 3p), 18 (4s + 3d + 4p), 18 (5s + 4d + 5p), 32 (6s + 4f + 5d + 6p) and 32 (7s + 5f + 6d + 7p).
Why is the f block drawn so oddly in most tables?
Written in its natural place, the f block makes the table 32 columns wide, which is awkward on a page. Most tables lift it out and put it underneath. This one leaves it in place so that the widths 2, 6, 10 and 14 can be seen together.
Where do the exceptions come from?
The rule is a good approximation of which subshell is lowest in energy, but the subshells in the d and f blocks are very close, and the interactions between electrons can tip the balance. Chromium and copper put an electron from 4s into 3d to make a half-full or full 3d subshell. The columns are the same whichever way it goes, because the elements are placed by the rule, and the chemistry is controlled by the outer electrons.
Why do elements in the same column behave alike?
Chemistry happens at the outside of the atom. Sodium, potassium and the rest of group 1 all have one electron in an outer s box, and a full shell beneath. They lose that electron easily and form the same kind of compound. The next section tests this directly, with numbers.
Work through it to see what it shows.
ExperimentsDoes the table predict anything?If the shape is real, properties that depend on the outer electrons should repeat with it.
Experiment 3 · Measured for every element
The sawtooth
Measure how hard it is to pull one electron off each element in turn, and plot it against atomic number. What shape is it?
For more than a century chemists had known that the elements fall into families that repeat: the alkali metals, the halogens, the noble gases. Mendeleev's periodic law (1869) put that on a page. If the table is the filling order, then something physical about the outer electron should repeat in step with the families. The first ionization energy, the energy to remove the outermost electron, is the cleanest thing to look at. Plot it for the first fifty-four elements and look for a pattern that comes back every row.
Hydrogen (H) · Z = 11s↑
Step 1 of 4 · The biggest dropDrag along the curve to neon (10), the highest point in its row, and then one step right to sodium (11). What happens, and why?The simplest atom: one electron, one proton.
Effective nuclear charge on the outer electron (Slater): Zeff = Z − S = 1 − 0.00 = 1.00
The simplest atom: one electron, one proton.
Questions you might have (4)Why do the noble gases have the highest energy?
Their outer subshell is full, so each outer electron feels the largest pull of any atom in its row, and there is no empty box to be tempted by. Removing an electron breaks a very stable arrangement, and it costs a lot.
Why is helium the highest of all?
Its two electrons are in the first shell, as close to the nucleus as electrons can get, with nothing in between to shield them. Further down a column the outer electron is farther away and more shielded, so the energy falls: neon, argon, krypton, and so on.
Why does the energy fall going down a column?
Each row adds a shell. The outer electron sits farther from the nucleus and has a whole extra shell of electrons between it and the nucleus, so it is held more loosely, even though the nucleus is bigger. Compare lithium, sodium, potassium and rubidium on the curve.
The d block (Sc to Zn) is almost flat. Why?
The electrons being added go into the inner 3d subshell, below the 4s electrons that come off first. They shield the 4s electrons nearly as much as the extra proton pulls on them, so the outer electrons feel little extra pull.
Run the experiment to see what it showed.
TheoryWhy the trends run the way they doSize, grip and the ions that result follow from the shells and the pull of the nucleus.
Theory 3 · Measured today
How big is an atom?
Why is lithium so much larger than fluorine, and sodium larger than lithium?
In 1920 W. L. Bragg used X-ray measurements of crystals to give each atom a size. Atoms have no edge, so a size has to be defined from how closely they sit when bonded, but the pattern in the sizes was plain. Last experiment you saw that the pull on the outer electron changes across the table. If that is right, atoms must shrink as the pull grows. Click around the table and watch an atom change size, to see which of two causes is at work.
Colour: the covalent radius, small (violet) to large (gold). Dashed outlines are noble gases, whose radii are estimates because they hardly bond. Click any element; the dashed circle is the one you came from.
Step 1 of 3 · Across a rowClick lithium (Li), then boron (B), then fluorine (F). They are in the same row, so they have the same number of shells. What happens to the size as the protons go up?
Covalent radius133 pm
Outer shell n2
Outer electrons2s1
Zeff on outer electron1.30
The maths
Size of an atom ≈ a × n² / Zeff: n = the outer shell, Zeff = the pull the outer electron feels
Questions you might have (3)How can you measure the size of an atom, when it has no edge?
You cannot, in the sense of a hard surface. The electron cloud thins out gradually. So chemists define it by how close two atoms of the same element sit when bonded: half of that distance is the covalent radius. Different definitions give somewhat different numbers, but the trends are the same.
Why do the radii change so slowly across the d block?
The electrons being added go into an inner 3d subshell, which shields the outer 4s electrons almost as much as the extra proton pulls on them. Over the ten elements from scandium to zinc the radius falls only from 148 pm to 110 pm and then creeps back up, where lithium to fluorine fall from 133 pm to 64 pm in seven.
Why are the noble gas radii flagged?
They almost never form bonds, so there is no bond length to halve. Their radii here are estimates. The trend you should rely on is the main-group elements around them.
Work through it to see what it shows.
Theory 4 · Pauling, 1932
Tug of war for electrons
When two atoms share a pair of electrons, who holds it, and how do you tell from the table?
In 1932 Linus Pauling noticed that a bond between two different atoms is stronger than the average of the bonds between two of each, and he blamed unfair sharing: one atom pulls the shared electrons harder. He gave every atom a number for its pulling power and called it electronegativity. You now know what sets the pull: shells and effective nuclear charge. Put two atoms in a bond, guess where the shared pair sits, and see how the numbers decide.
Chlorine (Cl) – Chlorine (Cl): nonpolar covalent
Step 1 of 4 · Two identical atomsChlorine bonded to chlorine. Drag the electron pair to where you think it sits, then check. Which atom should win?
Click an element to place it as
Colour: electronegativity, low (violet) to high (gold). Helium, neon and argon have no value because they form no bonds.
Difference in EN0.00
Bond typenonpolar covalent
Ionic character0 %
Pair sits50 % toward Cl
The maths
Electronegativity (EN) is how hard an atom pulls on electrons it shares. Pauling's scale runs from about 0.7 to 4.0.
Cl: EN = 3.16 Cl: EN = 3.16 Difference = |3.16 − 3.16| = 0.00
Ionic character = 1 − e^(−ΔEN² / 4) = 1 − e^(−0.00 / 4) = 0 %. Bond type by the usual cut-offs: nonpolar covalent.
Questions you might have (4)Where do the electronegativity numbers come from?
Linus Pauling worked them out in 1932 from bond energies. A bond between two different atoms is stronger than the average of the two bonds between like atoms, and the extra strength grows with the difference in the atoms' pull on the pair. He set fluorine, the greediest atom, at 4.0 and scaled the rest from it. They are unitless, and chemists have built other scales since that agree closely.
How is it related to ionization energy and electron affinity?
An atom that pulls hard on shared electrons is one that holds its own tightly (high ionization energy) and would like one more (a large electron affinity: the energy released when a gaseous atom gains an electron). Robert Mulliken defined an electronegativity as roughly the average of the two. Over the 51 elements to xenon that have both numbers, electronegativity and first ionization energy have a correlation of about 0.9.
Is 0.4 or 2.0 a real dividing line?
No. The cut-offs are convenient rules of thumb, and different books use slightly different ones. Bonding runs smoothly from fully shared to fully handed over, which is why the sharing here is a number and not a box.
Are these the actual electron positions?
The positions are an estimate: Pauling's empirical formula for ionic character turned into a shift of the pair. Real molecules have measured dipole moments that show the same ordering (hydrogen chloride is less polar than hydrogen fluoride, and sodium chloride far more than either), but the percentages are a guide, not an exact answer.
Work through it to see what it shows.
Theory 5 · Kossel, 1916
Losing and gaining electrons
Why does sodium form Na⁺ and not Na²⁺, and chlorine Cl⁻ and not Cl²⁻?
In 1916 Walther Kossel proposed that atoms bond by passing electrons from one to another until each has the electron count of a noble gas, so that ions form. Is that right, and why does each atom stop where it does? You have the evidence already: the energies from the first experiment of this unit show where an atom runs out of cheap electrons. Take electrons off and put them on in the bench, and see where each atom stops.
Atom:
Na
1s2 2s2 2p6 3s1
3s↑
Step 1 of 4 · Make sodium ionChoose sodium, then remove one electron. Which electron goes, and what does the ion's configuration now look like?
Same electrons, different protons:
After the reveal, the circles are drawn in proportion to 1 / Zeff: a model for particles with the same shells. Measured ionic radii follow the same order.
Charge0
Electrons11
Protons11
Outer electrons3s1
The maths
A cation loses its outermost electrons first (highest shell, then highest l). An anion adds electrons to the next free place in the filling order.
Na: 11 electrons 1s2 2s2 2p6 3s1 unpaired: 1
Questions you might have (4)Why do transition metals like iron form more than one ion?
Fe²⁺ and Fe³⁺ are both common. Try them above: the 4s electrons leave first (the highest shell), giving Fe²⁺ = 3d⁶, and a third electron can then come from 3d, because the 3d and 4s energies are close and there is no big jump to stop it. The ionization energies climb steadily instead of jumping, so several charges are possible.
Why does the 4s electron leave before 3d when 4s filled first?
Once 3d has electrons, 4s is the outer shell: it is farther out on average, and the 3d electrons shield it. So it is the easiest to remove, even though it was the first of the two to fill. For ions, remove from the highest shell first.
Why is an anion bigger than its atom?
The added electron joins the same outer subshell, and every electron there repels the others. The nuclear charge is the same, so the pull per electron drops and the cloud swells. Cl⁻ is noticeably bigger than Cl.
What does 'isoelectronic' mean?
Having the same number of electrons (and here the same arrangement). N³⁻, O²⁻, F⁻, Ne, Na⁺, Mg²⁺ and Al³⁺ all have the ten electrons of neon. They are different only in nuclear charge, so they isolate the effect of the protons on size.
Work through it to see what it shows.
PracticeProblemsNumbers from this unit, in the style of a college problem set.
Practice
Practice
Numbers from the experiments, in the style of a college problem set. Every problem has a hint and a worked solution.
Quantum numbers0 right of 0 · streak 0
What is the largest number of electrons that can occupy the shell with principal quantum number n = 5?