Light as evidence for electron energy changes

An electromagnetic spectrum is a range of electromagnetic radiation arranged by wavelength or frequency. Visible light makes up only a small part of this range: red light has a longer wavelength and lower frequency than violet light. From radio waves through microwaves, infrared, visible, ultraviolet, X-rays and gamma rays, wavelength gets shorter, while frequency and photon energy increase.

The relationship is

Photon energy is given by

A higher frequency therefore means higher energy. A longer wavelength means lower energy. That gives the colour–energy link: red has lower energy than blue or violet.

A photon is a packet of electromagnetic energy that is absorbed or emitted as a single unit. In atoms, photons come out when electrons lose energy. An excited state is an energy state of an atom in which at least one electron has more energy than it has in the lowest-energy arrangement. When an electron in an excited state drops back to a lower energy level, the atom emits a photon with energy equal to the difference between the two levels.

Continuous spectra and line spectra

A continuous spectrum is a spectrum containing an unbroken range of wavelengths, with no gaps between colours. Splitting white light with a prism produces this type of spectrum.

A line spectrum is a spectrum containing only particular wavelengths, shown as separate bright or dark lines. A hot, low-pressure gaseous element gives a line emission spectrum, a line spectrum with bright lines on a dark background because atoms emit photons at specific wavelengths. If a cold gas is placed in front of continuous light, it can produce an absorption spectrum, with dark lines where specific wavelengths have been absorbed.

The difference matters because a line emission spectrum acts as a fingerprint for an element. Different elements have different allowed electron energy changes, so they emit different sets of wavelengths. A spectroscope therefore shows more than our eyes can: two lamps may look similar in colour, while their line spectra can be quite different.

In a simple practical investigation, a gas discharge tube provides excited gaseous atoms, and a prism or diffraction grating separates the emitted light. Qualitative data include the colours and number of visible lines; quantitative data include measured wavelengths. Good records compare observed wavelengths with reference values, often by plotting observed wavelength against accepted wavelength and judging the accuracy from the agreement with a best-fit line.

Why hydrogen gives lines, not a smear

Hydrogen is the neatest place to begin because it has just one electron. Its line emission spectrum tells us something very specific: the electron cannot simply take any energy value. If every energy were allowed, the spectrum would appear continuous. Instead, the electron sits in discrete energy levels, which are allowed electron energy states separated by gaps in energy.

A ground state is the lowest-energy state available to an electron in an atom. In hydrogen, the electron is in the ground state when it is at the first energy level. If it absorbs exactly the right amount of energy, it can jump to a higher level. When it drops back down, it emits a photon. The photon energy equals the energy gap between the two levels, so each downward transition produces a particular wavelength.

At higher energies, the hydrogen energy levels crowd closer together. That is why the spectral lines bunch up, or converge, towards the high-frequency end of a series. For IB, don’t memorise the historical series names; focus on what the transitions mean.

Transitions to the first, second and third energy levels

Transitions that end at the first energy level release relatively large amounts of energy, so their photons fall in the ultraviolet region. Transitions ending at the second energy level include the visible hydrogen lines. The familiar red visible line comes from a smaller drop than the blue-violet lines, so it has a longer wavelength, lower frequency and lower energy. Transitions ending at the third energy level release less energy overall and lie in the infrared region.

This also gives a good nature-of-science example. Emission spectra provide evidence for different elements because each element has a different nuclear charge and electron arrangement, which creates a different pattern of allowed energy gaps. In astronomy, scientists can match dark absorption lines in starlight with known laboratory lines to infer which elements are present in stars.

When you study emission spectra with gas discharge tubes and prisms, the useful qualitative data are the colours, brightness and order of lines. The useful quantitative data are the wavelengths or frequencies of the lines. The instrument extends our senses: the naked eye may report “pink” or “blue-white”, but the spectrum gives a much more precise pattern.

The principal quantum number

A main energy level is a group of electron states in an atom with the same principal quantum number and a broadly similar distance from the nucleus. The principal quantum number, $n$, is a positive integer that labels a main energy level (unitless). The first level has $n = 1$, the second has $n = 2$, and so on.

A main energy level can hold a maximum of

electrons

So:

Be careful with the formula: it gives the capacity of the whole main energy level, not necessarily the number of electrons actually present in a ground-state atom.

Main energy level capacities calculated using 2n² for n = 1 to 4.

An atom’s highest occupied main energy level links to its period in the periodic table. For main-group elements, the period number gives the highest occupied value of $n$ in the ground-state atom. For example, elements in period 3 have their outer electrons in the third main energy level.

From orbits to orbitals

The Bohr model helps introduce discrete energy levels, but it only goes so far. Electrons are not tiny planets moving in fixed circular paths. In the quantum mechanical model, we talk about where an electron is likely to be found rather than the exact path it follows.

An atomic orbital is a region of space around a nucleus in which there is a high probability of finding an electron. The phrase “high probability” matters here. An orbital is not a shell, a track, or a container with a hard wall; it is a mathematical probability region, drawn as a boundary surface because that makes it easier to represent.

A sublevel is a subdivision of a main energy level containing orbitals of a particular type. Main energy levels split into s, p, d and f sublevels, with successively higher energy within a given main level. The number of sublevel types available in a main level equals $n$: for $n = 1$ only s is available; for $n = 2$, s and p; for $n = 3$, s, p and d; for $n = 4$, s, p, d and f.

Shapes you must recognize

An s orbital is an atomic orbital with spherical symmetry around the nucleus. Higher s orbitals are larger on average, but for IB shape recognition the key point is simple: s is spherical.

A p orbital is an atomic orbital with two lobes on opposite sides of the nucleus, often described as dumbbell-shaped. A p sublevel contains three p orbitals, oriented along the x, y and z axes. These are commonly labelled $p_x$, $p_y$ and $p_z$. Each orientation has the same shape, just pointing in a different direction.

The periodic table blocks link directly to sublevels. The s-block elements are filling s sublevels, the p-block elements are filling p sublevels, the d-block elements are filling d sublevels, and the f-block elements are filling f sublevels. That’s why the periodic table is not just a list: its shape reflects electron arrangements.

Orbitals, sublevels and spin

An electron configuration is a notation that shows how the electrons of an atom or ion are arranged in orbitals and sublevels. It matters because the outer, or valence, electrons have a strong influence on chemical properties.

For a given electron configuration and chemical environment, each orbital has a particular energy. Each orbital can hold a maximum of two electrons. The number of orbitals in each sublevel is fixed:

Number of orbitals and electron capacity for each sublevel.

Degenerate orbitals are orbitals in the same sublevel with the same energy. The three p orbitals in one p sublevel are degenerate; the five d orbitals in one d sublevel are too.

An orbital diagram shows electron occupancy using arrows in boxes: each box is one orbital, and each arrow is one electron. The relative vertical position of the boxes can show relative energy. Up and down arrows mean opposite spin. Don’t picture the electrons as literal spinning balls, but do use the notation correctly.

The Pauli exclusion principle says that an orbital can contain at most two electrons and, if there are two, they must have opposite spins. In an orbital diagram, a full orbital is written as $\uparrow\downarrow$, not $\uparrow\uparrow$.

The Hund’s rule says that electrons fill degenerate orbitals singly, with parallel spins, before they pair up. For example, a $p^3$ arrangement is $\uparrow\ \uparrow\ \uparrow$ across the three p boxes, not $\uparrow\downarrow\ \uparrow$ with one empty p orbital. This reduces electron–electron repulsion and gives a more stable arrangement.

The Aufbau principle says that electrons go into the lowest available energy orbitals before higher-energy orbitals are occupied. For atoms up to atomic number 36, use this filling order:

$1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p$

xxxFormulaEndxxx In a neutral atom, the number of electrons is also $Z$.

Writing configurations up to $Z = 36$

A full electron configuration is an electron configuration written from 1s onwards, with every occupied sublevel shown. Calcium has $Z = 20$, so its full configuration is:

$1s^2 2s^2 2p^6 3s^2 3p^6 4s^2$

A condensed electron configuration is an electron configuration that uses the symbol of the previous noble gas in square brackets to replace the inner electrons. For calcium, this is:

[Ar] $4s^2$

You’re expected to know both forms. The condensed version is not a shortcut for skipping the thinking; it separates the noble-gas core from the outer electrons.

For ions, start by working out the number of electrons. A positive ion has lost electrons; a negative ion has gained electrons. For main-group ions, add or remove electrons to match the ion’s electron count, then write the configuration. For transition-metal ions in this syllabus range, there’s one key rule: when cations form, 4s electrons are removed before 3d electrons. So Fe is [Ar] $4s^2 3d^6$, but Fe$^{2+}$ is [Ar] $3d^6$, not [Ar] $4s^2 3d^4$.

Chromium and copper

Chromium and copper are the two ground-state exceptions you need to know. From the Aufbau order, you would expect Cr: [Ar] $4s^2 3d^4$ and Cu: [Ar] $4s^2 3d^9$. The observed configurations are:

• Cr: [Ar] $4s^1 3d^5$
• Cu: [Ar] $4s^1 3d^{10}$

The reason is the extra stability linked to half-filled and completely filled d sublevels. In chromium, $3d^5$ gives five singly occupied d orbitals. In copper, $3d^{10}$ gives a full d sublevel. If copper forms Cu$^{2+}$, remove the 4s electron first and then one 3d electron, giving [Ar] $3d^9$.

First ionization energy and the convergence limit

First ionization energy is the energy needed to remove one electron from each atom in one mole of gaseous atoms, forming one mole of gaseous 1+ ions. It is usually reported in kJ mol$^{-1}$. The process is:

$X(g) \to X^+(g) + e^-$

The convergence limit is the high-frequency end of a set of spectral lines, where the lines get so close together that they merge. In an emission spectrum, this limit matches ionization: the electron has moved from a bound energy level into the continuum, where the atom no longer holds it.

If you know the convergence wavelength or frequency, you can calculate the first ionization energy. For one photon, use $E = hf$. If wavelength is given, combine this with $c = \lambda f$ to get $E = hc / \lambda$.

To change the energy for one atom into the energy for one mole, use

Dividing by 1000 converts J mol$^{-1}$ to kJ mol$^{-1}$.

A typical calculation route is:

1. If given $\lambda$, calculate $f$ using $c = \lambda f$, or go directly to $E = hc / \lambda$.
2. Calculate the energy for one atom in J.
3. Multiply by $N_A$ to get J mol$^{-1}$.
4. Divide by 1000 to get kJ mol$^{-1}$.

Trends across periods and down groups

First ionization energy generally increases across a period. As you move across, the number of protons increases, while added electrons go into the same main energy level. Shielding changes only slightly, so the outer electron feels a stronger attraction to the nucleus and becomes harder to remove.

First ionization energy generally decreases down a group. The outer electron is in a higher main energy level, farther from the nucleus, and inner electrons provide more shielding. This extra distance and shielding outweigh the higher nuclear charge, so less energy is needed to remove the outer electron.

These trends help explain the broad metal–non-metal pattern in the periodic table. Elements with low first ionization energies lose electrons more readily and tend to show metallic behaviour. Elements with high first ionization energies hold onto electrons more strongly and tend to show non-metallic behaviour.

The important discontinuities

There are two standard dips across a period that you need to explain using electron configurations.

Between group 2 and group 13, the first ionization energy drops. For example, Be is $1s^2\ 2s^2$, while B is $1s^2\ 2s^2\ 2p^1$. In boron, the electron removed is a 2p electron. It is slightly higher in energy and more shielded than a 2s electron, so it is easier to remove.

Between group 15 and group 16, the first ionization energy also drops. For example, N is $1s^2\ 2s^2\ 2p^3$, while O is $1s^2\ 2s^2\ 2p^4$. Nitrogen has three singly occupied p orbitals; oxygen has one paired set in a p orbital. Repulsion between the paired electrons in oxygen makes one electron easier to remove.

Log scales are useful when ionization energies cover a large range, just as they are useful for hydrogen ion concentration. A logarithmic plot compresses very large values while preserving patterns, so jumps and trends can be compared more clearly on one graph.

What successive ionization energies reveal

Successive ionization energies are the energies needed to remove electrons one by one from the same element, producing ions with increasing positive charge. The values always rise because, each time an electron is removed, the remaining electrons are still attracted by the same nucleus but have less electron–electron repulsion between them.

The pattern matters more than the steady rise itself. A very large jump means the next electron is coming from a lower main energy level, often once all the valence electrons have already gone. That jump shows how many electrons were in the outer shell, so you can use it to deduce the group.

For example, a largest jump between the first and second ionization energies means the atom had one outer electron, so it is in group 1. If the biggest jump comes between the third and fourth ionization energies, the atom had three outer electrons, so it is in group 13. For main-group elements, the method is simple: count the electrons removed before the big jump.

A database is an organized collection of retrievable data. In this topic, databases help because ionization energy data for many elements can be imported into a spreadsheet, graphed, and compared. Plotting successive ionization energy against ionization number makes the jumps visible; plotting $\log(\text{successive ionization energy})$ can make the pattern easier to inspect when the values differ by orders of magnitude.

Successive ionization energies also support the idea of energy levels and sublevels. Large jumps show changes between main energy levels, while smaller irregularities can point to sublevel structure. For transition elements, these patterns help explain variable oxidation states: $4s$ electrons are removed first, and then different numbers of $d$ electrons may be removed depending on the element and chemical environment.