Exercises: Periodic table and trends

At the end of the 19th century, Dmitri Mendeleev arranged the chemical elements in order of increasing atomic mass, which caused certain discrepancies in pairs such as tellurium ($Z = 52$, atomic mass $127.6\text{ u}$) and iodine ($Z = 53$, atomic mass $126.9\text{ u}$). Henry Moseley later resolved this anomaly using X-ray spectroscopy.

1. a) Explain the basis of Moseley's law and why the atomic number $Z$, and not the atomic mass, is the correct criterion for periodic ordering.
2. b) Justify the relative positions of tellurium and iodine in the current periodic table in terms of their valence electron configurations.

The ground-state electron configurations determine the position of chemical elements in the periodic table. Consider three neutral elements called A, B and C, whose valence shells are: A: $3s^2\,3p^1$; B: $[\mathrm{Ar}]\,4s^2$ (complete configuration except for the abbreviated core); C: $3s^2\,3p^5$.

1. a) Deduce the period and group (using the IUPAC numbering from 1 to 18) to which each of the three elements belongs.
2. b) Classify each element, giving reasons, as a metal, non-metal or metalloid.

Electronegativity quantifies the relative tendency of a bonded atom to attract the electron density of the chemical bond towards itself.

1. a) Explain how electronegativity varies across a period of the periodic table from left to right, justifying the behaviour in terms of effective nuclear charge and atomic radius.
2. b) Justify why noble gases generally lack defined electronegativity values on the usual Pauling scale.

Successive ionisation energies make it possible to identify the valence shell electron configuration of a chemical element. For a representative element $X$ in the third period of the periodic table, the values of its first four ionisation energies have been measured experimentally: $I_1 = 580\text{ kJ}\times\text{mol}^{-1}$, $I_2 = 1815\text{ kJ}\times\text{mol}^{-1}$, $I_3 = 2740\text{ kJ}\times\text{mol}^{-1}$ and $I_4 = 11600\text{ kJ}\times\text{mol}^{-1}$. The relative magnitude of these energies is illustrated in the following bar chart:

1. a) Justify, giving reasons, the most stable chemical valency or oxidation state of $X$ in its compounds with electronegative elements.
2. b) Deduce which group of the periodic table $X$ belongs to and identify the specific element.

You can do this on paper and send a photo with your answer.

You are given a series of isoelectronic monatomic chemical species: $\mathrm{N}^{3-}$ ($Z=7$), $\mathrm{O}^{2-}$ ($Z=8$), $\mathrm{F}^-$ ($Z=9$), $\mathrm{Na}^+$ ($Z=11$) and $\mathrm{Mg}^{2+}$ ($Z=12$).

1. a) Write the ground-state electron configuration common to all these species and order the ions, giving reasons, from smallest to largest ionic radius.
2. b) Explain why the radius of the cation $\mathrm{Na}^+$ is significantly smaller than the atomic radius of the neutral $\mathrm{Na}$ atom ($Z=11$), whereas the radius of the anion $\mathrm{F}^-$ is larger than that of the neutral $\mathrm{F}$ atom ($Z=9$).

The first ionisation energy generally increases across a period from left to right. However, occasional anomalies are observed when comparing adjacent elements in the second and third periods.

1. a) The first ionisation energy of magnesium ($Z = 12$) is $738\text{ kJ}\times\text{mol}^{-1}$, whereas that of aluminium ($Z = 13$) is $578\text{ kJ}\times\text{mol}^{-1}$. Justify this discrepancy based on the electron configurations and the type of sublevel from which the electron is removed.
2. b) The first ionisation energy of nitrogen ($Z = 7$) is $1402\text{ kJ}\times\text{mol}^{-1}$, higher than that of oxygen ($Z = 8$), which is $1314\text{ kJ}\times\text{mol}^{-1}$. Explain this using Hund's rule and the stability of half-filled sublevels.

Electron affinity measures the energy change when a neutral gaseous atom gains an electron: $\mathrm{X}_{\text{(g)}} + \mathrm{e}^- \to \mathrm{X}^-_{\text{(g)}}$. According to the usual thermodynamic convention, an exothermic process has $\Delta H_{\text{ae}} < 0$. The thermodynamic values for the halogens chlorine and fluorine are $\Delta H_{\text{ae}}(\mathrm{Cl}) = -349\text{ kJ}\times\text{mol}^{-1}$ and $\Delta H_{\text{ae}}(\mathrm{F}) = -328\text{ kJ}\times\text{mol}^{-1}$, while for gaseous nitrogen it is $\Delta H_{\text{ae}}(\mathrm{N}) \ge 0$.

1. a) Explain why adding an electron to fluorine releases less energy than for chlorine, even though fluorine has higher electronegativity and a smaller atomic radius.
2. b) Justify in detail why nitrogen has an unfavourable electron affinity (endothermic or zero) compared with carbon ($\Delta H_{\text{ae}} = -122\text{ kJ}\times\text{mol}^{-1}$).

Four representative chemical elements, labelled consecutively W, X, Y and Z, are analysed. The following experimental data are known:

1. W and X belong to the same period; W is an alkaline earth metal and X is a non-metal in group 16.

2. Y belongs to the same group as W and to the third period; its atomic radius is smaller than that of W.

3. Z has the electron configuration $[\mathrm{Ne}]\,3s^2\,3p^5$.

4. Element W has a first ionisation energy of $590\text{ kJ}\times\text{mol}^{-1}$ and an atomic radius of $197\text{ pm}$.

1. a) Deduce the identity and complete electron configuration of W, X, Y and Z, given that W belongs to the fourth period.
2. b) Arrange the four elements in order of increasing first ionisation energy and justify the order of electronegativity between X and Z.

The concept of effective nuclear charge ($Z_{\text{ef}} = Z - \sigma$, where $\sigma$ is the shielding constant) underpins the periodic variations in radius and redox reactivity.

1. a) Using the shielding model, justify why, going down group 1 of the alkali metals (from $\mathrm{Li}$ to $\mathrm{Cs}$), the atomic radius increases significantly despite the enormous increase in the total nuclear charge $Z$ (from $Z=3$ to $Z=55$).
2. b) Consider the ions $\mathrm{Na}^+$ ($Z=11$) and $\mathrm{K}^+$ ($Z=19$), with noble gas configurations of 10 and 18 electrons respectively. The ionic potential is defined here as $|z|/r$, where $z$ is the charge number and $r$ is the ionic radius. Justify which ion has the greater ionic potential and, according to a simple electrostatic model, which has the more negative enthalpy of hydration.

Topic challenge. How does quantum physics explain energy savings in our streets? Design a technical proposal for converting traditional street lighting to more efficient technologies.

1. Data collection: Go to the EirGrid (Smart Grid Dashboard, system demand) website and note down the electricity demand of your region at midnight compared with midday.

2. Sodium spectrum: Classic street lamps use $\text{Na}$ vapour ($Z=11$), with its characteristic doublet centred at $\lambda = 589.0\text{ nm}$. Calculate the energy of this photon ($E = h \times c / \lambda$) and give the four possible quantum numbers for the valence electron in its ground state and in the excited orbital to which it is promoted.

3. Trends and new materials: To replace it with LED luminaires, semiconductors containing gallium ($\text{Ga}$) or indium ($\text{In}$) are used. Write their electron configurations and explain, using effective nuclear charge and shielding, why their ionisation energy and atomic radius differ from those of sodium.

4. Reasoned proposal: Write a one-page report justifying the technological replacement, relating the electronic transition, the periodic properties of the proposed material and the potential impact on EirGrid's night-time demand curve.

Assessment will consider:

• Accuracy in quantum-energy calculations and rigorous assignment of quantum numbers.

• Coherent justification of periodic properties according to the quantum-mechanical model.

• Ability to connect the EirGrid data with atomic structure and energy efficiency.