Briefly describe each of the following ideas, methods, or phenomena: (a) diagonal relationship; (b) preparation of deionized water by ion exchange; (c) thermite reaction; (d) inert pair effect.

The periodic table is not only a display of elements sorted by increasing atomic number but also a remarkable layout revealing much about the chemical behavior of elements. Among its many patterns is the 'diagonal relationship', an intriguing phenomenon seen across periods and groups.

The diagonal relationship surfaces between certain pairs of elements that, despite not being in the same group, show strikingly similar properties. This kinship is best illustrated by elements in the second and third periods, like lithium (Li) and magnesium (Mg), or beryllium (Be) and aluminum (Al). But why does this happen? It's a nuanced balance of ionic sizes and charge intensities that fosters resemblances in reactivity and bonding tendencies. Essentially, as we move across a period, atoms shrink, and down a group, they enlarge. Diagonally adjacent elements fall in a sweet spot where their sizes and charges align closely, leading to similarities that they wouldn't share with their immediate neighbors.

When it comes to laboratory experiments or industrial applications, the presence of ions in water can impede processes and skew results. This is where deionized (DI) water comes into play – it's essentially water in an ultra-pure state, stripped of mineral ions.

To attain this purity, a method known as ion exchange is employed. Here's how it unfolds: An ion-exchange resin is bathed in water; this resin is like a chemical sponge, designed to grab onto mineral ions from the water, such as sodium (Na+), calcium (Ca2+), and various anions like chloride (Cl-). These resins swap these unwanted ions with harmless hydrogen (H+) and hydroxide (OH-) ions. When H+ and OH- meet, they rejoice, forming water (H₂O) and leaving behind a more purified state. This deionization not only removes ions but also eradicates unwanted conductivity in water – an essential requirement for precision in numerous scientific exercises.

Think of the thermite reaction as a pyrotechnic handshake between a metal and a metal oxide that releases a burst of heat and light. This reaction is exothermic, meaning it liberates energy, which it does so fiercely that it can weld railway tracks or slice through metal like butter.

At the heart of the reaction lies a mix comprising a metal, such as iron (Fe), and a metal oxide, like aluminum oxide (Al₂O₃). When ignited, the aluminum strips the oxygen away from iron oxide with great gusto, in the process creating molten iron and aluminum oxide. The result is a scalding mix that can reach temperatures over 2500°C (4500°F). It's a dramatic dance of reduction and oxidation serving practical purposes: from welding to metal cutting, and it's all thanks to the metals' strong desire to rearrange their oxidative states, turning reactants into hefty energy and new products.

Diving into the heavier realms of the periodic table, certain elements hold on to their electrons with a tighter grip than predicted. This phenomenon is termed the 'inert pair effect', where the inner s-electrons develop stage fright, refusing to participate readily in chemical bonding. As a result, elements like lead (Pb) often exhibit oxidation states that are two less than what their group position would suggest.

Typically, we'd expect lead to valiantly show its +4 oxidation state, but it tends to lean towards a cozier +2 state. This reluctance can be ascribed to the increasing effective nuclear charge and relativistic effects as we migrate down the periodic table, making the inner s-electrons more stabilized and thus less reactive. This has a profound impact on the chemistry of such elements, often resulting in a reduced number of available oxidation states – a curious quirk that underscores the nuances that come into play when protons and electrons congregate in heavier atoms.