Arrange the elements in order of increasing ionization energy: Ga, In, F, Si, N.

The periodic table is an invaluable tool in understanding various properties of elements, including ionization energy. Elements are arranged in order of increasing atomic number, laid out in a manner that groups elements with similar properties into columns and varying properties into rows or periods.

Ionization energy, which is the amount of energy needed to remove an electron from an atom, shows a noticeable trend within the periodic table. Across a period, ionization energy generally increases. This is due to the higher effective nuclear charge experienced by electrons as you move from left to right. The addition of protons in the nucleus as we go across a period means that electrons are more strongly attracted to the center, making them harder to remove.

In contrast, if you look at elements down a group, the ionization energy decreases even though the number of protons increases. This is because as we move down a group, new electron shells are added, increasing the distance between the nucleus and the outermost electrons, and thus reducing the effective nuclear charge felt by these electrons. Moreover, there's also an increase in electron shielding, with inner electrons repelling the outermost ones, making it easier to ionize atoms.

The atomic radius is the distance from the nucleus of an atom to the boundary of its surrounding cloud of electrons. Understanding atomic radius helps explain trends in ionization energy. A smaller atomic radius generally indicates that electrons are located closer to the nucleus and more tightly held by the nucleus's positive charge. As such, a greater ionization energy is required to remove these electrons.

Conversely, a larger atomic radius means that valence electrons are further from the nucleus and can be removed with less energy. This effect is seen when comparing elements within a group; as atomic radius increases down a group, the ionization energy decreases. This can be attributed to the addition of electron shells, increasing the distance over which the nuclear charge must work to hold on to an electron.

It's worth mentioning that cations, which are positively charged ions, generally have a smaller radius compared to their neutral atoms due to the loss of the outermost electron(s), whereas anions, with added electrons, have a larger radius. These changes in radius affect the ionization energy nonetheless.

Nuclear charge refers to the total charge of the nucleus due to the presence of protons, which is a contributor to an element's ionization energy. An important concept tied to nuclear charge is effective nuclear charge, which is the net positive charge experienced by electrons in the valence shell. It takes into account not only the total nuclear charge but also the shielding effect of electrons in inner shells.

The greater the effective nuclear charge, the tighter the nucleus holds onto its electrons, and as a result, higher is the ionization energy. In a period from left to right, for instance, the effective nuclear charge increases because additional protons in the nucleus attract the electrons more strongly. However, in a group from top to bottom, this effective charge decreases for the valence electrons due to increasing distance from the nucleus and the additional electron shielding from inner layers.

When we consider elements such as Ga and In from the exercise, despite both having a significant nuclear charge, the effective nuclear charge felt by the outer electrons in Ga is greater than in In. This is because Ga has fewer inner electron shells shielding the valence electrons from the nucleus, making its ionization energy relatively higher.