Within our model and referring to the Coulombic Potential Energy expression, explain why the \(\mathrm{IE}_{1}\) increases from left to right across a row of the periodic table.

The periodic table is a systematic arrangement of elements that reveals a lot about the properties of atoms and how they interact. When we examine the periodic table horizontally, moving from left to right across a row (also known as a period), we see a notable increase in ionization energy (IE).

This trend can be attributed to several factors. As we move across a period, each successive element has one more proton in its nucleus and one more electron in the surrounding cloud than the previous element. This steady increase in the number of protons amplifies the nuclear charge, which consequently enhances the nucleus’s ability to hold onto its electrons. Moreover, the additional electrons with each step are added to the same principal energy level, which does not significantly increase the average distance of the outer electrons from the nucleus.

As a result, the increased positive charge has a stronger pull on the electrons in the same energy level, which means more energy is needed to remove an electron, and thus the ionization energy rises. It's like a tug-of-war, where each step to the right adds a stronger puller on the nucleus' team, making it harder for an electron to break free.

Coulomb's Law is a cornerstone of electromagnetism and provides critical insight into why ionization energy trends behave as they do on the periodic table. It states that the electrostatic force between two charged objects is directly proportional to the product of the charges, and inversely proportional to the square of their separation distance (\( F \text{ is proportional to } \frac{q_1 \times q_2}{r^2} \text{, where } F \text{ is force, } q \text{ is charge, and } r \text{ is distance}\)).

When we consider Coulomb's Law in the context of atoms, it explains the interaction between positively charged protons in the nucleus and negatively charged electrons. If we increase the charge by adding protons (which we do when moving across a period), the force of attraction between the nucleus and the electron cloud grows stronger, assuming the distance doesn’t change significantly. This stronger force means that higher ionization energy is required to overcome it and remove an electron from the atom. This law effectively elucidates the underlying physics of why it's tougher to strip electrons from atoms with higher atomic numbers within the same period.

Nuclear charge is essentially the total charge of the protons in the nucleus of an atom. It has a profound influence on ionization energy because it's one of the main factors that determines the force holding the electrons in the atom. As the nuclear charge increases, this stronger positive charge exerts a greater force on the electron cloud, especially on the valence (outermost) electrons, pulling them closer to the nucleus.

This converging effect not only impacts ionization energy but also size of the atom. Higher nuclear charge without a corresponding increase in electron shielding or increase in the principal energy level leads to smaller atomic radii. Additionally, in chemical reactions, elements with higher nuclear charges typically have a stronger ability to attract electrons from other atoms, which is critical in bond formation.

In summary, the nuclear charge is a pivotal factor that dictates not only the ionization energy but various other chemical properties. It’s the increasing nuclear charge across a period that reinforces the upward trend of ionization energies.