Comprehensive Guide to Atomic Structure and Quantum Chemistry Notes

Fundamental Properties of Elements: Atomic and Nucleon Numbers

In 1913, the scientist Moseley conducted experiments observing that when different elements were bombarded with cathode rays, they produced X-rays of specific characteristic frequencies. Moseley determined that the square root of the frequency of these X-rays, denoted as $\sqrt{\nu}$, is directly proportional to the atomic number of the element ($Z$). This led to the conclusion that the atomic number is a fundamental property of an element, more so than atomic mass. The atomic number, also referred to as the proton number, represents the total number of protons within the nucleus. The identity of an element is defined by this number.

The nucleus also contains neutrons, and together, protons and neutrons are known as nucleons. The total number of these particles is the nucleon number ($A$), which is also known as the mass number. The relationship between these values is expressed by the equation $A = Z + N$, where $N$ represents the number of neutrons. To find the number of neutrons in an atom, the formula is rearranged to $N = A - Z$. For example, an atom of Aluminum ($Al$) has an atomic number ($Z$) of $13$ and a mass number ($A$) of $27$. Therefore, the number of neutrons is $27 - 13 = 14$. This notation is often written as ${}_{13}^{27}Al$.

The concept of atomic and nucleon numbers also applies to ions. When an atom loses or gains electrons, its identity remains the same because the proton number ($Z$) does not change. For instance, when a neutral Aluminum atom loses three electrons to become an $Al^{3+}$ ion, the number of protons remains $13$ and neutrons remain $14$, but the number of electrons decreases to $13 - 3 = 10$. Conversely, when a Chlorine atom gains an electron to form a $Cl^{-}$ ion, the number of protons is $17$ and neutrons is $18$, but the electron count increases to $17 + 1 = 18$. Other examples include oxide ions ($O^{2-}$), sulfide ions ($S^{2-}$), and phosphide ions ($P^{3-}$), where the change in electron count is reflected in the ion's charge while proton and neutron numbers remain consistent with the neutral species.

Behavior of Fundamental Particles in an Electric Field

The fundamental particles of an atom—protons, electrons, and neutrons—behave differently when passed through an electric field at the same speed. This behavior is determined by their respective masses and electrical charges. Neutrons, being electrically neutral, are not deflected by the electric field and continue to travel in a straight path perpendicular to the direction of the field. Protons, which carry a positive charge, are deflected toward the negative plate of the electric field. Electrons, carrying a negative charge, are deflected toward the positive plate.

The extent of deflection varies significantly between protons and electrons. Electrons exhibit a much greater degree of deflection than protons because they are approximately $1836$ times lighter. The deviation from the original path can be quantified in two ways. The angle of deflection is directly proportional to the ratio of charge to mass ($\text{Angle of deflection} \propto \frac{\text{charge}}{\text{mass}}$). Alternatively, if we imagine the particle moving in a circular path after deflection, the radius of deflection is proportional to the ratio of mass to charge ($\text{Radius of deflection} \propto \frac{\text{mass}}{\text{charge}}$). These factors indicate that lighter or more highly charged particles will experience more significant path changes.

Ionization Energy and Electronic Shell Structure

Ionization energy is the energy required to remove an electron from an atom and serves as a primary tool for investigating electronic configurations. This can be studied through successive ionization energies of a single element or the first ionization energies across different elements. Successive ionization energies involve removing electrons one by one until only the nucleus remains ($X \rightarrow X^{+} + e^{-}$, then $X^{+} \rightarrow X^{2+} + e^{-}$, and so on). Because all ionization processes are endothermic, they require an input of energy. The trend in successive ionization energies provides evidence for the arrangement of electrons in shells.

Using Magnesium ($Mg$) as a case study, a plot of successive ionization energies against the number of electrons removed reveals distinct jumps. The first and second electrons are removed from the valence (third) shell with relatively low energy. However, a large increase (jump) occurs when removing the third electron, as it must be taken from the second shell, which is much closer to the nucleus and experiences a stronger nuclear pull. The next seven electrons (from the 3rd to 9th removed) show a gradual increase as they are all from the same second shell. A second massive jump occurs when removing the 11th and 12th electrons, which reside in the first shell, right next to the nucleus. These two large jumps confirm that Magnesium electrons exist in three distinct shells.

First ionization energy trends across the periodic table further illuminate atomic structure. As one moves down a group (e.g., from Helium to Neon, or Lithium to Sodium), the first ionization energy decreases. This is because the addition of electronic shells increases the distance between the nucleus and valence electrons, weakening the nuclear attraction. Conversely, across a period, the ionization energy generally increases. In a period, the shell number remains the same, but the proton number increases, leading to a stronger nuclear pull on the electrons. Consequently, alkali metals in Group 1 have the lowest ionization energies in their periods, while noble gases have the highest.

Quantum Numbers and Orbital Descriptions

While the Bohr model utilized a single quantum number ($n$) to describe one-dimensional orbits, the Schrödinger model requires three quantum numbers to define orbitals in three-dimensional space: the principal ($n$), azimuthal ($l$), and magnetic ($m$) quantum numbers. A fourth, the spin quantum number ($s$), is needed to distinguish between two electrons in the same orbital. The principal quantum number ($n$) takes positive integer values ($1, 2, 3, ...$) associated with shells $K, L, M, N, ...$. It defines the size and energy of the orbital; as $n$ increases, the electron is further from the nucleus and less tightly bound. The maximum number of electrons a shell can hold is calculated using the formula $2n^2$, resulting in capacities of $2, 8, 18,$ and $32$ for the first four shells.

The azimuthal quantum number ($l$) determines the shape of the orbital and can have integer values from $0$ to $(n-1)$. These values correspond to subshells labeled $s, p, d,$ and $f$ (representing sharp, principal, diffused, and fundamental). If $l = 0$, the orbital is an $s$ subshell; $l = 1$ is $p$; $l = 2$ is $d$; and $l = 3$ is $f$. The number of electrons in a subshell is given by $2(2l + 1)$, meaning $s$ holds $2$, $p$ holds $6$, $d$ holds $10$, and $f$ holds $14$. The magnetic quantum number ($m$) describes the spatial orientation of an orbital and has values ranging from $-l$ to $+l$, including zero. This means an $s$ subshell ($l = 0$) has one orbital, a $p$ subshell ($l = 1$) has three orbitals ($m = -1, 0, +1$), a $d$ subshell ($l = 2$) has five orbitals, and an $f$ subshell ($l = 3$) has seven orbitals. Orbitals within the same subshell have the same energy and are called degenerate orbitals.

The spin quantum number ($s$) accounts for the electron's rotation around its own axis, which generates a magnetic field. There are only two possible spin states: clockwise, represented by an upward arrow ($\uparrow$) with a value of $+1/2$, and anticlockwise, represented by a downward arrow ($\downarrow$) with a value of $-1/2$. In any given shell $n$, there are $n$ subshells, $n^2$ orbitals, and a maximum of $2n^2$ electrons.

Shapes of Atomic Orbitals and Filling Rules

Atomic orbitals are regions in space where the probability of finding an electron is highest. The $s$-orbital ($l = 0$) is spherically symmetrical, with electron density distributed uniformly. As the principal quantum number $n$ increases, the size of the $s$-orbital grows (1s < 2s < 3s). The $p$-orbital ($l = 1$) consists of two lobes and is not spherically symmetric. There are three $p$-orbitals—$p_x, p_y,$ and $p_z$—oriented along the mutually perpendicular x, y, and z axes. The $d$-orbitals ($l = 2$) have five orientations: $d_{xy}, d_{xz},$ and $d_{yz}$ (lying in the respective planes), $d_{x^2-y^2}$ (lobes along the axes), and $d_{z^2}$ (two lobes along the z-axis with a central doughnut shape). The $f$-orbitals ($l = 3$) have seven complex orientations.

The filling of these orbitals follows specific principles. The Aufbau principle states that subshells are filled in order of increasing energy based on the $(n + l)$ rule. The subshell with the lower $(n + l)$ value is filled first. If two subshells have the same $(n + l)$ value, the one with the lower $n$ value is filled first. For example, a $4s$ orbital ($4 + 0 = 4$) is filled before a $3d$ orbital ($3 + 2 = 5$). Similarly, $4p$ ($4 + 1 = 5$) and $3d$ ($5$) are compared, and $3d$ is filled first due to the lower $n$. The general order is 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 6s < 4f < 5d < 6p < 7s.

Two other critical rules govern electron distribution: Pauli's Exclusion Principle and Hund's Rule. Pauli's Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers, which implies that an orbital can hold a maximum of two electrons with opposite spins. Hund's Rule applies to degenerate orbitals; it states that when filling orbitals of equal energy, electrons should be placed singly in each orbital with the same spin before any pairing occurs. For example, in a Nitrogen atom ($Z = 7$, $1s^2 2s^2 2p^3$), each of the three $2p$ orbitals ($p_x, p_y, p_z$) receives one electron with a parallel spin.

Questions & Discussion

During the examination of atomic structure, several specific scenarios and data points were discussed. It was noted that the second ionization energy of Magnesium ($I_2$) is much smaller than its third ($I_3$) because the third electron must be removed from a stable, inner shell (2nd shell), whereas the second electron is still in the valence shell. In the context of first ionization energies for Li, K, Ca, S, and Kr, Potassium ($K$) has the lowest because it is an alkali metal with the largest atomic radius in this set, while Krypton ($Kr$) has the highest due to its stable noble gas configuration and high nuclear charge for its period.

For the element Potassium ($Z = 19$), the full electronic configuration is written as $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1$. The $4s$ subshell fills before $3d$ because its $(n+l)$ value is lower (4 < 5). In another example, an atom with $Z = 17$ and $A = 35$ (Chlorine) contains $17$ protons, $18$ neutrons, and $17$ electrons; if it forms a $-1$ charge ion, it will have $18$ electrons while proton and neutron counts remain unchanged. For Mercury ($Hg, Z = 80$), specific shell occupancy includes $18$ electrons in the $n=3$ shell and $10$ electrons in the $4d$ subshell. Any single $4p$ orbital (like $4p_y$) can hold only $2$ electrons. In a phosphorus atom ($Z = 15$), following Hund's rule, the valence shell ($3s^2 3p^3$) features three unpaired electrons in the $3p$ orbitals. Finally, an unknown element with ionization energies $I_1 = 896$, $I_2 = 1752$, and a massive jump to $I_3 = 14807$ implies it has two valence electrons and thus belongs to the alkaline earth metal family (Group 2).