In the **Bohr model**, an electron orbits the positively charged nucleus in the same way that the earth orbits the Sun. Electrostatic attraction pulls the electron toward the nucleus and the electron orbits at high speed to prevent it from crashing into the nucleus. The electron can orbit at different energy levels (n=1, n=2, n=3) and the higher the energy level, the larger the radius from the nucleus. When an electron transitions from a higher energy level to a lower energy level, it emits electromagnetic radiation. The emission spectrum of hydrogen consists of sharp, distinct lines.^{1} The distinct lines of the emission spectrum prove that electron energy is quantized into energy levels. If electron energy is not quantized, then a continuous spectrum would be observed. The energy of the energy levels is governed by:

*, *where E is energy and n is the energy level.

The equation is negative, so all energies are negative. Negative energies mean that it is energy that contributes to the “stability” of the system – the electron binding energy. The more negative (lower) the energy, the more stable the orbit, the harder it is to knock out the electron. The less negative (higher) the energy, the less stable the orbit and the easier it is to knock out the electron. At the highest energy, 0 eV, there is no binding energy, so the electron dissociates. For atoms other than hydrogen, the shape of the energy level curve stays the same. However, the numerator is a constant other than 13.6 eV. The precise relationship for atoms other than hydrogen is:

*, *where Z is the atomic number.

Higher Z values give more negative binding energy (more stable) because the more charge, the more electrostatic attraction. The wavelength of the emitted or absorbed radiation is governed by the Rydberg formula:

, where lambda is the wavelength, n_{f} is the final energy level, n_{i} is the initial energy level, and R is the Rydberg constant.^{1}

The energy of the emitted or absorbed radiation is:

, where E is energy, f and v both mean frequency and c is the speed of light.

Energy is emitted for transitions to lower energy levels (n_{f} < n_{i}) and energy is absorbed for transitions to higher energy levels (n_{f} > n_{i}).

**Aufbau principle** states that shells / subshells of lower energy gets filled first.^{4} For example, 1s fills first, then 2s, then 2p and so on. Note, however, that the d subshells get filled after the s.

**Hund’s rule** state that when you fill a subshell with more than 1 orbital (p, d, f), you first fill each orbital with a single electron and with the same spin The reason for Hund’s rule is that electron-electron repulsion in doubly occupied orbitals make them higher in energy than singly occupied orbitals.

**Pauli Exclusion Principle** states that 2 electrons in the same orbital must be of different spins. Modern atomic theory postulates that any electron in an atom can be completely described by four quantum numbers: *n*, *l*, *m _{l}*, and

*m*

_{s}. Furthermore, according to the Pauli Exclusion Principle, no two electrons in a given atom can possess the same set of four quantum numbers. The position and energy of an electron described by its quantum numbers is known as its

**energy state**. The value of

*n*limits the values of

*l*, which in turn limit the values of

*m*. In other words, for a given value of

_{l}*n*, only particular values of

*l*are permissible; given a value of

*l*, only particular values of

*m*are permissible. The values of the quantum numbers qualitatively give information about the orientation of the orbitals.

_{l }The first quantum number is commonly known as the **principal quantum number** and is denoted by the letter ** n**.

^{1}This is the quantum number used in Bohr’s model that can theoretically take on any positive integer value. The larger the integer value of

*n,*the higher the energy level and radius of the electron’s

**shell**. The second quantum number is called the

**azimuthal**(

**angular momentum**)

**quantum number**and is designated by the letter

**. The second quantum number refers to the shape and number of**

*l***subshells**within a given principal energy level (shell). The third quantum number is the

**magnetic quantum number**and is designated

**. The magnetic quantum number specifies the particular orbital within a subshell where an electron is most likely to be found at a given moment in time. Each orbital can hold a maximum of two electrons. The fourth quantum number is called the**

*ml***spin quantum number**and is denoted by

*m**s*.

^{4}In classical mechanics, an object spinning about its axis has an infinite number of possible values for its angular momentum. However, this does not apply to the electron, which has two spin orientations designated (+1/2) and (-1/2). Whenever two electrons are in the same orbital, they must have opposite spins. In this case, they are often referred to as being paired. Electrons in different orbitals with the same ms values are said to have parallel spins. For a given atom or ion, the pattern by which subshells are filled, as well as the number of electrons within each principal energy level and subshell, are designated by its electron configuration. Electron configurations use spectroscopic notation, wherein the first number denotes the principal energy level, the letter designates the subshell, and the superscript gives the number of electrons in that subshell. For example, 2

*p*

^{4}indicates that there are four electrons in the second (

*p*) subshell of the second principal energy level. This also implies that the energy levels below 2

*p*(that is, 1

*s*and 2

*s*) have already been filled, as shown below:

Bohr came to describe the structure of the hydrogen atom as a nucleus with one proton forming a dense core, around which a single electron revolved in a defined pathway (**orbit**) at a discrete energy value. If one could transfer an amount of energy exactly equal to the difference between one orbit and another, this could result in the electron “jumping” from one orbit to a higher-energy one. These orbits had increasing radii, and the orbit with the smallest, lowest-energy radius was defined as the ground state (*n* = 1). More generally, the **ground state** of an atom is the state of lowest energy, in which all electrons are in the lowest possible orbitals. In Bohr’s model, the electron was promoted to an orbit with a larger radius (higher energy), the atom was said to be in the excited state. In general, an atom is in an **excited** **state** when at least one electron has moved to a subshell of higher than normal energy. Bohr likened his model of the hydrogen atom to the planets orbiting the sun, in which each planet traveled along a roughly circular pathway at set distances (and energy values) from the sun. Bohr’s model was reconsidered over the next two decades, but remains an important conceptualization of atomic behavior. In particular, we now know that electrons are not restricted to specific pathways, but tend to be localized in certain regions of space.

At room temperature, the majority of atoms in a sample are in the ground state. However, electrons can be excited to higher energy levels by heat or other energy forms to yield excited states. Because the lifetime of an excited state is brief, the electrons will return rapidly to the ground state, resulting in the emission of discrete amounts of energy in the form of photons.

* *

The electromagnetic energy of these photons can be determined using the following equation:

*E = hc/λ*

where h is **Planck’s constant**, c is the speed of light in a vacuum and λ is the wavelength of the radiation. The electrons in an atom can be excited to different energy levels. When these electrons return to their ground states, each will emit a photon with a wavelength characteristic of the specific energy transition it undergoes. These energy transitions do not form a continuum, but rather are quantized to certain values. Thus, the spectrum is composed of light at specified frequencies. It is sometimes called a **line spectrum**, where each line on the emission spectrum corresponds to a specific electron transition. Because each element can have its electrons excited to a different set of distinct energy levels, each possesses a unique **atomic** **emission spectrum**, which can be used as a fingerprint for the element. When an electron is excited to a higher energy level, it must absorb exactly the right amount of energy to make that transition. This means that exciting the electrons of a particular element results in energy absorption at specific wavelengths. Thus, in addition to a unique emission spectrum, every element possesses a characteristic absorption spectrum. Not surprisingly, the wavelengths of absorption correspond exactly to the wavelengths of emission because the difference in energy between levels remains unchanged. Identification of elements in the gas phase requires absorption spectra. Absorption is the basis for the color of compounds. We see the color of the light that is not absorbed by the compound.

Electrons have and this spin generates a very, very tiny magnetic field. Atoms or molecules that have at least one unpaired electron are known as paramagnetic. If all electrons are paired, chemists refer to the compound as diamagnetic. When a magnetic field is applied a paramagnetic substance will be attracted to the field, while diamagnetic molecules will be repelled from the field. The most popular type of magnetism (the kind that keeps magnets on the fridge) is known as ferromagnetism. Ferromagnets are permanent magnets, as they generate their own magnetic field. Ferromagnets have unpaired electrons (we can say that ferromagnets are paramagnetic, not diamagnetic), but ferromagnets have one additional key trait: the unpaired electron spins are all aligned in the same direction, which generates a permanent magnetic field.

While Bohr’s model marked a significant advancement in the understanding of the structure of atoms, his model ultimately proved inadequate to explain the structure and behavior of atoms containing more than one electron. Bohr’s model states that electrons follow a clearly defined circular pathway or orbit at a fixed distance from the nucleus but we now understand that electrons move rapidly and are localized within regions of space around the nucleus called **orbitals**. In the current quantum mechanical model, it is impossible to pinpoint exactly where an electron is at any given moment in time. This is expressed best by the **Heisenberg uncertainty principle** which states that it is impossible to simultaneously determine, with perfect accuracy, the momentum and the position of an electron.^{4} If we want to assess the position of an electron, the electron has to stop (thereby removing its momentum) and if we want to assess its momentum, the electron has to be moving (thereby changing its position).

Many properties of atoms depend on electron configuration and on how strongly the outer electrons in the atoms are attracted to the nucleus. **Coulomb’s law** tells us that the strength of the interaction between two electrical charges depends on the magnitudes of the charges and on the distance between them. Thus, the attractive force between an electron and the nucleus depends on the magnitude of the nuclear charge and on the average distance between the nucleus and the electron. The force increases as the nuclear charge increases and decreases as the electron moves farther from the nucleus. In a many-electron atom, each electron is simultaneously attracted to the nucleus and repelled by the other electrons. There are so many electron–electron repulsions that we cannot analyze the situation exactly. We can, however, estimate the attractive force between any one electron and the nucleus by considering how the electron interacts with the average environment created by the nucleus and the other electrons in the atom. We treat each electron as though it were moving in the net electric field created by the nucleus and the electron density of the other electrons. We view this net electric field as if it results from a single positive charge located at the nucleus, called the **effective nuclear charge**, *Z*_{eff}. The effective nuclear charge acting on an electron in an atom is smaller than the actual nuclear charge (Z_{eff} < Z) because the effective nuclear charge includes the effect of the other electrons in the atom.^{3} In any many-electron atom, the inner electrons partially screen outer electrons from the attraction of the nucleus, and the relationship between Z_{eff} and the number of protons in the nucleus Z is:

*Effective Nuclear Charge, Z _{eff }= Z – S*

where S is a positive number called the **screening constant**. It represents the portion of the nuclear charge that is screened from a valence electron by the other electrons in the atom. Because core electrons are most effective at screening a valence electron from the nucleus, the value of S is usually close to the number of core electrons in an atom. The effective nuclear charge increases from left to right across any period of the periodic table. Although the number of core electrons stays the same across the period, the number of protons increases. The valence electrons added to counterbalance the increasing nuclear charge screen one another ineffectively. Thus, Zeff increases steadily.

When light of a sufficiently high frequency (typically, blue to ultraviolet light) is incident on a metal in a vacuum, the metal atoms emit electrons. This phenomenon is called the **photoelectric effect**. Electrons liberated from the metal by the photoelectric effect will produce a net charge flow per unit time, or **current**. Provided that the light beam’s frequency is above the threshold frequency of the metal, light beams of greater intensity produce larger current in this way. The higher the intensity of the light beam, the greater the number of photons per unit time that fall on an electrode, producing a greater number of electrons per unit time liberated from the metal. When the light’s frequency is above the threshold frequency, the magnitude of the resulting current is directly proportional to the intensity (and amplitude) of the light beam. The minimum frequency of light that causes ejection of electrons is known as the **threshold frequency** (*f*T).^{3} The threshold frequency depends on the type of metal being exposed to the radiation. The photoelectric effect is basically an all-or-nothing response. If the frequency of the incident photon is less than the threshold frequency (*f* < *f*_{T}), then no electron will be ejected because the photons do not have sufficient energy to dislodge the electron from its atom. But if the frequency of the incident photon is greater than the threshold frequency (*f* > *f*_{T}), then an electron will be ejected, and the maximum kinetic energy of the ejected electron will be equal to the difference between hf and hf_{T} (also called the work function). Einstein’s explanation of these results was that the light beam consists of an integral number of photons and the energy of each photon is proportional to the frequency of the light:

*E** = hf*

Where *E* is the energy of the photon of light, *h* is Planck’s constant (6.626 × 10^{−34} J **·** s), and *f* is the frequency of the light.^{3} Once we know the frequency, we can easily find the wavelength *λ*, according to the equation:

*c = f λ*

**References**

3) Javadi, H. (2011). *Effective Nuclear Charge.* Retrieved from

http://www.gsjournal.net/old/science/javadi11e.pdf