Rank The Following Orbitals In Terms Of Energy

In the realm of quantum mechanics, understanding the energy levels of atomic orbitals is fundamental to comprehending the behavior of atoms and molecules. Ranking orbitals in terms of energy allows us to predict how electrons will fill them, influencing the chemical properties of elements. This article delves into the principles governing orbital energy, providing a comprehensive guide to ranking them accurately.

Understanding Atomic Orbitals

Atoms, the building blocks of matter, consist of a nucleus containing protons and neutrons, surrounded by electrons that exist in specific energy levels and regions of space called orbitals. Atomic orbitals are mathematical functions that describe the probability of finding an electron in a particular region around the nucleus. Each orbital has a unique shape and energy level, which are determined by a set of quantum numbers.

Quantum Numbers

Four quantum numbers are used to describe the state of an electron in an atom:

• Principal Quantum Number (n): This number determines the energy level of the electron and can be any positive integer (n = 1, 2, 3, ...). Higher values of n indicate higher energy levels and greater distance from the nucleus.
• Azimuthal Quantum Number (l): Also known as the angular momentum or orbital shape quantum number, l describes the shape of the orbital and ranges from 0 to n - 1. The values of l correspond to different orbital shapes:
  • l = 0: s orbital (spherical)
  • l = 1: p orbital (dumbbell-shaped)
  • l = 2: d orbital (more complex shapes)
  • l = 3: f orbital (even more complex shapes)
• Magnetic Quantum Number (ml): This number specifies the orientation of the orbital in space and can take integer values from -l to +l, including 0. For example, a p orbital (l = 1) has three possible orientations (ml = -1, 0, +1), corresponding to the px, py, and pz orbitals.
• Spin Quantum Number (ms): This number describes the intrinsic angular momentum of the electron, which is quantized and referred to as spin. Electrons behave as if they are spinning, creating a magnetic dipole moment. The spin quantum number can be either +1/2 (spin up) or -1/2 (spin down).

Orbital Energy Levels

The energy of an atomic orbital is primarily determined by the principal quantum number (n). Orbitals with lower values of n have lower energy and are closer to the nucleus. For a given value of n, the energy of the orbitals also depends on the azimuthal quantum number (l). In general, for a given n, the energy of the orbitals increases with increasing l:

s < p < d < f

However, it is important to note that this order is not always followed exactly, especially for atoms with multiple electrons.

Factors Affecting Orbital Energy

Several factors influence the energy levels of atomic orbitals, leading to deviations from the simple n + l rule:

Nuclear Charge

The nuclear charge, which is the number of protons in the nucleus, has a significant impact on orbital energy. A higher nuclear charge results in a stronger attraction between the nucleus and the electrons, lowering the energy of all orbitals. This effect is particularly pronounced for inner electrons, which experience the full nuclear charge.

Shielding

In multi-electron atoms, the inner electrons shield the outer electrons from the full nuclear charge. This shielding effect reduces the effective nuclear charge experienced by the outer electrons, raising their energy levels. The extent of shielding depends on the number and distribution of inner electrons.

Penetration

Penetration refers to the ability of an electron to penetrate the electron cloud of inner electrons and get closer to the nucleus. Orbitals with higher penetration experience a greater effective nuclear charge and have lower energy. The penetration effect is more pronounced for s orbitals than for p orbitals, and for p orbitals than for d orbitals.

Electron-Electron Repulsion

Electron-electron repulsion also affects orbital energy. The repulsion between electrons in the same orbital or in different orbitals increases the energy of the orbitals. This effect is particularly significant for orbitals with high electron density.

Rules for Ranking Orbitals in Terms of Energy

Based on the factors discussed above, we can establish a set of rules for ranking orbitals in terms of energy:

1. Principal Quantum Number (n): Orbitals with lower values of n have lower energy. For example, 1s < 2s < 3s.
2. Azimuthal Quantum Number (l): For a given value of n, orbitals with lower values of l have lower energy. For example, 2s < 2p, 3s < 3p < 3d.
3. n + l Rule: For orbitals with the same value of n + l, the orbital with the lower value of n has lower energy. For example, 3d (n = 3, l = 2, n + l = 5) has lower energy than 4p (n = 4, l = 1, n + l = 5).
4. Hund's Rule: Within a subshell (orbitals with the same n and l), electrons will individually occupy each orbital before doubling up in any one orbital. This minimizes electron-electron repulsion and leads to a lower energy state.

Examples of Ranking Orbitals

Let's illustrate the rules for ranking orbitals with some examples:

Example 1: Ranking Orbitals in Hydrogen (H)

Hydrogen has only one electron, so there is no electron-electron repulsion or shielding. The energy of the orbitals is determined solely by the principal quantum number (n). Therefore, the ranking of orbitals in hydrogen is:

1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < ...

Example 2: Ranking Orbitals in Oxygen (O)

Oxygen has eight electrons, so shielding and electron-electron repulsion become important. The ranking of orbitals in oxygen is:

1s < 2s < 2p < 3s < 3p < 4s < 3d < ...

Note that the 4s orbital is lower in energy than the 3d orbital due to the penetration effect. The 4s electrons penetrate closer to the nucleus than the 3d electrons, experiencing a greater effective nuclear charge and lowering their energy.

Example 3: Ranking Orbitals in Iron (Fe)

Iron has 26 electrons, so the effects of shielding and electron-electron repulsion are even more pronounced. The ranking of orbitals in iron is:

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < ...

Again, the 4s orbital is lower in energy than the 3d orbital due to the penetration effect.

The Aufbau Principle and Electron Configuration

The Aufbau principle, which means "building up" in German, is a guideline used to determine the electron configuration of an atom. It states that electrons first fill the lowest energy orbitals available before filling higher energy orbitals. The Aufbau principle, along with Hund's rule and the Pauli exclusion principle (which states that no two electrons in an atom can have the same set of four quantum numbers), provides a systematic way to predict the electron configuration of an atom.

Steps to Determine Electron Configuration:

1. Determine the number of electrons: The number of electrons in a neutral atom is equal to the number of protons in the nucleus, which is the atomic number.
2. Identify the lowest energy orbital: Start with the 1s orbital, which has the lowest energy.
3. Fill the orbital with electrons: Each orbital can hold a maximum of two electrons, according to the Pauli exclusion principle.
4. Move to the next higher energy orbital: Continue filling orbitals in order of increasing energy, following the rules outlined above.
5. Apply Hund's rule: When filling a subshell (orbitals with the same n and l), electrons will individually occupy each orbital before doubling up in any one orbital.

Example: Electron Configuration of Carbon (C)

Carbon has six electrons. Following the Aufbau principle, the electron configuration of carbon is:

1s² 2s² 2p²

This means that carbon has two electrons in the 1s orbital, two electrons in the 2s orbital, and two electrons in the 2p orbitals. According to Hund's rule, the two electrons in the 2p orbitals will occupy separate orbitals with parallel spins.

Exceptions to the Aufbau Principle

There are some exceptions to the Aufbau principle, particularly for transition metals. These exceptions arise due to the small energy differences between certain orbitals and the tendency of atoms to achieve half-filled or fully-filled d subshells, which are particularly stable.

Chromium (Cr)

Chromium has 24 electrons. According to the Aufbau principle, the expected electron configuration of chromium is:

1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁴

However, the actual electron configuration of chromium is:

1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d⁵

This is because a half-filled d subshell (3d⁵) is more stable than a partially filled d subshell (3d⁴) with a filled 4s orbital. One electron from the 4s orbital is promoted to the 3d orbital to achieve this more stable configuration.

Copper (Cu)

Copper has 29 electrons. According to the Aufbau principle, the expected electron configuration of copper is:

1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁹

However, the actual electron configuration of copper is:

1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d¹⁰

This is because a fully-filled d subshell (3d¹⁰) is more stable than a partially filled d subshell (3d⁹) with a filled 4s orbital. One electron from the 4s orbital is promoted to the 3d orbital to achieve this more stable configuration.

Implications for Chemical Properties

The electron configuration of an atom, which is determined by the ranking of its orbitals in terms of energy, has a profound impact on its chemical properties. The valence electrons, which are the electrons in the outermost shell, are primarily responsible for chemical bonding and reactivity. The number of valence electrons and their arrangement in orbitals determine how an atom will interact with other atoms to form molecules and compounds.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom in the gaseous phase. The ionization energy is related to the energy of the highest occupied orbital. Atoms with loosely held valence electrons have lower ionization energies and are more likely to form positive ions (cations).

Electronegativity

Electronegativity is a measure of the ability of an atom to attract electrons in a chemical bond. Electronegativity is related to the energy of the valence orbitals. Atoms with high electronegativity have a strong attraction for electrons and are more likely to form negative ions (anions).

Chemical Bonding

The type of chemical bond that forms between two atoms depends on the electron configurations and electronegativity values of the atoms. Atoms with similar electronegativity values tend to form covalent bonds, in which electrons are shared between the atoms. Atoms with significantly different electronegativity values tend to form ionic bonds, in which electrons are transferred from one atom to the other.

Conclusion

Ranking orbitals in terms of energy is crucial for understanding the electronic structure of atoms and molecules. The rules governing orbital energy, including the principal quantum number, azimuthal quantum number, n + l rule, and Hund's rule, provide a framework for predicting the order in which electrons fill orbitals. While there are exceptions to the Aufbau principle, particularly for transition metals, the general principles remain valid. The electron configuration of an atom, which is determined by the ranking of its orbitals in terms of energy, has a significant impact on its chemical properties, including ionization energy, electronegativity, and chemical bonding. A thorough understanding of these concepts is essential for anyone studying chemistry, physics, or materials science.