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2.6.3 - Energies of Orbitals

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Energy Levels of Orbitals in Hydrogen

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Teacher
Teacher

Today, we will explore how the energy levels of electrons in hydrogen are determined. In hydrogen, the energy of an electron is influenced entirely by its principal quantum number, n. Can anyone tell me how these energy levels are arranged?

Student 1
Student 1

The energy levels are arranged from lowest to highest as 1s, then 2s, 2p, and so forth.

Teacher
Teacher

Exactly! The order goes as follows: 1s < 2s = 2p < 3s = 3p = 3d < 4s and so on. What does this imply about the shapes of the 2s and 2p orbitals?

Student 2
Student 2

It means that even though they have different shapes, they hold the same energy.

Teacher
Teacher

That's correct! These orbitals, even though defined differently geometrically, are degenerate when it comes to energy. Can anyone remember what 'degenerate' means?

Student 3
Student 3

Degenerate orbitals have the same energy.

Teacher
Teacher

Precisely! Understanding this concept will help us later on when we discuss multi-electron atoms.

Teacher
Teacher

In summary, in hydrogen, electrons occupy energy levels defined only by n, leading to simple arrangements of energy states.

Energy Levels of Orbitals in Multi-Electron Atoms

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Teacher
Teacher

Now, let's move on to multi-electron atoms. Unlike hydrogen, the energy of the orbitals cannot be determined by n alone. Can someone explain why?

Student 1
Student 1

Because in multi-electron atoms, the presence of other electrons affects the energy level due to shielding.

Teacher
Teacher

Great point! Indeed, in multi-electron systems, energy levels depend on both the principal (n) and the azimuthal quantum number (l). This leads to the energy ordering: **s < p < d < f**. Who can explain why this ordering exists?

Student 2
Student 2

It’s due to the interactions between the inner and outer electrons. Inner electrons shield the outer electrons from the nuclear charge.

Teacher
Teacher

Exactly! This shielding effect creates variations in energy levels and introduces the concept of effective nuclear charge, or Z_eff. Remember the formula: Z_eff = Z - S, where S represents the screening constant. What implications do these shielding effects have on electron configurations?

Student 3
Student 3

It means that electrons in outer shells will fill up different orbitals based on their energies and the repulsion from inner electrons.

Teacher
Teacher

Exactly! This understanding of shielding is crucial for mastering the concept of electronic configurations.

Teacher
Teacher

In summary, knowing how shielding and effective nuclear charge affect electron energy levels is key to understanding multi-electron atoms.

Filling of Orbitals and the Pauli Exclusion Principle

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Teacher
Teacher

Next, we dive into how electrons actually fill these orbitals. This is governed by the Aufbau principle, Pauli Exclusion Principle, and Hund’s Rule. Can someone explain the Aufbau principle?

Student 1
Student 1

The Aufbau principle states that electrons fill orbitals starting from the lowest energy to the highest.

Teacher
Teacher

Excellent! This is important to remember. Can anyone elaborate on the Pauli Exclusion Principle?

Student 2
Student 2

It states that no two electrons can have the same set of all four quantum numbers; essentially, two electrons can occupy the same orbital only if they have opposite spins.

Teacher
Teacher

Precisely! And what about Hund’s Rule?

Student 3
Student 3

It says that electrons will fill degenerate orbitals singly before doubling up, to maximize unpaired electrons.

Teacher
Teacher

Exactly! The idea of maximizing unpaired electrons adds stability to the atom. Can someone summarize how these principles guide the filling of electrons in orbitals?

Student 4
Student 4

Electrons fill starting from the lowest energy orbital, no two can have same quantum numbers in the same orbital, and they try to occupy degenerate orbitals singly before pairing up.

Teacher
Teacher

Great summary! This understanding allows us to predict the electronic structure of atoms accurately.

Introduction & Overview

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Quick Overview

This section discusses the energy levels associated with different atomic orbitals in hydrogen and multi-electron atoms.

Standard

The energy of electrons in atomic orbitals varies between hydrogen and multi-electron atoms. In hydrogen, the energy is solely based on the principal quantum number, while in multi-electron atoms, both the principal and azimuthal quantum numbers influence orbital energy. Additionally, concepts such as shielding and effective nuclear charge are introduced to explain electron interactions within atoms.

Detailed

Detailed Summary

The understanding of atomic structure is significantly influenced by the energies of orbitals, which refer to the defined energy states that electrons can occupy in an atom.

  1. Hydrogen Atom: For hydrogen, the energy of an electron only depends on its principal quantum number (n). The energy levels increase in the following order:
  2. 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f... This means that while 2s and 2p orbitals have different shapes, they share the same energy level.
  3. Multi-Electron Atoms: In multi-electron atoms, orbital energy is influenced by both the principal (n) and azimuthal (l) quantum numbers. The energies increase according to the order: s < p < d < f. Here, shielding occurs, where electrons in inner shells repel those in outer shells, effectively lowering the nuclear charge experienced by the outer electrons known as Effective Nuclear Charge (Z_eff).
  4. Shielding: Because of electron-electron interactions, specifically mutual repulsion in multi-electron atoms, outer electrons do not experience the full charge of the nucleus. This effect creates a complexity in energy levels not observed in hydrogen, necessitating calculations using the n + l rule. Lower values of (n + l) equate to lower energy levels.

Understanding these principles is crucial for predicting how electrons fill orbitals and the resulting chemical behavior of elements, thereby reinforcing the foundational knowledge required for further studies in quantum chemistry.

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Audio Book

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Energy Levels in Hydrogen Atom

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The energy of an electron in a hydrogen atom is determined solely by the principal quantum number. Thus the energy of the orbitals in hydrogen atom increases as follows: 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f.

Detailed Explanation

In hydrogen, the energy levels of the electron are determined by the principal quantum number (n), which characterizes the size and energy of the orbital. As n increases, the energy levels become higher (less negative). For hydrogen, the energy levels can be listed in ascending order: the 1s orbital has the lowest energy, followed by the 2s and 2p orbitals, which are equal in energy, and so on for the 3s, 3p, and 3d orbitals. Higher principal quantum numbers correspond to orbitals that are further from the nucleus and have higher energies.

Examples & Analogies

Imagine a ladder where each step represents an energy level. The bottom step is the most stable and has the least potential energy (like the 1s orbital). As you climb each ladder rung (stepping up to 2s, 2p, etc.), you move away from the ground (the nucleus) and gain energy until eventually, you're at the top of the ladder where the potential energy is the highest. This visual helps understand that the higher you go, the less stable and more energetic your position becomes.

Degenerate Orbitals

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Although the shapes of 2s and 2p orbitals are different, an electron has the same energy when it is in the 2s orbital as when it is present in 2p orbital. The orbitals having the same energy are called degenerate.

Detailed Explanation

Degenerate orbitals are orbitals that have the same energy level despite having different shapes. In the case of the 2s and 2p orbitals, they are both considered to be at the same energy level even though 2s is spherical and 2p has a dumbbell shape. This means that when an electron occupies either the 2s or 2p orbital, it experiences the same energy, which is significant for understanding electron configurations.

Examples & Analogies

Think of a playground with a slide and a swing that are both at the same height from the ground. Though the slide and the swing are shaped differently, they give you the same potential energy when you sit on them at that height. Similarly, the 2s and 2p orbitals are at the same energy level, allowing electrons to occupy either one without additional energy cost.

Energy Dependence in Multi-Electron Atoms

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The energy of an electron in a multi-electron atom, unlike that of the hydrogen atom, depends not only on its principal quantum number (shell), but also on its azimuthal quantum number (subshell).

Detailed Explanation

In multi-electron systems, electrons are subjected to more complex interactions due to the presence of multiple nuclei and electron repulsion. Although the principal quantum number (n) indicates the energy level, the azimuthal quantum number (l) introduces further differentiation in energy levels among orbitals. For instance, in the same principal shell, the energy order is typically s < p < d < f due to variances in how effective the shielding from the nucleus is among the electrons in different orbital types.

Examples & Analogies

Consider a set of stairs where each floor represents an energy level determined mainly by how far you are from the ground (n). However, if you also think about the type of flooring (l) like hardwood, carpet, or tile, you realize that each type affects how comfortable it feels to stand on. Similarly, the atomic orbitals have varied energies based on both their distance from the nucleus and their type, meaning that s orbitals are 'lower' or 'more comfortable' than p orbitals at the same level.

Effective Nuclear Charge and Shielding

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Due to the presence of electrons in the inner shells, the electron in the outer shell will not experience the full positive charge of the nucleus (Ze). The effect will be lowered due to the partial screening of positive charge on the nucleus by the inner shell electrons.

Detailed Explanation

The concept of effective nuclear charge (Z_eff) encapsulates how much of the nucleus's charge an outer-shell electron actually experiences. Inner-shell electrons repel outer-shell electrons and diminish the full nuclear charge they feel, resulting in a lower effective nuclear charge. This dynamic explains why electrons further from the nucleus possess higher energy states due to reduced attraction, facilitating chemical behavior like bonding and ionization.

Examples & Analogies

Imagine being in a crowded stadium. The cheers (positive nuclear charge) come from everywhere, but if you are surrounded by a cluster of fans (inner-shell electrons), it becomes hard to hear the crowd. The effective cheering you perceive is less because of all the noise around you, much like how outer electrons perceive less nuclear pull when crowded by inner ones.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Orbitals: Regions in an atom where electrons are likely to be found.

  • Hydrogen Energy Levels: Energy in hydrogen depends solely on the principal quantum number.

  • Multi-Electron Atom Energy Levels: Energy levels depend on both principal and azimuthal quantum numbers.

  • Effective Nuclear Charge: The net positive charge experienced by an outer electron in a multi-electron atom due to shielding.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • The 1s orbital is closest to the nucleus and has the lowest energy level, while the 3p orbitals have higher energies and are further from the nucleus.

  • In Lithium (

  • Li), the 1s2 2s1 configuration shows how electrons occupy the lowest energy levels first.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • From low to high, orbitals go, Aufbau shows us the energy flow.

📖 Fascinating Stories

  • Imagine a row of seats where the first ones filled are those closest to the stage; the further ones fill as the closer ones fill up. This represents how electrons should ideally fill orbitals.

🧠 Other Memory Gems

  • Remember: 'Pauli said, opposite spin, when in the same bin!'.

🎯 Super Acronyms

HAP – Hund’s, Aufbau, Pauli for the principles of electron arrangement.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Principal Quantum Number (n)

    Definition:

    A positive integer that determines the size and energy of an atomic orbital.

  • Term: Azimuthal Quantum Number (l)

    Definition:

    Describes the shape of the orbital and can take values from 0 to n - 1.

  • Term: Effective Nuclear Charge (Z_eff)

    Definition:

    The net positive charge experienced by an electron in a multi-electron atom.

  • Term: Aufbau Principle

    Definition:

    Electrons fill orbitals starting from the lowest energy level moving up to higher levels.

  • Term: Pauli Exclusion Principle

    Definition:

    No two electrons can have the same set of four quantum numbers in an atom.

  • Term: Hund’s Rule

    Definition:

    Electrons will fill degenerate orbitals singly before pairing up to maximize the number of unpaired electrons.