17.4 Hydrogen atom and atomic orbitals
4 min read•july 30, 2024
Quantum mechanics unveils the mysterious world of atoms, focusing on the hydrogen atom as the simplest model. By solving the , we uncover quantized energy levels and wavefunctions that describe electron behavior around the nucleus.
Atomic orbitals, characterized by , reveal the intricate shapes and orientations of electron probability distributions. Understanding these orbitals is crucial for grasping atomic structure and chemical bonding in more complex systems.
Hydrogen Atom Orbitals
Solving the Schrödinger Equation
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- The Schrödinger equation for the hydrogen atom describes the behavior of an electron in the presence of a proton
- Solving the Schrödinger equation yields the energy levels (En) and wavefunctions (ψn,l,ml) of the atomic orbitals
- The energy levels of the hydrogen atom are quantized and depend on the principal quantum number (n) given by the formula
- The wavefunctions of the atomic orbitals are characterized by the quantum numbers n, l, and ml which determine the energy, shape, and orientation of the orbitals
Probability and Wavefunctions
- The probability of finding an electron at a specific point in space is proportional to the square of the wavefunction () at that point
- The total probability of finding an electron in all space must equal 1 as the electron must exist somewhere
- Probability density plots can be used to visualize the spatial distribution of electrons in an atom or molecule (e.g., high density near the nucleus, lower density farther away)
Atomic Orbital Visualization
s Orbitals
- s orbitals (l = 0) are spherically symmetric and have no angular dependence
- s orbitals increase in size and complexity with increasing principal quantum number (n) (e.g., 1s, 2s, 3s)
- The number of radial nodes (regions where the radial probability distribution equals zero) in an is equal to n-1
p Orbitals
- p orbitals (l = 1) have three possible orientations (ml = -1, 0, 1) and are shaped like dumbbells
- p orbitals are oriented along the x, y, and z axes (px, py, pz) and increase in size and complexity with increasing n (e.g., 2p, 3p)
- The number of radial nodes in a is equal to n-2
d Orbitals
- d orbitals (l = 2) have five possible orientations (ml = -2, -1, 0, 1, 2) and have more complex shapes (cloverleaf, dumbbell, double dumbbell)
- d orbitals are oriented in various planes (xy, xz, yz, x^2-y^2, z^2) and increase in size and complexity with increasing n (e.g., 3d, 4d)
- The number of radial nodes in a is equal to n-3
Electron Density and Probability
Electron Density
- Electron density is a measure of the probability of finding an electron at a specific point in space
- The electron density is proportional to the square of the wavefunction () at that point
- Regions with high electron density have a higher probability of finding an electron, while regions with low electron density have a lower probability
Visualization
- Electron density plots can be used to visualize the spatial distribution of electrons in an atom or molecule
- Probability density plots show the likelihood of finding an electron in a given region of space
- Isosurface plots display surfaces of constant probability density, helping to visualize the shape of atomic orbitals (e.g., spherical s orbitals, dumbbell-shaped p orbitals)
Quantum Numbers and Orbital Properties
Principal Quantum Number (n)
- The principal quantum number (n) determines the energy and size of the orbital
- n can take positive integer values (n = 1, 2, 3, ...) with larger n corresponding to higher energy and larger orbitals
- The energy of an orbital depends primarily on n, with smaller contributions from the angular quantum number (l)
Angular Quantum Number (l)
- The angular quantum number (l) determines the shape of the orbital
- l can take integer values from 0 to n-1 (l = 0, 1, 2, ..., n-1) with each value corresponding to a specific orbital shape (s, p, d, f)
- For each value of n, there are n^2 possible combinations of l and ml, corresponding to the number of orbitals in each energy level
Magnetic Quantum Number (ml)
- The magnetic quantum number (ml) determines the orientation of the orbital
- ml can take integer values from -l to +l (ml = -l, ..., -1, 0, 1, ..., +l) with each value corresponding to a specific orbital orientation (px, py, pz; dxy, dxz, dyz, dx^2-y^2, dz^2)
- The number of possible ml values for a given l is equal to 2l+1
Radial and Angular Probability Distributions
Radial Probability Distribution
- The radial probability distribution describes the probability of finding an electron at a certain distance from the nucleus, regardless of direction
- The radial probability distribution is obtained by integrating the square of the wavefunction () over all angles (θ and φ) at a fixed distance (r) from the nucleus
- The number of radial nodes (regions where the radial probability distribution equals zero) is equal to n-l-1
- Radial probability distributions help visualize the most probable distances of electrons from the nucleus for different orbitals (e.g., 1s, 2s, 2p)
Angular Probability Distribution
- The angular probability distribution describes the probability of finding an electron in a specific direction, regardless of distance from the nucleus
- The angular probability distribution is obtained by integrating the square of the wavefunction () over all distances (r) at fixed angles (θ and φ)
- The angular probability distribution is responsible for the characteristic shapes of the s, p, and d orbitals
- Angular probability distributions help visualize the spatial orientation and shape of atomic orbitals (e.g., spherical s orbitals, dumbbell-shaped p orbitals, cloverleaf-shaped d orbitals)
Key Terms to Review (19)
Bohr Model: The Bohr Model is a fundamental theory in atomic physics that describes the behavior of electrons in hydrogen atoms. It introduces the concept of quantized energy levels, where electrons orbit the nucleus at specific distances and energies, leading to distinct spectral lines. This model laid the groundwork for understanding atomic structure and the subsequent development of quantum mechanics.
D orbital: The d orbital is a type of atomic orbital that can hold a maximum of 10 electrons and is characterized by its unique shapes and orientations. It plays a significant role in the electron configuration of transition metals and is essential for understanding their chemical properties and bonding behavior. The presence of d orbitals influences the magnetic properties, color, and reactivity of these elements.
Electron cloud model: The electron cloud model describes the regions around an atom's nucleus where electrons are likely to be found, emphasizing that electrons do not travel in fixed orbits but rather exist in a 'cloud' of probability. This model helps explain the behavior of electrons in atoms, particularly for hydrogen, as it reflects their wave-like nature and the uncertainty of their exact positions.
Emission spectra: Emission spectra are the spectrum of light released from excited atoms or molecules as they return to lower energy states. When electrons in an atom absorb energy, they can jump to higher energy levels, and when they fall back to their original positions, they emit light at specific wavelengths, creating unique patterns of colors. These spectra serve as a fingerprint for elements, allowing scientists to identify substances and understand their properties.
Erwin Schrödinger: Erwin Schrödinger was an Austrian physicist best known for formulating the wave equation that describes how quantum systems evolve over time. His work laid the foundation for modern quantum mechanics and introduced the concept of wave functions, which are central to understanding the behavior of particles at the atomic level.
Excitation: Excitation refers to the process by which an electron in an atom absorbs energy and moves from a lower energy level to a higher energy level. This process is crucial in understanding the behavior of electrons in hydrogen atoms, as it explains how they can occupy different atomic orbitals when energy is supplied, such as through heat or electromagnetic radiation.
Hydrogen spectrum: The hydrogen spectrum refers to the set of electromagnetic radiation wavelengths emitted or absorbed by hydrogen atoms, which occurs when electrons transition between energy levels. These spectral lines are unique to hydrogen and provide essential insights into atomic structure and the behavior of electrons within an atom.
Hydrogenic atoms: Hydrogenic atoms are ions that contain only one electron, similar to the hydrogen atom, allowing them to be analyzed using the same quantum mechanical principles. This includes species like He+, Li2+, and other ions where the nucleus contains more than one proton but the electron count remains at one. Their behavior is modeled using solutions to the Schrödinger equation for hydrogen-like systems, showcasing distinct energy levels and atomic orbitals.
Niels Bohr: Niels Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory. He is best known for his model of the hydrogen atom, which incorporated quantum mechanics to explain how electrons exist in discrete energy levels, leading to the development of atomic orbitals and the rigid rotor model in quantum mechanics.
Orbital angular momentum: Orbital angular momentum is a quantum property of electrons in atoms that describes the motion of an electron as it orbits the nucleus. It is quantized, meaning it can only take on specific values, which are determined by the electron's quantum numbers. This property is critical for understanding the shape and orientation of atomic orbitals, influencing how atoms interact and bond with each other.
Orbital hybridization: Orbital hybridization is the concept in chemistry where atomic orbitals combine to form new, equivalent hybrid orbitals that are oriented to minimize electron pair repulsion and maximize bonding. This process explains the geometry of molecular structures by describing how atomic orbitals, such as s and p orbitals, mix to create hybrid orbitals that are used in bonding, allowing for a better understanding of molecular shape and bond angles.
P orbital: A p orbital is a type of atomic orbital that has a distinctive dumbbell shape and can hold a maximum of two electrons. These orbitals are essential in understanding the behavior of electrons in an atom, particularly in the hydrogen atom, where they play a crucial role in defining the electron's energy levels and spatial distribution.
Probability density function: A probability density function (PDF) is a statistical function that describes the likelihood of a random variable taking on a particular value. It provides a way to visualize the distribution of probabilities across possible outcomes, with the area under the curve representing the total probability of all outcomes. In the context of atomic orbitals, the PDF helps to illustrate where an electron is most likely to be found around the nucleus of an atom.
Quantization: Quantization is the concept that certain physical properties, such as energy, can only take on discrete values rather than a continuous range. This idea is fundamental in understanding atomic and subatomic systems, particularly in relation to the behavior of electrons in atoms and how they occupy specific energy levels or orbitals.
Quantum Numbers: Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. They provide important information about the size, shape, and orientation of atomic orbitals, as well as the spin of electrons. Understanding quantum numbers is essential for explaining the behavior of electrons in atoms and how they relate to the energy levels and structure of atoms.
S orbital: An s orbital is a type of atomic orbital that has a spherical shape and can hold a maximum of two electrons with opposite spins. These orbitals are the lowest energy level and are found in all electron shells, playing a fundamental role in the structure of atoms, particularly in the hydrogen atom where they house the electron closest to the nucleus.
Schrödinger Equation: The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It connects the concept of wave functions to the probabilistic nature of particles, laying the groundwork for understanding the behavior of subatomic particles and their interactions.
Spin quantum number: The spin quantum number is a fundamental property of electrons that describes their intrinsic angular momentum, taking on values of +1/2 or -1/2. This characteristic allows for the differentiation between two possible spin states of an electron, which is crucial for understanding how electrons occupy atomic orbitals and interact within atoms, particularly in the hydrogen atom where these principles are foundational for its electron configuration and energy levels.
Wave function: A wave function is a mathematical description of the quantum state of a system, typically denoted by the Greek letter psi (\(\psi\)). It contains all the information about the system's physical properties and can be used to calculate probabilities of finding particles in various states. The wave function is central to understanding phenomena in quantum mechanics, especially in relation to time-dependent and time-independent equations, as well as atomic structure.