Chapter 7: Problem 110
In an atom of barium, how many electrons have (a) \(\ell=0\) and \((\mathbf{b}) m_{\ell}=1 ?\)
Short Answer
Expert verified
(a) Barium has 12 electrons with \(\ell=0\). (b) Barium has 8 electrons with \(m_{\ell}=1\).
Step by step solution
01
Understanding Quantum Numbers
In quantum mechanics, the quantum number \(\ell\) represents the orbital angular momentum of an electron in an atom, and it can take on any integer value from 0 to \(n-1\), where \(n\) is the principal quantum number. For \(\ell=0\), the electrons are in an s-orbital, which is spherical in shape.
02
Identify the S-Orbitals in Barium
Barium has an atomic number of 56, which corresponds to 56 electrons. For every principal quantum level \(n\), there is one s-orbital (\(\ell=0\)). Barium's electron configuration up to the principal quantum number 6 is: \(1s^2, 2s^2, 3s^2, 4s^2, 5s^2, 6s^2\). Thus, there are 2 electrons for each s-orbital from \(1s\) to \(6s\).
03
Counting the Electrons with \(\ell=0\)
To find out how many electrons in barium have \(\ell=0\), add up all the electrons in the s-orbitals. There are 6 s-orbitals with 2 electrons each, giving a total of \(6 \times 2 = 12\) electrons.
04
Understanding Magnetic Quantum Number \(m_{\ell}\)
The magnetic quantum number \(m_{\ell}\) can take on values from \(\ell\) to \(\-\ell\), including zero. It specifies the orientation of the orbital in space. For \(m_{\ell}=1\), we are interested in orbitals where \(\ell \geq 1\), which excludes s-orbitals (for which \(\ell=0\)).
05
Identify Orbitals with \(m_{\ell}=1\)
Electrons in p, d, and f orbitals can have \(m_{\ell}=1\). The p-orbitals start from \(n=2\) and include \(2p, 3p, 4p, 5p\), d-orbitals start from \(n=3\) and include \(3d, 4d, 5d\), but for barium, the \(5d\) orbitals are not filled in the ground state. There are no filled f-orbitals in the ground state of barium.
06
Counting the Electrons with \(m_{\ell}=1\)
Each p-orbital can hold up to 6 electrons (with \(m_{\ell}\) values of \-1, 0, and +1), but only one electron in each set can have \(m_{\ell}=1\). There are 4 filled p-orbitals, each contributing 1 electron with \(m_{\ell}=1\). Each d-orbital set has 5 d-orbitals, which can hold a total of 10 electrons, with 2 of them having \(m_{\ell}=1\). Barium has only the \(3d\) and \(4d\) orbitals filled. This adds up to \(4 + 2 \times 2 = 8\) electrons with \(m_{\ell}=1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Orbital Angular Momentum
Imagine an electron as a tiny planet orbiting a vast sun; much like planets possess momentum from spinning on their axes and orbiting the sun, electrons have angular momentum due to their motion around an atom's nucleus. This momentum, known as the orbital angular momentum, is a fundamental concept in quantum mechanics, symbolized by the quantum number \(\ell\). What's fascinating about \(\ell\) is its quantized nature, meaning that it can only take specific values. These values are integers ranging from 0 up to \(n-1\), where \(n\) is the principal quantum number that signifies the electron's energy level.
Picture the different values of \(\ell\) as signposts indicating various types of orbitals: for \(\ell=0\), you have s-orbitals, which are spherical; as you increase \(\ell\), you enter the world of p-orbitals (when \(\ell=1\)), d-orbitals \(\ell=2\), and f-orbitals \(\ell=3\), each with increasingly complex shapes. The value of \(\ell\) not only determines the shape of the orbital but also the electron's energy and, in turn, the fine structure of an atom's spectral lines.
To visualize orbital angular momentum, think about a figure skater spinning with outstretched arms; the skater's angular momentum changes when they pull their arms close or fling them wide. Similarly, an electron's angular momentum changes depending on the value of \(\ell\), profoundly affecting atomic behavior under the mysterious rules of quantum mechanics.
Picture the different values of \(\ell\) as signposts indicating various types of orbitals: for \(\ell=0\), you have s-orbitals, which are spherical; as you increase \(\ell\), you enter the world of p-orbitals (when \(\ell=1\)), d-orbitals \(\ell=2\), and f-orbitals \(\ell=3\), each with increasingly complex shapes. The value of \(\ell\) not only determines the shape of the orbital but also the electron's energy and, in turn, the fine structure of an atom's spectral lines.
To visualize orbital angular momentum, think about a figure skater spinning with outstretched arms; the skater's angular momentum changes when they pull their arms close or fling them wide. Similarly, an electron's angular momentum changes depending on the value of \(\ell\), profoundly affecting atomic behavior under the mysterious rules of quantum mechanics.
Electron Configuration
The way electrons are distributed among the orbitals of an atom is described by its electron configuration. Much like a game of musical chairs, electrons fill available seats based on a set of rules. These rules, derived from the principles of quantum mechanics, dictate the most stable arrangements for electrons within an atom.
The simplest rule is the 'Aufbau Principle', which instructs electrons to fill the lowest energy orbitals first. Think of a hotel building - guests (electrons) will generally prefer to occupy the lower floors (lower energy orbitals) before moving to higher levels. In this metaphor, the floors represent principal energy levels, while the rooms on each floor correspond to different orbitals (s, p, d, f).
To specify an atom's electron configuration, scientists use notations such as \(1s^2\), which indicates that two electrons are occupying the first energy level's s-orbital. In our example, Barium, with 56 electrons, fills up its hotel starting from the 1s orbital all the way to the 6s orbital. The electron configuration can be likened to a map, providing insight into the chemical behavior and properties of the element, such as its valency, magnetism, and reactivity.
An ion, or atom with excess charge due to gain or loss of electrons, will have an electron configuration that differs from the neutral state, reflecting the new balance of its electronic structure. Understanding the electron configurations not only paints a picture of the atom's structure but also empowers chemists to predict how it will behave in chemical reactions.
The simplest rule is the 'Aufbau Principle', which instructs electrons to fill the lowest energy orbitals first. Think of a hotel building - guests (electrons) will generally prefer to occupy the lower floors (lower energy orbitals) before moving to higher levels. In this metaphor, the floors represent principal energy levels, while the rooms on each floor correspond to different orbitals (s, p, d, f).
To specify an atom's electron configuration, scientists use notations such as \(1s^2\), which indicates that two electrons are occupying the first energy level's s-orbital. In our example, Barium, with 56 electrons, fills up its hotel starting from the 1s orbital all the way to the 6s orbital. The electron configuration can be likened to a map, providing insight into the chemical behavior and properties of the element, such as its valency, magnetism, and reactivity.
An ion, or atom with excess charge due to gain or loss of electrons, will have an electron configuration that differs from the neutral state, reflecting the new balance of its electronic structure. Understanding the electron configurations not only paints a picture of the atom's structure but also empowers chemists to predict how it will behave in chemical reactions.
Magnetic Quantum Number
The orientation of an orbital around an atom's nucleus is determined by the magnetic quantum number, symbolized by \(m_{\ell}\). If you return to the notion of the electron being like a planet, \(m_{\ell}\) tells us the direction in which this 'planet' is orbiting. It can take on a set of values ranging from \(\ell\) to \(\-\ell\), which includes zero. This range of values is what allows electrons to occupy different orientations in space, akin to the different tilts and angles in which Earth's axis could hypothetically be oriented.
Let's take p-orbitals as an example. They have three orientations: one along each of the x, y, and z axes in three-dimensional space. These are distinguished by \(m_{\ell}\) values of -1, 0, and +1. For an orbital to have \(m_{\ell}=1\), it must have an \(\ell\) value of at least 1, automatically excluding s-orbitals, which have \(\ell=0\).
In practical terms, understanding the magnetic quantum number can shed light on how atomic orbitals are affected by magnetic fields, further explaining phenomena such as electron spin resonance and the Zeeman effect. In spectroscopy, \(m_{\ell}\) reveals how atomic energy levels split when an external magnetic field is applied, demonstrating yet another quantum mechanical principle governing the micro-realm of the atom.
Let's take p-orbitals as an example. They have three orientations: one along each of the x, y, and z axes in three-dimensional space. These are distinguished by \(m_{\ell}\) values of -1, 0, and +1. For an orbital to have \(m_{\ell}=1\), it must have an \(\ell\) value of at least 1, automatically excluding s-orbitals, which have \(\ell=0\).
In practical terms, understanding the magnetic quantum number can shed light on how atomic orbitals are affected by magnetic fields, further explaining phenomena such as electron spin resonance and the Zeeman effect. In spectroscopy, \(m_{\ell}\) reveals how atomic energy levels split when an external magnetic field is applied, demonstrating yet another quantum mechanical principle governing the micro-realm of the atom.
