While Quantum Theory gives exact equations describing the H-atom, which has only one electron, it runs into problems trying to give exact equations of atoms with many electrons. This is because in addition to the electrostatic attraction between the electron and the positively charged nucleus, there are electrostatic repulsions between electrons. The problem starts to get complicated quickly. In spite of this problem, approximate solutions can be obtained, which can, in fact, be quite accurate. For a multi-electron atom the energy of a particular electron in the atom is given by $\large E = - Ry \displaystyle \frac{Z_\text{eff}^2}{n^2}$ Which looks the same as for a single electron atom except that now we use an effective charge for the positively charged nucleus. The effective charge is reduced from the full charge due to the shielding of the nuclear charge by other electron in the atom.  The effective nuclear charge equates the number of protons in the nucleus, Z, minus the average number of electrons, S, between the nucleus and the electron of interest. In a multi-electron atom it turns out that the effective charge, Zeff, decreases with increasing value of ℓ, the azmuthial quantum number. This is because electrons in the s-orbital have a greater probability of being near the nucleus than a p-orbital, so the s-orbital is less shielded than a p-orbital. Likewise, a p-orbital is less shielded than a d-orbital. In a multi-electron atom, the energy of an orbital increases with increasing value of ℓ for a given value of n.
Aufbau PrincipleThe energy structure of a many-electron atom is obtained by filling the orbitals one-electron at a time, in order of increasing energy starting with the lowest energy. This is called the Aufbau principle. The ordering of orbital energy levels is
These are two ways to indicate the electronic structure of an atom: the Electronic Configuration and the Orbital Diagram. 1. Electronic Configuratione.g., for the H-atom, the electronic configuration is 1s1. This notation is compact, and describes how the electrons are distributed with principal, n, and azimuthal, ℓ, quantum numbers, but does indicate the magnetic, mℓ and spin, ms quantum numbers for the electrons. 2. Orbital DiagramOrbital Diagrams give a more complete indication of the electron quantum numbers. Each orbital represented by a box and each electron by a half-arrow. For example Hydrogen has 1 electron. That electron goes into the lowest energy orbital, that is the 1s orbital. Thus we write... For Boron we start with 5 electrons. Again we start by filling the lowest energy orbital 1s, then the 2s orbital, and finally putting one electron in the 2p orbital... Now let's look at Carbon, which has 6 electrons.
Therefore for carbon we have Another example, Neon has 10 electrons.
For Neon, both the n=1 and n=2 shells are completely full. Let's look at Sodium, which has 11 electrons. Na has one electron its outermost shell. For convenience we often represent the electron configuration of the closed shells with the corresponding noble gas symbol. So for sodium we would write:
[Ne] 3 s1 where [Ne] represents 1s2 2s2 2p6 In fact we distinguish between electrons depending on whether they part of the closed shell or in the outermost shell.
If you look at the periodic table you will notice that elements in the same group have the same number of valence electrons. That is the main reason why elements in the same group have such similar chemical and physical properties. Let's look at Argon, which has 18 electrons. It has the configuration Now you might be tempted for Potassium (the next element) to put the extra electron in the 3d orbital, however, it turns out that the 4s orbital is slightly lower in energy than the 3d, so the electron configuration of Potassium is: After the 4s orbital is filled, then you can start to fill the 3d orbital. For example, Titanium has 22 electrons. It's configuration is
The triangular diagram below can be used to simplify memorizing the order in which the orbitals are filled.
6.59 6.61, 6.63, 6.65, 6.67, 6.69, 6.71, 6.73, 7.9
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Travis C. Chemistry 101 11 months, 3 weeks ago
For an atom or ion having single electron, compare the energies of the following orbitals: $\mathrm{S}_{1}=$ a spherically symmetrical orbital having two spherical nodes. $\mathrm{S}_{2}=$ an orbital which is double dumb-bell and has no radial node. $\mathrm{S}_{3}=$ an orbital with orbital angular momentum zero and three radial nodes. $\mathrm{S}_{a}=$ an orbital having one planar and one radial node. (a) $\mathrm{S}_{1}=\mathrm{S}_{2}=\mathrm{S}_{3}=\mathrm{S}_{4}$ (b) $\mathrm{S}_{1}=\mathrm{S}_{2}=\mathrm{S}_{4}<\mathrm{S}_{3}$ (c) $\mathrm{S}_{1}>\mathrm{S}_{2}>\mathrm{S}_{3}>\mathrm{S}_{4}$ (d) $\mathrm{S}_{1}<\mathrm{S}_{4}<\mathrm{S}_{3}<\mathrm{S}_{2}$
The quantum mechanical model allowed us to determine the energies of the hydrogen atomic orbitals; now we would like to extend this to describe the electronic structure of every element in the Periodic Table. The process of describing each atom’s electronic structure consists, essentially, of beginning with hydrogen and adding one proton and one electron at a time to create the next heavier element in the table; however, interactions between electrons make this process a bit more complicated than it sounds. Before demonstrating how to do this, however, we must introduce the concept of electron spin and the Pauli principle.
Unlike in hydrogen-like atoms with only one electron, in multielectron atoms the values of quantum numbers n and l determine the energies of an orbital. The energies of the different orbitals for a typical multielectron atom are shown in Figure \(\PageIndex{1}\). Within a given principal shell of a multielectron atom, the orbital energies increase with increasing l. An ns orbital always lies below the corresponding np orbital, which in turn lies below the nd orbital. These energy differences are caused by the effects of shielding and penetration, the extent to which a given orbital lies inside other filled orbitals. For example, an electron in the 2s orbital penetrates inside a filled 1s orbital more than an electron in a 2p orbital does. Since electrons, all being negatively charged, repel each other, an electron closer to the nucleus partially shields an electron farther from the nucleus from the attractive effect of the positively charged nucleus. Hence in an atom with a filled 1s orbital, the effective nuclear charge (Zeff) experienced by a 2s electron is greater than the Zeff experienced by a 2p electron. Consequently, the 2s electron is more tightly bound to the nucleus and has a lower energy, consistent with the order of energies shown in Figure \(\PageIndex{1}\).
Notice in Figure \(\PageIndex{1}\) that the difference in energies between subshells can be so large that the energies of orbitals from different principal shells can become approximately equal. For example, the energy of the 3d orbitals in most atoms is actually between the energies of the 4s and the 4p orbitals.
When scientists analyzed the emission and absorption spectra of the elements more closely, they saw that for elements having more than one electron, nearly all the lines in the spectra were actually pairs of very closely spaced lines. Because each line represents an energy level available to electrons in the atom, there are twice as many energy levels available as would be predicted solely based on the quantum numbers \(n\), \(l\), and \(m_l\). Scientists also discovered that applying a magnetic field caused the lines in the pairs to split farther apart. In 1925, two graduate students in physics in the Netherlands, George Uhlenbeck (1900–1988) and Samuel Goudsmit (1902–1978), proposed that the splittings were caused by an electron spinning about its axis, much as Earth spins about its axis. When an electrically charged object spins, it produces a magnetic moment parallel to the axis of rotation, making it behave like a magnet. Although the electron cannot be viewed solely as a particle, spinning or otherwise, it is indisputable that it does have a magnetic moment. This magnetic moment is called electron spin. In an external magnetic field, the electron has two possible orientations (Figure Figure \(\PageIndex{2}\)). These are described by a fourth quantum number (ms), which for any electron can have only two possible values, designated +½ (up) and −½ (down) to indicate that the two orientations are opposites; the subscript s is for spin. An electron behaves like a magnet that has one of two possible orientations, aligned either with the magnetic field or against it.
The implications of electron spin for chemistry were recognized almost immediately by an Austrian physicist, Wolfgang Pauli (1900–1958; Nobel Prize in Physics, 1945), who determined that each orbital can contain no more than two electrons. He developed the Pauli exclusion principle: No two electrons in an atom can have the same values of all four quantum numbers (n, l, ml, ms). By giving the values of n, l, and ml, we also specify a particular orbital (e.g., 1s with n = 1, l = 0, ml = 0). Because ms has only two possible values (+½ or −½), two electrons, and only two electrons, can occupy any given orbital, one with spin up and one with spin down. With this information, we can proceed to construct the entire periodic table, which was originally based on the physical and chemical properties of the known elements.
List all the allowed combinations of the four quantum numbers (n, l, ml, ms) for electrons in a 2p orbital and predict the maximum number of electrons the 2p subshell can accommodate. Given: orbital Asked for: allowed quantum numbers and maximum number of electrons in orbital Strategy:
A For a 2p orbital, we know that n = 2, l = n − 1 = 1, and ml = −l, (−l +1),…, (l − 1), l. There are only three possible combinations of (n, l, ml): (2, 1, 1), (2, 1, 0), and (2, 1, −1). B Because ms is independent of the other quantum numbers and can have values of only +½ and −½, there are six possible combinations of (n, l, ml, ms): (2, 1, 1, +½), (2, 1, 1, −½), (2, 1, 0, +½), (2, 1, 0, −½), (2, 1, −1, +½), and (2, 1, −1, −½). C Hence the 2p subshell, which consists of three 2p orbitals (2px, 2py, and 2pz), can contain a total of six electrons, two in each orbital.
List all the allowed combinations of the four quantum numbers (n, l, ml, ms) for a 6s orbital, and predict the total number of electrons it can contain. Answer(6, 0, 0, +½), (6, 0, 0, −½); two electrons Magnetic Quantum Number (ml) & Spin Quantum Number (ms): Magnetic Quantum Number (ml) & Spin Quantum Number (ms), YouTube(opens in new window) [youtu.be] (opens in new window)
The arrangement of atoms in the periodic table arises from the lowest energy arrangement of electrons in the valence shell. In addition to the three quantum numbers (n, l, ml) dictated by quantum mechanics, a fourth quantum number is required to explain certain properties of atoms. This is the electron spin quantum number (ms), which can have values of +½ or −½ for any electron, corresponding to the two possible orientations of an electron in a magnetic field. The concept of electron spin has important consequences for chemistry because the Pauli exclusion principle implies that no orbital can contain more than two electrons (with opposite spin). |
For a multielectron atom a 2 s orbital lies lower in energy than a 2 p orbital because
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