1.3: Electronic Structure of Elements

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Wave functions of electrons in an atom are called atomic orbitals. An atomic orbital is expressed using three quantum numbers; the principal quantum number, n; the azimuthal quantum number, l; and the magnetic quantum number, m_\(ell\). For a principal quantum number n, there are n azimuthal quantum numbers l ranging from 0 to n-1, and each corresponds to the following orbitals.

\[\begin{split} \ell & : 0, 1, 2, 3, 4, \ldots \\ & \; \; \; s, p, d, f, g, \ldots \end{split}\]

An atomic orbital is expressed by the combination of n and l. For example, n is 3 and l is 2 for a 3d orbital. There are 2l+1 m_\(ell\) values, namely l, l-1, l-2,..., -l. Consequently there are one s orbital, three p orbitals, five d orbitals and seven f orbitals. The three aforementioned quantum numbers are used to express the distribution of the electrons in hydrogen-type atom, and another quantum number m_s (1/2, -1/2) which describes the direction of an electron spin is necessary to completely describe an electronic state. Therefore, an electronic state is defined by four quantum numbers (n, l, m_\(ell\), m_s).

The wave function \(\psi\) which determines the orbital shape can be expressed as the product of a radial wavefunction R and an angular wave function Y as follows.

\[\psi_{n, l, m_{l}} = R_{n, l} (r) Y_{l, m_{l}} (\theta, \phi)\]

R is a function of distance from the nucleus, and Y expresses the angular component of the orbital. Orbital shapes are shown in Figure \(\PageIndex{1}\). Since the probability of the electron’s existence is proportional to the square of the wave function, an electron density map resembles that of a wave function. The following conditions must be satisfied when each orbital is filled with electrons.

[The conditions of electron filling]

Pauli principle

The number of electrons that are allowed to occupy an orbital must be limited to one or two, and, for the latter case, their spins must be anti-parallel (different direction).

Hund's rule

When there are equal-energy orbitals, electrons occupy separate orbitals nd their spins are parallel (same direction).

The order of orbital energy of a neutral atom is

\[1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p \ldots\]

and the electron configuration is determined as electrons occupy orbitals in this order according to the Pauli principle and Hund's rule. An s orbital with one m_\(ell\) can accommodate 2 electrons, a p orbital with three m_\(ell\) 6 electrons, and a d orbital with five m_\(ell\) 0 electrons.

Exercise \(\PageIndex{1}\)

Describe the electron configuration of a C atom, an Fe atom, and a Au atom.

Answer

Electrons equal to the atomic number are arranged in the order of orbital energies. Since the electrons inside the valence shell take the rare gas configuration, they may be denoted by the symbol of a rare gas element in brackets.

\[\begin{split} C & : 1s^{2} 2s^{2} 2p^{2} \quad or \quad [He] 2s^{2} 2p^{2} \\ Fe & : 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 3d^{6} 4s^{2} \quad or \quad [Ar] 3d^{6} 4s^{2} \\ Au & : 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 3d^{10} 4s^{2} 4p^{6} 4d^{10} 4f^{14} 5s^{2} 5p^{6} 5d^{10} 6s^{1} \quad or \quad [Xe] 4f^{14} 5d^{10} 6s^{1} \end{split}\]

Figure \(\PageIndex{1}\): - Shapes of s, p, and d orbitals.

Table \(\PageIndex{1}\) Periodic table of elements. The values are atomic weights
	1	2	3	4	5	6	7	8	9
1	1.008 ₁H
2	6.941 ₃Li	9.012 ₄Be
3	22.99 ₁₁Na	24.31 ₁₂Mg
4	39.10 ₁₉K	40.08 ₂₀Ca	44.96 ₂₁Sc	47.87 ₂₂Ti	50.94 ₂₃V	52.00 ₂₄Cr	54.94 ₂₅Mn	55.85 ₂₆Fe	58.93 ₂₇Co
5	85.47 ₃₇Rb	87.62 ₃₈Sr	88.91 ₃₉Y	91.22 ₄₀Z	92.91 ₄₁Nb	95.94 ₄₂Mo	(99) ₄₃Tc	101.1 ₄₄Ru	102.9 ₄₅Rh
6	132.9 ₅₅Cs	137.3 ₅₆Ba	Lanthanoid	178.5 ₇₂Hf	180.9 ₇₃Ta	183.8 ₇₄W	186.2 ₇₅Re	190.2 ₇₆Os	192.2 ₇₇Ir
7	(223) ₈₇Fr	(226) ₈₈Ra	Actinoid
Lanthan-oid			138.9 ₅₇La	140.1 ₅₈Ce	140.9 ₅₉Pr	144.2 ₆₀Nd	(145) ₆₁Pm	150.4 ₆₂Sm	152.0 ₆₃Eu
Actinoid			(227) ₈₉Ac	232.0 ₉₀Th	231.0 ₉₁Pa	238.0 ₉₂U	(237) ₉₃Np	(239) ₉₄Pu	(243) ₉₅Am

10	11	12	13	14	15	16	17	18
								4.003 ₂He
			10.81 ₅B	12.01 ₆C	14.01 ₇N	16.00 ₈O	19.00 ₉F	20.18 ₁₀Ne
			26.98 ₁₃Al	28.09 ₁₄Si	30.97 ₁₅P	32.07 ₁₆S	35.45 ₁₇Cl	39.95 ₁₈Ar
58.69 ₂₈Ni	63.55 ₂₉Cu	65.39 ₃₀Zn	69.72 ₃₁Ga	72.61 ₃₂Ge	74.92 ₃₃As	78.96 ₃₄Se	79.90 ₃₅Br	83.80 ₃₆Kr
106.4 ₄₆Pd	107.9 ₄₇Ag	112.4 ₄₈Cd	114.8 ₄₉In	118.7 ₅₀Sn	121.8 ₅₁Sb	127.6 ₅₂Te	126.9 ₅₃I	131.3 ₅₄Xe
195.1 ₇₈Pt	197.0 ₇₉Au	200.3 ₈₀Hg	204.4 ₈₁Tl	207.2 ₈₂Pb	209.0 ₈₃Bi	(210) ₈₄Po	(210) ₈₅At	(222) ₈₆Rn
157.3 ₆₄Gd	158.9 ₆₅Tb	162.5 ₆₆Dy	164.9 ₆₇Ho	167.3 ₆₈Er	168.9 ₆₉Tm	173.0 ₇₀Yb	175.0 ₇₁Lu
(247) ₉₆Cm	(247) ₉₇Bk	(252) ₉₈Cf	(252) ₉₉Es	(257) ₁₀₀Fm	(258) ₁₀₁Md	(259) ₁₀₂No	(262) ₁₀₃Lr