# Building the Periodic Table

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Building the Periodic Table

The four quantum numbers of Table 41-1 identify the quantum states of individual electrons in a multielectron atom. The wave functions for these states. however. are not the same as the wave functions for the corresponding states of the hydrogen  atom because. in multielectron atoms. the potential energy associated with a given electron is determined not only by the charge and position of the atom’ s nucleus but also by the charges and positions of all the other electrons in the atom. Solutions of  Schrtldinger’s equation for multielectron atoms can be carried out numerically-in principle at least-using a computer.  As we discussed in Section 40-8. all states with the same values of the quantum numbers n and 1form a subshell. For a given value of I. there are 21 + I possible values of the magnetic quantum number ml and. for each there are two possible values for the spin quantum number m,. Thus. there are 2(21 + I) states in a subshell.
It turns out that all states in Q given subshell have the same energy. its value being determined primarily by the value of n and to a lesser extent by the value of I. For the purpose of labeling subshells. the values of 1 are represented by letters: 1=0 2 3 4 5

S,p,d,f,g,h

For example. then = 3. 1 = 2 subshell would be labeled the 3d subshell. When we assign electrons to states in a multielectron atom. we must be guided by the Pauli exclusion principle of  that is. no two electrons in an atom can have the same set of the quantum numbers n. I. mi. and m. If this important principle did not hold. all the electrons in any atom could jump to the atom’s lowest energy level. which would eliminate the chemistry of atoms and molecules. and thus also biochemistry. Let us examine the atoms of a few elements to see how the Pauli exclusion principle operates in the building up of the periodic table.

Neon

The neon atom has 10 electrons. Only two of them fit into the lowest energy subshell, the Is subshell. These two electrons both have n = I, / = 0, and m, = 0, but one has m. = +t and the other has m. = -to The Is subshell, according to Table 41-1,  contains 2(2/ + 1) = 2 states. Because this subshell then contains all the electrons permitted by the Pauli principle, it is said to be closed. Two of the remaining eight electrons fill the next lowest energy subshell, the
2s subshell. The last six electrons just fill the 2p subshell which, with / = I, holds 2(2/ + 1) = 6 states. In a closed subshell, all allowed z projections of the orbital angular momentum vector rare present and, as you can  verify from Fig. 41-5, these projections cancel for the sub  as a whole; for every positive projection there is a corresponding
negative projection of the same magnitude. Similarly, the z projections of the spin angular momenta also cancel. Thus, a closed sub she has no angular momentum and no magnetic moment of any kind. Furthermore, its probability density is spherically symmetric. Then neon with its three closed sub s (Is, 2s, and 2p) has no “loosely dangling electrons” to encourage chemical interaction with other atoms.  Neon, like the other noble gases that form the right-hand column of the periodic table, is chemically inert.

Sodium

Next after neon in the periodic table comes sodium, with II electrons. Ten of them  form a closed neonate core, which, as we have seen, has zero angular momentum.The remaining electron is largely outside this inert core, in the 3s subshell-the next lowest energy subshell. Because this valence electron of sodium is in a state with / = 0 (that is, an s state), the sodium atom’s angular momentum and magnetic dipole moment must be due entirely to the spin of this single electron.  Sodium readily combines with other atoms that have a “vacancy” into which sodium’s loosely bound valence electron can fit. Sodium, like the other alkali metals that form the left-hand column of the periodic table, is chemically active .

Chlorine

The chlorine atom, which has 17 electrons, has a closed lO-electron, unlike core, with 7 electrons left over. Two of them fill the 3s subshell, leaving five to be assigned to the 3p subshell, which is the  next lowest in energy. This sub  which has / = I, can hold 2(2/ + I) = 6 electrons, so there is a vacancy, or a hole, in this subshe l.  Chlorine is receptive to interacting with other atoms that have a valence electron hat might fill this hole. Sodium chloride (NaC]), for example, is a very stable compound. Chlorine, like the other halogens that form column VIl A of the periodic table, is chemically active.

Iron

The arrangement of the 26 electrons of the iron atom can be represented as follows: The subshells are listed in numerical order and, following convention, a superscript gives the number of electrons in each subshell.  we can see that ants-subshell can hold 2 electrons, a p-subsheU 6, and subshe. Thus, iron’s first 1 8 electrons form the five filled subshells that are marked off by the bracket, leaving8 electrons to be accounted for. Six of the eight go into the 3d subshell and the  remaining two go into the 4s subshell. he last two electrons do not also go into the 3d subshell (which can hold 10 electrons) because the 3d6 4S2 configuration results in a lower energy state for the atom as a whole than would the 3tf8 configuration. An iron atom with 8 electrons (rather than 6) in the 3d sub shell would quickly make a transition to the 3tf> 4S2configuration, emitting electromagnetic radiation in the process. The lesson here is that except for the simplest elements, the states may not be filled in what we might think of as their “logical sequence.

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