Nuclear effective charges

The observed trend in atomic radius as we move to the right across a row, however, is a bit more complex. To understand this trend, we must revisit some concepts from Section 8.3, including effective nuclear charge and shielding. 

As we saw in Section 8.3, any one electron in a multielectron atom experiences both the positive charge of the nucleus (which is attractive) and the negative charges of the other electrons (which are repulsive). Consider again the outermost electron in the lithium atom  

Even though the 2x orbital penetrates into the lx orbital to some degree, the majority of the 2s orbital is outside of the lx orbital. Therefore the electron in the 2s orbital is partially 

As we have seen, we can define the average or net charge experienced by an electron as the effective nuclear charge. The effective nuclear charge experienced by a particular electron in an atom is the actual nuclear charge (Z) minus the charge shielded by other electrons (S)  

In many introductory courses, cr is taken to be an integer representing the number of inner-core, or shielding, electrons. For example, consider neon, which has an electronic configuration of 2P 2p. To calculate the effective nuclear charge acting on a valence 2s ov2p electron, we note that there are two (Ir) inner-core electrons. 

An electron in an atom is attracted to the nucleus but also repelled by other electrons, hence it does not feel the full strength of the nuclear attraction. On the other hand, some atoms have positive electron affinity, i.e. attract an electron as if they possessed a net positive charge. Hence the electron shells do not completely compensate the nuclear charge Z, so that an approaching electron experiences the attraction by a certain effective nuclear charge, Z 0. In 1930 Slater [37] suggested the following approximate solution to this problem. An electron at the radius r from the nucleus is shielded (screened) from its attraction by the electron density within the sphere of 

Numerous attempts to improve the technique of calculating Z, bear witness to the importance of this problem for theoretical chemistry [75-79]. Most of these attempts concentrated on deriving more adequate screening constants for d- and/-electrons. Slater s over-estimation of those being obvious on comparison with experimental 

Waldron et al. [86] proposed to use the screening percentage , 1 - Z /Z, as a more evident indicator of the fraction of the nuclear charge that is shielded from a 

As we have seen, the electron configurations of the elements show a periodic variation with increasing atomic number. In this and the next few sections, we will examine how electron configuration explains the periodic variation of physical and chemical properties of the elements. We begin by introducing the concept of effective nuclear charge. 

Although all the electrons in an atom shield one another to some extent, those that are most effective at shielding are the core electrons. As a result, the value of 2 increases steadily from 

Many of the periodic trends in properties of the elements can be explained using Coulomb s law, which states that the force (F) between two charged objects (Qi and is directly proportional to the product of the two charges and inversely proportional to the distance (d) between the objects squared Force is inversely proportional to d, whereas energy is inversely proportional to d [W Section 5.1]. The SI unit of force is the newton (N = m kg/s ) and the SI unit of energy is the joule (J = m kg/s ). 

TABLE 7.2 I. - tlracuvc lorcc Bdlvveen 1 pposiiet) C hargsJ Objeci.s ai a Fixed D tunce (t/ It from Fach Other 

The product of a positive number and a negative number is a negative number. When we ate simply comparing the magnitudes of attractive forces, however, it is unnecessary to include the sign. 

TABLE 7.1 Comparison of the Properties of Eka-Silicon Predicted by Mendeleevwith the Observed Properties of Germanium 

In 1913, two years after Rutherford proposed the nuclear model of the atom (Section 2.2), English physicist Henry Moseley (1887-1915) developed the concept of atomic numbers. Bombarding different elements with high-energy electrons, Moseley found that each element produced X-rays of a unique frequency and that the frequency generally increased as the atomic mass increased. He arranged the X-ray frequencies in order by assigning a unique whole number, called an atomic number, to each element. Moseley correctly identified the atomic number as the number of protons in the nucleus of the atom. (Section 2.3) 

The concept of atomic number clarified some problems in the periodic table of Moseley s day, which was based on atomic weights. For example, the atomic weight of Ar (atomic number 18) is greater than that of K (atomic number 19), yet the chemical and physical properties of Ar are much more like those of Ne and Kr than like those of Na and Rb. However, when the elements are arranged in order of increasing atomic number, rather than increasing atomic weight, Ar and K appear in their correct places in the table. Moseley s studies also made it possible to identify holes in the periodic table, which led to the discovery of previously unknown elements. 

Arranging the elements by atomic weight leads to an order slightly different from that in a modern periodic table, where the arrangement is by atomic number. 

Why does this happen Looking at the periodic tabie on the inside front cover, can you find an example other than Ar and K where the order of the elements would be different if the elements were arranged in order of increasing atomic weight  

---

Electronegativity x is the relative attraction of an atom for the valence electrons in a covalent bond. It is proportional to the effective nuclear charge and inversely proportional to the covalent radius ... 

Later methods, especially that of Gordy (1955), and later Allred and Rochow (1958) make use of screening constants of the electron strucmre for the nuclear charge of each atom. This determines die attraction between the nucleus of the atom and an electron outside the normal electron complement, and is die effective nuclear charge. The empirical equation for the values of electronegativity obtained in this manner by Allred and Rochow is... 

The decrease in atomic radius moving across the periodic table can be explained in a similar manner. Consider, for example, the third period, where electrons are being added to the third principal energy level. The added electrons should be relatively poor shields for each other because they are all at about the same distance from the nucleus. Only the ten core electrons in inner, filled levels (n = 1, n = 2) are expected to shield the outer electrons from the nucleus. This means that the charge felt by an outer electron, called the effective nuclear charge, should increase steadily with atomic number as we move across the period. As effective nuclear charge increases, the outermost electrons are pulled in more tightly, and atomic radius decreases. 

See Standard oxidation voltage See Standard reduction voltage Effective nuclear charge Positive charge felt by the outermost electrons in an atom approximately equal to the atomic number minus the number of electrons in inner, complete levels, 154 Efflorescence Loss of water by a hydrate, 66 Effusion Movement of gas molecules through a pinhole or capillary,... 

Fig. 1-16. Moseley plot for Ka2 lines. The curvature at high Z is due to a change in the effective nuclear charge (Z — 1). The insert shows the atomic number Z to be more fundamental than the atomic weight M. X-rays made possible the first experimental determinations of Z. Crosses = atomic weight dots = atomic number.

As well as being attracted to the nucleus, each electron in a many-electron atom is repelled by the other electrons present. As a result, it is less tightly bound to the nucleus than it would be if those other electrons were absent. We say that each electron is shielded from the full attraction of the nucleus by the other electrons in the atom. The shielding effectively reduces the pull of the nucleus on an electron. The effective nuclear charge, Z lle, experienced by the electron is always less than the actual nuclear charge, Ze, because the electron-electron repulsions work against the pull of the nucleus. A very approximate form of the energy of an electron in a many-electron atom is a version of Eq. 14b in which the true atomic number is replaced by the effective atomic number ... 

FIGURE 1.45 The variation ol the effective nuclear charge for the outermost valence electron with atomic number. Notice that the effective nuclear charge increases from left to right across a period but drops when the outer electrons occupy a new shell. (The effective nuclear charge is actually Zc,tfe, hut Zal itself is commonly referred to as the charge.)... 

FIGURE 1.47 The periodic variation in the atomic radii of the elements. The variation across a period can be explained in terms of the effect of increasing effective nuclear charge that down a group by the occupation of shells with inc reasing principal quantum number. 

SOLUTION The smaller member of a pair of isoelectronic ions in the same period will be an ion of an element that lies farther to the right in a period, because that ion has the greater effective nuclear charge. If the two ions are in the same group, the smaller ion will be the one that lies higher in the group, because its outermost electrons are closer to the nucleus. Check your answer against the values in Appendix 2C. 

Account for the fact that the ionization energy of potassium is less than that of sodium despite the latter having the smaller effective nuclear charge. 

Sodium is in Group 1 of the periodic table and can be expected to form a +1 ion. However, the valence electron is tightly held by the effective nuclear charge—... 

All the elements in a main group have in common a characteristic valence electron configuration. The electron configuration controls the valence of the element (the number of bonds that it can form) and affects its chemical and physical properties. Five atomic properties are principally responsible for the characteristic properties of each element atomic radius, ionization energy, electron affinity, electronegativity, and polarizability. All five properties are related to trends in the effective nuclear charge experienced by the valence electrons and their distance from the nucleus. 

As the value of n increases, d- and /-electrons become less effective at shielding the outermost, highest-energy elec-tron(s) from the attractive charge of the nucleus. This higher effective nuclear charge makes it more difficult to oxidize the metal atom or ion. 

A similar model for many-electron atoms has been developed,6 by considering each electron to be hydrogen-like, but under the influence of an effective nuclear charge (Z — Ss)e, in which Ss is called the size-screening constant. It is found that atoms and ions containing only 5 electrons (with the quantum number l equal to zero) and completed sub-groups of... 

As a first approximation, each electron in a many-electron atom can be considered to have the distribution in space of a hydrogen-like electron under the action of the effective nuclear charge (Z—Ss)e, in which 5s represents the screening effect of inner electrons. In the course of a previous investigation,6 values of S5 for a large number of ions were derived. 

The observed inter-atomic distances (Table XIX) for the beryllium salts are somewhat smaller than those calculated. This indicates that there is more deformation in these crystals than in the sodium chloride type crystals, despite the smaller effective nuclear charge of the two-shell cation and points to the existence of an increased tendency to deformation... 

A question which has been keenly argued for a number of years is the following if it were possible continuously to vary one or more of the parameters determining the nature of a system such as a molecule or a crystal, say the effective nuclear charges, then would the transition from one extreme bond type to another take place continuously, or would it show discontinuities For example, are there possible all intermediate bond types between the pure ionic bond and the pure electron-pair bond With the development of our knowledge of the nature of the chemical bond it has become evident that this question and others like it cannot be answered categorically. It is necessary to define the terms used and to indicate the point of view adopted and then it may turn out, as with this question, that no statement of universal application can be made. 

It is customary to express the empirical data concerning term values in the X-ray region by introducing an effective nuclear charge Zeff e in the place of the true nuclear charge Ze in an equation theoretically applicable only to a hydrogen-like atom. Often a screening constant S is used, defined by the equation... 

---