Radius variation with atomic number

Figure 1. Variation with atomic number of the inverse cube radius of valence shell electrons. Reproduced from Ref. 1 by permission of the American Institute of Physics.

Figure 1.46 shows some atomic radii, and Fig. 1.47 shows the variation in atomic radius with atomic number. Note the periodic, sawtooth pattern in the latter plot. Atomic radius generally decreases from left to right across a period and increases down a group. 

The periodic structure of the elements is evident for many physical and chemical properties, including chemical valence, atomic radius, electronegativity, melting point, density, and hardness. Two classic prototypes for periodic behavior are the variations of the first ionization energy and the atomic radius with atomic number. These are plotted in Figs. 9.4 and 9.5. 

Use the data from Appendix E to graph the variation of atomic radius with atomic number for the rare-earth elements from lanthanum to lutetium. 

The graph in the margin represents the variation of Z ff and atomic radius with atomic number. Which axis and correspondingly colored line corresponds to Zeff and which to atomic radius ... 

The variation of polarizability with atomic number (solid black line) closely resembles that of atomic volume (dashed red line). The atomic volume is calculated as (4/3)7rf, where r is the atomic radius as defined in Figure 9-4. Both polarizability and atomic volume decrease from left to right across a period and increase from top to bottom in a group. 

The following graph shows the variation in atomic radius with increasing atomic number ... 

A The general trend to more exothermic values with increasing atomic number is attributable to the decrease in ionic radius across the period because, as the anion-cation separation becomes smaller, the lattice enthalpy increases (equation 3.3). Superimposed on this trend is the effect of CFSE values. These are small in comparison to the overall magnitude of A// , but nonetheless have a significant effect. The double dip in the plot may be accounted for in terms of the variation in high-spin CFSE values across the first-row d-block. as shown in Figure 6.3. This shows respective CFSE contributions to of -(YsfA j for Ti-", -(V Aq for... 

Comparison of the predictions of this scheme with the data for the diamagnetic sesquioxides of lanthanum and lutetium (193) suggests that if all of the entropy variation is to be ascribed to the cation, the contributions would have to decrease with increasing mass (or atomic number) from 15.2 for lanthanum to 13.0 for lutetium. As an approximation, the decrease is taken proportional to that of the cationic radius obtained by x-ray diffraction measurements. The following (improved) values then apply ... 

Based on simple electrostatic theory one would predict a more or less direct relationship between the formation constants and say the atomic numbers or the ionic radii of the lanthanides, as the ionic radii varies more or less monotonically along the series and do not show (Fig. 4) any strong gadolinium break (12). However, as we have seen from the plots in Fig. 3 that the variation of the formation constants are no simple function of the atomic numbers or the ionic radii. Thus Grenthe (19) writes The experimental results proved that there was no simple variation of the measured properties with the crystallographic ionic radius - on the contrary a double series was observed. The author feels that at this point some comments on the practice of plotting the formation constants vs. the ionic radii or any function of the ionic radii (polarizability) should be made. 

The effect of varying nuclear charge on ionic radii is seen in the variation in radius in an isoelectronic series of ions. The term isoekctronic means that flie ions possess the same number of electrons. For example, each ion in the series 0, F , Na, Mg, and Al has 10 electrons. The nuclear chaige in this series increases steadily in the order listed. (Recall that the charge on the nucleus of an atom or monatomic ion is given by the atomic number of the element.) Because the number of electrons remains constant, the radius of the ion decreases with... 

All atoms, except for those of the p-block and of the element palladium, have an outer shell of electrons. Atoms of the so-called rf-block have a penultimate d -shell. Variation of atomic radius, within such a series with a uniform outer shell, is almost continuous. Discontinuity occurs where the number of electrons in the outer shell differs from the general s". ... 

The general geometrical problem of the packing of spheres has not been solved. An example of closest packing of atoms with some variation in effective radius is the icosahedral packing found (13) in the intermetallic compound Mg3B(Al,Zn) (Fig. 1). The successive layers in this structure contain 1, 12, 32, and 117 spheres. These numbers are reproduced (to within 1) by the empirical equation (12)... 

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