Atomic volume calculated

Fig. 7. A plot of atomic volume of the rare earths against atomic number, o atomic volume calculated from lattice constants of the stable structure at room temperature or above, atomic volume for bcc structure...

It is improbable that phosphorus is trivalent in POCl3, and further, in such compounds it is exercising its maximum valency. Since the atomic volume calculated on the assumption of a double bond between the phosphorus and the oxygen agrees most closely with the atomic volume of elementary phosphorus, it is probable that in the liquid element as well as in POCl3 phosphorus is exercising this maximum valency, which includes mixed bonds, thus... 

Here represents the radius of a sphere containing x atoms, each having the atomic volume calculated from the molar volume of the solid and W is the work function of the corresponding bulk material in the condensed (solid or liquid) phase. This theory works well for the rare gases and the data extrapolate to the limit Ip = W oi the bulk materials. 

Thus, polarizability decreases from left to right across a period and increases from top to bottom within a group. The polarizability of an atom is similar in magnitude to the atomic volume calculated from atomic radii, as suggested by Figure 9-18. 

In this paper, the electronic structure of disordered Cu-Zn alloys are studied by calculations on models with Cu and Zn atoms distributed randomly on the sites of fee and bcc lattices. Concentrations of 10%, 25%, 50%, 75%, and 90% are used. The lattice spacings are the same for all the bcc models, 5.5 Bohr radii, and for all the fee models, 6.9 Bohr radii. With these lattice constants, the atomic volumes of the atoms are essentially the same in the two different crystal structures. Most of the bcc models contain 432 atoms and the fee models contain 500 atoms. These clusters are periodically reproduced to fill all space. Some of these calculations have been described previously. The test that is used to demonstrate that these clusters are large enough to be self-averaging is to repeat selected calculations with models that have the same concentration but a completely different arrangement of Cu and Zn atoms. We found differences that are quite small, and will be specified below in the discussions of specific properties. 

Calculate the ratio of the number of electrons in a neutral xenon atom to the number in a neutral neon atom. Compare this number to the ratio of the atomic volumes of these two elements. On the basis of these two ratios, discuss the effects of electron-electron repulsions and electron-nuclear attractions on atomic size. 

The volume per mole of atoms of some fourth-row elements (in the solid state) are as follows K, 45.3 Ca, 25.9 Sc, 18.0 Br, 23.5 and Kr, 32.2 ml/mole of atoms. Calculate the atomic volumes (volume per mole of atoms) for each of the fourth-row transition metals. Plot these atomic volumes and those of the elements given above against atomic numbers. 

That this is not the case follows from the experimental data discussed by A. Russell (9), and F. Koref (10) has attempted to calculate the change of frequency of an element when it enters into combination by means of the alteration of melting-point and atomic volume. According to Lindemann s equation, for the combined atom ... 

V4 is the molecular volume of the solute (m /kmol). Values for simple molecules are given in Table 10.4. For more complex molecules, V. is calculated by summation of the atomic volume and other contributions given in Table 10.4. 

The German physicist Lothar Meyer observed a periodicity in the physical properties of the elements at about the same time as Mendeleev was working on their chemical properties. Some of Meyer s observations can be reproduced by examining the molar volume for the solid element as a function of atomic number. Calculate the molar volumes for the elements in Periods 2 and 3 from the densities of the elements found in Appendix 2D and the following solid densities (g-cuU ) nitrogen, 0.88 fluorine, 1.11 neon, 1.21. Plot your results as a function of atomic number and describe any variations that you observe. 

From the number of particles deflected and the pattern of deflection, Rutherford calculated the fraction of the atomic volume occupied by the positive nucleus. That fraction is 1 part in lO. To give you an idea of what that means, an atom the size of a baseball stadium would have a nucleus the size of a pea. The density of the nucleus is so great that a nucleus the size of a pea would have a mass of more than 250 million tons, as much as 33 million elephants ... 

From van der Waals volumes calculated from atomic increments [57],... 

As its name implies, AIM enables us to calculate such properties of atoms in a molecule as atomic charge, atomic volume, and atomic dipole. Indeed it shows us that the classical picture of a bond as an entity that is apparently independent of the atoms, like a Lewis bond line or a stick in a ball-and-stick model, is misleading. There are no bonds in molecules that are independent of the atoms. AIM identifies a bond as the line between two nuclei. 

Volume of the unit cell (VcM). This may be calculated from the unit cell constants by using the following formulae. From this the density (/>) of the compound may be obtained p = Mcc /FccM, where MceH is the mass of the matter in the unit cell, that is the sum of the masses of all the atoms contained. The mass of an individual atom is calculated in grams by the ratio of the atomic molar mass and the Avogadro number. 

Concluding this section, we may mention a paper by Daams and Villars (1993) concerning an atomic environment classification of the chemical elements. Critically evaluated crystallographic data for all element modifications (and recommended atomic volumes) have been reported. Special structural stability diagrams were used to separate AET stability domains and to predict the structure (in terms of environment types) of hitherto unknown high-pressure and high-temperature modifications. Reference to the use of AET in thermodynamic (CALPHAD) modelling and calculation has been made by Ferro and Cacciamani (2002). 

The atomic volume can actually be calculated without recourse to, or use of, crystallography. The atomic density n of atoms per cubic centimeter is expressed as... 

Silvery metal density 13.51 g/cm (calculated) atomic volume IScm /mole melts in the range 1,300 to 1,380°C magnetic susceptibihty 12.2xl0-8cgs units/mole at 25°C dissolves in mineral acids. 

The average chemical potential reduction (driving force) per atom for decay of the grooved profile we have considered is the order of Ap=yD/h, where Q is the atomic volume. The decay rate we have calculated is very non-linear in Ap (e.g. it decreases exponentially with h) and contrasts markedly with theories" based on a linear dependence of the decay rate on Ap. 

The metallic radii listed in Table 2 are defined as one half the bond-length in the first coordination sphere of the metal structure. They are obtained from the atomic volumes Vat, calculated at room temperature for all phases of a metal (see Chap. C), by means of the equation ... 

The first part of the chapter is devoted to an analysis of these correlations, as well as to the presentation of the most important experimental results. In a second part the following stage of development is reviewed, i.e. the introduction of more quantitative theories mostly based on bond structure calculations. These theories are given a thermodynamic form (equation of states at zero temperature), and explain the typical behaviour of such ground state properties as cohesive energies, atomic volumes, and bulk moduli across the series. They employ in their simplest form the Friedel model extended from the d- to the 5f-itinerant state. The Mott transition (between plutonium and americium metals) finds a good justification within this frame. 

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