Calculation from atomic volumes

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

Amino Acids. The values for the refractive indices and molar refractions of the amino acids calculated from the refractive indices by Lorenz-Lorentz Equations 1 and 2 are recorded in Table I. Values for molar refractions of the aliphatic amino acids are in good agreement with values calculated from atomic refraction factors. However, the molar refractions of tryptophan, tyrosine, phenylalanine, and histidine are larger than those calculated from atomic refraction factors and larger than might be expected from their comparative specific volumes. 

Kralft found the increase of Vm per CH2 at the m.p. in paraffin hydrocarbons 18 5-17 6 for 11 to 35 in C H2 +2, the average being 18. He calculated the atomic volumes H 1 5, C 15 and from the fatty acids with C12 to C18 he calculated the atomic volume of oxygen as 6, OH and CO oxygen being assumed equal. The same value was found from the ketones with C15 to C27. 

The awkwardness of a molecule can be simply expressed as the ratio of its surface area to its volume. This parameter is easily calculated from atomic coordinates and van der Waals radii and has been invoked in studies aimed at establishing trends in molecular packing. ... 

Values for the steric parameter V/lOO were calculated from atomic parachors, through a characteristic molecular volume (Vx) to a corresponding intrinsic molecular volume (Vj). 

Polar interactions, which result from dispersion forces (van der Waals interactions) are best described by molar refractiv-ity MR, which is a refractive index-corrected molar volume. Thus, it has also been used to describe steric features of substituents. Like log P, MR is an additive constitutive molecular property which can be calculated from atom and group contributions. " ... 

The excess of NaOH is titrated with standard acid using phenolphthalein indicator. The amount of phosphate in the given solution is calculated from the volume of the standard alkali consumed, using the reaction (8.14) which indicates that 1 gram-atom of phosphorus is equivalent to 23 gram-moles of NaOH. 

A variety of physical properties of several liquids and polymers can be quantitatively described by considering a crystalline arrangement of molecules or other volume elements bound by non-directional forces. In some cases we propose that the intermolec-ular forces in the liquid or solid under question are simply due to van der Waals interactions. In those cases we demonstrate that physical properties such as surface energy cohesive strength compressibility thermal expansion and work of vaporization can be calculated from atomic constants and related to one another by the proposed model. In the case of other liquids and polymers intermolecular forces cannot be described in terms of van der Waals binding alone and other (directional) forces such as dipole-dipole binding must be included. It will be shown that a variety of the calculated properties favorably compare with experimental results. 

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. 

The partial molar volume of an organic compound in dilute aqueous solution can be calculated from the volume increments for atoms and/or atomic groups, assuming additivity and allowing corrections for covolume, ring formation, and electrostriction (ionization) ... 

Comment enzyme substrate use of Eqs. (31), (33), (34) and Tables 1, 2 and 8. For experimental values see Table 8. Calculation from the volume increments for atoms and ions (Table 1) ... 

The number of Auger electrons from a particular element emitted from a volume of material under electron bombardment is proportional to the number of atoms of that element in the volume. However it is seldom possible to make a basic, first principles calculation of the concentration of a particular species from an Auger spectrum. Instead, sensitivity factors are used to account for the unknown parameters in the measurement and applied to the signals of all of the species present which are then summed and each divided by the total to calculate the relative atomic percentages present. 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

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 ... 

V is the volume, and F is a factor of proportionality, which is calculable from the elastic properties of the solid. The connection with elasticity was in fact suspected by Sutherland in 1910 (Phil, May., 20, 657), who found that the infra-red frequency of a solid was of the same order as the frequency of an elastic transversal vibration with a wave length equal to the distance between two neighbouring atoms. To every degree of freedom Debye assigns an amount of energy ... 

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. 

Calculate the apparent volume (in pnt1) and radius (in pm) of a helium atom as determined from the van der Waals parameters,... 

Identify a process to solve the problem. The question asks about the volume of one silver atom. Mass and volume are related through density p — mj V. From this equation, we can calculate the total volume of the silver atoms. The problem also gives the total number of silver atoms transferred from the wire to the spoon. The volume of a single atom is the total volume divided by the number of atoms. Oftentimes, a flow chart helps to summarize the process ... 

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 ... 

It is at this point that the bulk of the CPU time is utilized, since the number of distance calculations increases with the square of the number of chain segments. To reduce this time, an abbreviated check is made to determine if a given atomic cartesian coordinate is within a specified distance (e.g., 12 A) of the newly appended atom. Thus, the vast majority of existing segments are disposed of with one, or at most three, subtraction calculations rather than full calculations of the exact distance. Terms in the second moment from atoms outside this cubic volume are not included in the summation. This is valid because of the rapidity at which the magnitude of a term decreases with increasing separation. 

---