Partial molar atomic volume

4 Partial molar (atomic) volume. Another useful and general way of discussing the actual trend in an alloy system of the average molar dimensions, as represented for instance by the molar (average) volumes, is through the definition and use of the partial molar (atomic) volume. 

The partial molar quantity of a molar quantity Q, related to the component A, is generally written as 0A and is defined by  

The following rules generally apply to the partial molar quantities  

In a binary A-B system, from the graph of the integral quantity vs. composition, the partial values may be obtained, for each composition z, from the intercepts on the A and B axes of the tangent, at the composition z, to the integral curve. The construction is very simple in the special case that, in a certain range of compositions, the curve of the integral quantity is close to a straight line. 

As an example for the specific case of vanadium alloys with palladium, the trend of the average atomic volume of the alloys is shown in Fig. 4.20 and compared with the phase diagram. These data were obtained by Ellner (2004) who studied the solid solutions of several metals (Ti, V, Cr, Mn, Fe, Co and Ni) in palladium. The alloys were heat treated at 800°C and water-quenched. From the unit cell parameters measured by X-ray diffraction methods, the average atomic volume was obtained Vat = c 14 (see Table 4.3). These data together with those of the literature were reported in a graph, and the partial molar (atomic) value of the vanadium volume in Pd solid solution (Fv) 

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Complete the table, calculating the atom fraction of zinc, the number of g-atoms in 100 g of alloy, the volume of 100 g of the alloy, and the volume occupied by one g-atom of the alloy. Also, find the partial molar (atomic) volumes of zinc and aluminium at 25°C in an alloy containing 0.75 atom fraction of zinc. Atomic weights of Zn and A1 are 65.38 and 26.96 respectively. 

Careful consideration was taken in the parameterization process to insure that the parameters were deemed reasonable for the atom types, using the OPLS-AA force field atom types as a comparison. As one of the goals of this project was to ensure that robustness was achieved in many different calculated properties of the newly developed model, several sets of simulations were also performed to ensure that the parameters could achieve a reasonable agreement with experiment. Some of the properties calculated included the gas phase density, the partial molar volume in aqueous solution, and the bulk solvent structure as well. The calculation of the solubility was discussed in the previous section for the parameterization process and the viewing of these results, the solubility will be reported in log S values, as many of the literature values are reported as log S values, and therefore, the comparison would not lose any sensitivity due to rounding error from the log value. 

The composition dependence of the total volume of a solution at constant temperature and pressure is expressed in terms of the partial molar volumes of the solute and the solvent. Since we are concerned with solvation properties, the quantities which we need to discuss are the partial molar volumes in infinite dilution of the solute so that solute-solute interactions make no contribution. In practice, partial molar volumes are obtained indirectly from precise density measurements. The partial molar volumes at infinite dilution of the amino acids are compiled in Table 2 [7]. It is apparent from these data that an approximately linear correlation exists between the partial molar volume and the number of carbon atoms in the backbone. The data indicate volume contributions from the polar head group (NH, COj) and from the CH2 group and to be about... 

Other molecular properties have been also proposed to model the hydrophobic interactions. The parachor, which is related to the surface tension of a compound (139, 140) represents mainly the intermolecular interactions in a liquid. The Hildebrand-Scott solubility parameter, 6, (141) is related to intermolecular van der Waals forces and the closely related molar attraction constant, F, is obtained by multiplying 6 by the molar volume (142). The partition coefficient between two solvents can be obtained from the solubility parameters and the molar volumes of the solute and the solvents (193). This relationship is based on regular solution theory (194) and the assumption that the partial molar volumes of the solute is not different from its molar volume. Recently this has been criticized and a new derivation was proposed (195) in which the partial molar volumes are taken into account. The molar refractivity, MR, is related to dispersion forces and can be obtained as a sum of the partial molar refractivi-ties assigned to atoms and bonds (140, 143). These parameters have been compared (144) to establish their relative applicability to correlations with biological activity. The conclusion was that logP and molecular refractivity were the best parameters. Parameters obtained from high pressure liquid chromatography (144,... 

The simplest material that can change composition is a binary mixture of two atomic species an example would be a copper-tin alloy. Let a mixture of this type contain atomic species A and B then for the chemical potential of species A, dn fdC = RTfC, where is the mole fraction or number ratio of atoms of A to total number of atoms in a sample, n l(n -l- n ), and dfi /dP = the partial molar volume of species A. If stress is nonhydrostatic, the associated equilibrium potential of A for direction n follows a similar relation djx jda = V. ... 

In this expression Vatom and Vcavities are the volumes of the atoms and the cavities respectively and AVhydrat.on is the volume change of the solution resulting from the interactions of the protein molecule with the solvent. More defined models for the partial molar volumes of proteins are discussed by Chalikian [27,28]. Care should be taken if quantities derived Ifom the volume (such as compressibility and thermal expansion) are interpreted on the molecular level. The experimental results may depend on the sensitivity range of the method used. Global measurements such as ultrasonics detect the whole molar volume, while some local probes may feel only the change of the protein interior volume. 

Pressure and temperature effects on the reaction rate of dimethyl l,2,4,5-tetrazine-3,6-dicarboxylate with 1-hexene have been investigated. The activation volume (—26.7cm moP, 298.1 K) is in agreement with the conservation of all four nitrogen atoms in the transition state. Densitometry, H NMR, and calorimetric studies of the reaction indicate nitrogen molecule loss by the intermediate just after its formation. Partial molar volumes in acetone of diene (127.2), 1-hexene (127.6), and the resulting adduct (206.9cm moP ) have been determined <1999T12201, 1999DOK498>. 

Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (K-B) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. 

We plot AFm = M.exp — kM(14i,exp) IS the experimentally measured value and Vm the theoretically calculated value) against the number of atoms in the amino acids in Fig.3-14. It is obvious that the discrepancy AVm increases as the number of atoms N becomes larger. There can be two major reasons for this. The first one is the ideal fluctuation volume discussed in 4.2. The partial molar volume can be decomposed into two terms, the ideal contribution VJJj and the excess quantity The former can further be decomposed into the ideal-gas contribution and the fluctuation volume... 

The internal fluctuation of atoms in a solute molecule is entirely frozen by the constraints expressed by the Dirac delta-functions in the RISM theory. If all the constraints were removed, the atoms would move freely, giving the ideal contribution NtiTk T to the partial molar volume. Therefore, a reasonable conjecture is 0 < < N — l)KTkBT and... 

By combining the K-B theory with the RISM theory, we have derived the equation for calculating the partial molar volume of a polyatomic solute in solvent. We have calculated the VM-values of the 20 amino acids, constituents of natural proteins. The calculated values are always smaller than the corresponding experimental values. Moreover, the discrepancy becomes larger as the number of the atoms in the amino-... 

Because of the simplicity of the functions of state of the ideal gas, they serve well as models for other mixing experiments. Dilute solutions, for example, can be modeled as ideal gases with the empty space between the gas atoms being filled with a second component, the solvent. In this case, the ideal condition can be maintained as long as the overall interaction between solvent and solute is negligible. Deviations from the ideal mixing are treated by evaluation of the partial molar quantities, as illustrated on the example of volume, V, in Fig. 2.25. The first row of equations gives the definitions of the partial molar volumes and Vg and shows the addition... 

It is important to note that the above formulas represent fluctuations (8X=X - (X)) in the properties of the whole system, that is, bulk fluctuations. They are useful expressions but provide no information concerning fluctuations in the local vicinity of atoms or molecules. These latter quantities will prove to be most useful and informative. One can also derive expressions for partial molar quantities by taking appropriate first (to give the chemical potential) and second (to give partial molar volume and enthalpy) derivatives of the expressions presented in Equation 1.28. However, these do not typically lead to useful simple formulas that can be applied directly to theory or simulation. For instance, while it is straightforward to calculate the compressibility, thermal expansion, and heat capacity from simulation, the determination of chemical potentials is much more involved (especially for large molecules and high densities). 

Up to now, the characterization of amino acids by theoretical structural descriptors has not received wide attention. The study reported in [35] employs for predicting the partial molar volumes (pMV) of 17 amino acids (AA) that include some heterocycHc molecules, and appear Usted in Table 3 together with the numerical values for x and Three of the compounds have unknown values for the experimental property (isoleucine, threonine, lysine). This particular molecular set involves four optimizable parameters for each type of atom x (carbon),y (oxygen), z (nitrogen), and w (sulfur). As a starting point in the search for the optimal values of the four parameters, it is assumed that all the variables have zero as the initial value. The simple... 

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