Thomson equation

Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

The relative lowering of the saturated vapour pressure of water is described by the Thomson equation ... 

If two or more crystalline forms of a substance exist, they are called polymorphic forms if the substance is an element (C, S, P) they are called allotropic forms. The Thomson equation obviously applies to this case ... 

NMR cryoporometry relies on the melting point depression, i.e., the difference in the melting point of crystals with a finite size d, Tm(d), relative to the value of the bulk liquid Tm, which is given by the simplified Gibbs-Thomson equation [16] ... 

Isothermal crystallization was carried out at some range of degree of supercooling (AT = 3.3-14 K). AT was defined by AT = T - Tc, where Tj is the equilibrium melting temperature and Tc is the crystallization temperature. T s was estimated by applying the Gibbs-Thomson equation. It was confirmed that the crystals were isolated from each other by means of a polarizing optical microscope (POM). 

They were evaluated from our analysis of the primary nucleation and lateral growth rates and that of the l dependence to the melting temperature Tm using the Gibbs-Thomson equation. Insertion of the parameters given by Eq. 20 into Eq. 6 shows that the shape of a nucleus is a long thin rectangular parallelepiped with the ratio of... 

The volume of one vapor molecule may be computed from the gas law. pvy = kT, and the volume vL may be neglected compared with vY. The substitution and integration give the Thomson Equation (T2)... 

The activities of precipitate particle components depend explicitly on their size and form. The quantitative relation which describes this fact is the so-called Gibbs-Thomson equation... 

Up to this point we have dealt with the thermodynamics of planar boundaries. Let us add several relations for curved interfaces. First, we have to establish an equivalent to the Gibbs-Thomson equation which holds for curved external surfaces in a multi-component system. For incoherent (fluid-like) interfaces, this can be done by considering Figure 10-5. From the equilibrium condition at constant P and T, one has... 

The surface energy per area, 7, has the same units as a force per length and for some interfacial geometries can lead to an interfacial net force that is balanced by a difference in pressure between the two adjacent phases. If 7 is isotropic, this pressure difference is directly proportional to the interfacial curvature through the the Gibbs-Thomson equation (see Sections C.2.1 and C.4.1),... 

C.4 EQUILIBRIUM AT A CURVED INTERFACE C.4.1 Gibbs-Thomson Equation... 

Equation C.23 is the form of the Gibbs-Thomson equation introduced in Eq. C.17. It is a conditionfor mechanical equilibrium in a two-phase system with a curved interface. The phase located on the side of the interface toward its center of curvature (e.g. the (3 phase in Fig. C.5), has the higher pressure. Note also that for a flat interface, Eq. C.23 gives Pa = P0, as expected. 

Another approach to estimating the size of a latent image center uses redox buffers to determine the size at the transition potential between dissolution and growth. These buffers either develop the latent image centers, bleach them, or do neither. Konstantinov and associates (203,204) used the Gibbs-Thomson equation to analyze results obtained by this method. 

They arrived at a size of the order of 100 silver atoms for a bare metal nucleus, but only 10 atoms for a nucleus embedded in gelatin. Moisar and associates (205) estimated the minimum size as four silver atoms. Ronde (206) revised their calculations and obtained 10 for an unsensitized silver bromide emulsion and six for a S-sensitized emulsion. The validity of the Gibbs-Thomson equation for such small sizes, however, is questionable. 

The spatial microstructure of the interface is strongly influenced by its surface energy, which appears in the Gibbs-Thomson equation (87) for the melting temperature of a curved interface... 

The relationship between particle size, solubility and supersaturation is expressed by the well-known Gibbs-Thomson equation [5a] ... 

As shown in Equation 10.4, the depression of the melting point of a given confined solvent is related to the geometry of the pores of the confining material. In principle, measurement of AT can give access to the pore size. Three main techniques have been developed to measure porosity in solids via the use of the Gibbs-Thomson equation thermoporosimetry, NMR cryporometry and surface force apparatus. These techniques are secondary methods since they require pre-... 

In 2006, McKenna and co-workers reviewed recent work on the use of TPM to study polymer heterogeneity [35]. In this work, uncrosslinked and crosslinked polysisoprene was studied with benzene and hexadecane as swelling solvents. The authors were able to distinguish contributions coming from the confinement as described by the Gibbs-Thomson equation and contributions from polymer-solvent interactions described by the Flory-Huggins (FH) theory [36, 37]. For the first time, it was shown that for an uncrosslinked sample an excess shift AT is... 

The next step is the scattering by an atom. This effect is basically the addition of the scattering of the electron cloud around the nucleus, since each electron in the atom scatters part of the incident radiation in a coherent form in agreement with the Thomson equation. Owing to the fact that the electrons in an atom are located at different points within the atom, and the fact that the x-ray wavelength is of the same order as the atomic dimensions, there will be path differences between waves scattered by different electrons if these path differences are less than one wavelength, then the interference will be partially destructive [20,22,26], To describe this effect, the parameter/is defined, also called the atomic scattering factor, which is the ratio of the amplitude scattered by an atom, Aa, to the amplitude scattered by an electron, Ae, that is [21]... 

If we examine the competitive growth of multiple-size crystallites in solutions, we find the growth occurring along concentration gradients. The concentration gradients around the particles are explained with the Gibbs-Thomson equation ... 

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