Modulus molar volume

Mcdifler, in glass structure, 59, 62 Modulus of elasticity, see elasticity modulus Molar volume, 82—83 Molybdenum silicide, 345, 346, 416 Monolithic lining, 391 Monticell ite, 240 Monfmorillonite, 33 Mullife, 38-42, 232, 372 in porcelain, 304—306 refractories, 373 Multi-chamber kiln, 298—299... 

E is Young s modulus and Vav is die average molar volume. Summing these terms the total Gibbs energy change accompanying the fluctuation, AG is... 

In cellulose II with a chain modulus of 88 GPa the likely shear planes are the 110 and 020 lattice planes, both with a spacing of dc=0.41 nm [26]. The periodic spacing of the force centres in the shear direction along the chain axis is the distance between the interchain hydrogen bonds p=c/2=0.51 nm (c chain axis). There are four monomers in the unit cell with a volume Vcen=68-10-30 m3. The activation energy for creep of rayon yarns has been determined by Halsey et al. [37]. They found at a relative humidity (RH) of 57% that Wa=86.6 kj mole-1, at an RH of 4% Wa =97.5 kj mole 1 and at an RH of <0.5% Wa= 102.5 kj mole-1. Extrapolation to an RH of 65% gives Wa=86 kj mole-1 (the molar volume of cellulose taken by Halsey in his model for creep is equal to the volume of the unit cell instead of one fourth thereof). 

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence ... 

The material properties used in the simulations pertain to a new X70/X80 steel with an acicular ferrite microstructure and a uniaxial stress-strain curve described by er, =tr0(l + / )", where ep is the plastic strain, tr0 = 595 MPa is the yield stress, e0=ff0l E the yield strain, and n = 0.059 the work hardening coefficient. The Poisson s ratio is 0.3 and Young s modulus 201.88 OPa. The system s temperature is 0 = 300 K. We assume the hydrogen lattice diffusion coefficient at this temperature to be D = 1.271x10 m2/s. The partial molar volume of hydrogen in solid solution is... 

The bulk modulus of a mineral, k, represents the pressure-dependence of its molar volume, V, defined by... 

Another simple model that can explain how the excess volume in deformations can lead to a reduction of the enthalpy of formation in non-metallic materials is the Birch-Mumaghan (BM) equations of state that give the molar energy as a function of the equilibrium molar energy Eg, the equilibrium molar volume Vo, the actual molar volume V, and the bulk modulus at equilibrium Bo [12]. At OK, the enthalpy of formation coincides with the molar energy of formation between the two deformed materials. 

Straightforward measurements of elastic properties of materials can be made via high-pressure static compression experiments, in which X-ray diffraction (XRD) is used to measure the molar volume (V), or equivalently the density (p), of a material as a function of pressure (P). The pressure dependence of volume is expressed by the incompressibility or isothermal bulk modulus (Kt), where Kp = —V(bP/bV)p. 

Here the porosity and the diffusivity vary with conversion of solid vs and v are the reactant and product molar volumes. A Thiele modulus < > and dimensionless time 9 can be defined, e.g., for a rate second-order in A and first-order in S ... 

Elastic constants of solids can be related to the fundamental interaction energy terms. In fact it can be shown easily that the bulk modulus, K scales as the energy density, Ulr for an ionic material interacting through Bom-Mayer potential (see later in this section). Thus the molar volume which increases with the presence of larger ions in glasses can be expected to cause a decrease in Young s modulus. 

Figure 3.1 8 Bulk modulus of PMMA at different pressures vs. a function of molar volume. The straight line is the prediction of Eq. (3.26) for a van der Waals solid with = 137kJ mol. ...

Here, >i is the volume fraction of the solvent in the swollen polymer, Oxx is the stress developed within the polymer, Dn is the mutual diffusion coefficient, % is the polymer-solvent interaction parameter, Vi is the molar volume of the solvent, T is the temperature, E is the modulus of die polymer and T is the viscosity of the polymer. Equation (1) is valid in the region between x s R and x = S. 

The ratio of properties in Equation 9.24 is called the Krichevskii function, and identified at the solvent critical point as the Krichevskii parameter (Levelt Sengers 1991). The first EST correlation for partial molar volumes of gases in liquids was done by Brelvi (Brelvi and O Connell 1975c), using characteristic properties for his correlation of the reduced bulk modulus (Brelvi and O Connell 1972). Recent work (Ellegaard, Abildskov, and O Connell 2011) has used the form of Equation 9.4 for partial molar volumes of gases in ILs. The comparisons with data for these systems seem not to be as successful as for their compressibilities and phase equilibria, for reasons that are not apparent. 

A sample of vulcanized natural rubber, d5-polyisoprene, swells to five times its volume in toluene. What is Young s modulus of the unswollen elastomer at 25°C (The interaction parameter, Xi, is 0.39 for the system c/5-polyisoprene-toluene.) [Hint See equation (9.92) the molar volume of toluene is 106.3 cm /moL]... 

Fig. 5. Electronic density-of-states function N(E) derived from experimental values at 600 K of the bulk modulus B, partial molar volume Vjj and concentration dependence of AH by means of Eq. 39. The theoretical curves were obtained from refs. 22 and 28 by means of a convolution of calculated N(E) with an appropriate thermal function (600 K). For details see ref. 40.

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