Specific volume: defined, 94 equation

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

V, = Specific volume at inlet conditions, m /kg. Other nomenclature is as defined after Equation (4), above. 

The equilibrium state is generated by minimizing the Gibbs free energy of the system at a given temperature and pressure. In [57], the method is described as the modified equilibrium constant approach. The reaction products are obtained from a data base that contains information on the enthalpy of formation, the heat capacity, the specific enthalpy, the specific entropy, and the specific volume of substances. The desired gaseous equation of state can be chosen. The conditions of the decomposition reaction are chosen by defining the value of a pair of variables (e.g., p and T, V and T). The requirements for input are ... 

The MOLWT-II program calculates the molecular weight of species in retention volume v(M(v)), where v is one of 256 equivalent volumes defined by a convenient data acquisition time which spans elution of the sample. I oment of the molecular weight distribution (e.g., Mz. Mw. Mn ) are calculated from summation across the chromatogram. Along with injected mass and chromatographic data, such as the flow rate and LALLS instruments constants, one needs to supply a value for the optical constant K (Equation la), and second virial coefficient Ag (Equation 1). The value of K was calculated for each of the samples after determination of the specific refractive index increment (dn/dc) for the sample in the appropriate solvent. Values of Ag were derived from off-line (static) determinations of Mw. 

The rate of sedimentation is defined by the sedimentation constant 5, which is directly proportional to the polymer mass m, solution density p, and specific volume of the polymer V, and inversely proportional to the square of the angular velocity of rotation o), the distance from the center of rotation to the point of observation in the cell r, and the fractional coefficient /, which is inversely related to the diffusion coefficient D extrapolated to infinite dilution. These relationships are shown in the following equations in which (1 — Vp) is called the buoyancy factor since it determines the direction of macromolecular transport in the cell. 

For a transmitted shock wave advancing into any gas at an initial pressure pe of 1 atm, the RH (Rankine-Hugoniot) equation defines a functional relationship between pressure p and particle velocity w behind the wave S3, involving initial pressure, initial specific volume v, and equations of state of the target medium. Similarly, the conditions behind the reflected wave S2 and close to the product-target interface are expressible by means either of the shock wave equations or the Rie-mann adiabatic wave equations in terms of any one such variable and the conditions... 

Before examining an example which shows the effects of density, the unit "specific volume" must be defined. Specific volume is defined as volume per unit mass as shown in Equation 3-1. 

These equations relate the undisturbed explosive lying at rest with pressure Pq = 0 and specific volume Vq = to the state behind the detonation front, which is characterized by a pressure P, a specific volume V, and a particle flow velocity u. Both u and the detonation velocity, D, are measured in the reference frame of the undisturbed material. Because Pq and Vq are known, the Rankine-Hugoniot relations are a set of three equations for the four unknowns, u, D, P, and V. The first relation determines u in terms of D, P, and Vi, which leaves two equations with three unknowns. The first of the remaining equations, Eq. (4b) defines the Rayleigh line while Eq. (4c) defines the Hugoniot curve. The problem is formally determined by selecting the solution of Eqs. (4b) and (4c) that corresponds to the minimum value of D for an unsupported detonation. This additional condition is the Chapman—Jouguet hypothesis, which was put on a firmer foundation by Zel dovich. ... 

Specific Enthalpy (h) is defined by the equation h = u + pv where p is the pressure and v is the specific volume. Specific enthalpy is measured in J/kg. When a small amount of heat is added to a substance at constant pressure, the increase in specific enthalpy is given by -dh = cp dT, where cp is the specific heat capacity at constant pressure. 

Since v p is defined as the specific volume at close-packed state and p is equal to e /v, i.e., the cohesive energy density at close-packed state [17], the specific volume at 0 K corresponds to vsp, and the cohesive energy density at 0 K to p. The T is obtained by inserting the values ofp, v, and simulated (T, vsp) data at room temperature into the lattice fluid theory. The absolute values of simulated equation-of-state parameters may not be the same as the experimental ones as shown in Table 1, because the procedures obtaining the parameters are differ-... 

Once the two thermodynamic variables, pressure and temperature, have been defined, it is possible to evaluate all the other thermodynamic variables characterizing the stagnation state. For example, stagnation specific volume emerges from equations (14.43) and (14.44) as... 

If equations of state (or specific volume data) are available for both the pure fluid and the mixture, the integral can be evaluated. However, for liquid mixtures not describable by an equation of state, common thermodynamic notation is to define an activity coefficient /i(7, P, x), which is a function of temperature, pressure, and composition, by the equation... 

Figure 11.15b shows a comparison of various fractional free volumes in ER6. The total free volume is defined as the difference between the total, V, and the van der Waals specific volume, Vvdw, which was calculated from the group contributions given by van Krevelen [1993]. In the temperature range from Tg (PVT) = 325 to 470 K,/f varies between 0.316 and 0.373. Its value for a hexagonal close-packed (hep) structure is 0.26. The Bondi free volume [Bondi, 1968] assumes a occupied volume of 1.3Vvdw-/Bondi varies between 0.111 and 0.185. It is distinctly larger than the hole free volume determined from the S-S equation of state of PVT (and from PALS) experiments, h = Vf IV, which increases from 0.0576 to 0.134. 

There are generally two responses of proteins towards the application of pressure. They result from the property that pressure favours states with a smaller specific volume. This is described by the standard equations which define the reaction volume AF and activation volume of a thermodynamic reaction ... 

While accurate calculation of the entropy of pure water turned out to be a formidable task, it is, however, not hard to rationalize why the entropy of liquid water is much smaller than that in the ideal gas limit. First, the translational entropy is lower because of the excluded-volume effect. The volume available to a water molecule is defined by its neighbors and the specific volume in the Sackur-Tetrode equation is to be replaced by the free volume available to individual water molecules. When this is taken into account, we obtain a contribution of 11.8 caFK -mor from translational entropy. Second, the rotational contribution also gets reduced because of the restriction ftiat rotational motion experiences in liquid water due to hydrogen-bonding. The reduced value of the entropy... 

The prediction of miscibility requires knowledge of the parameters T" (the characteristic temperature), p (the characteristic pressure) and V (the characteristic specific volume) of the corresponding equation of state which can be calculated from the density, thermal expansivity and isothermal compressibility. The isobaric thermal expansivity and the isothermal compressibility can be determined experimentally from p-V-Tmeasurements where these values can be calculated from V T) and V(p)j. The characteristic temperature T is a measure of the interaction energy per mer, V is the densely packed mer volume so that p is defined as the interaction energy per... 

P, V and T are the actual pressure, specific volume and temperature while the values with asterisks are characteristic for a given material. Moreover, the free volume is defined as = v — v. The Hartmann equation has been demonstrated to work well for both polymer solids and melts. Figure 12.4 shows that the combination of equations (12.1)-(12.3) provides the capability of prediction of the shift factor aj values both below and above the glass transition temperature. 

The basic assumption made by Brout was that metallic uranium obeyed a universal law known as the principle of corresponding states. Briefly, the principle of corresponding states is that all substances obey the same equation of state in terms of the reduced values of pressure, temperature, and specific volume, the reduced values being defined as the absolute value divided by the critical value. Therefore, after making the basic assumption that UO2 obeys the principle of corresponding states (for nonmetals), critical constants for UO2 must be estimated before the equation of state calculations can be done. 

In Equation 17, v defines the time-dependent, instantaneous value of the specific volume, and v the corresponding equilibrium value at the temperature, T. 

In these two equations, xp/r) are the molecular orbitals (MOs), p(r) is the electronic density, v, (r) is the external potential felt by electrons, is the set point for the constraint, and w(r) is a weight function that defines the constraint property. The Coulomb and exchange-correlation energy contributions are denoted by T[p] and Exclp]- The cDFT energy equation can be applied, for example, to constrain a given number of electrons to occupy a specific volume Q. Alternatively, if the... 

Suppose the particle is subject to a gravitational force, = mg -vp, where g is the gravitation constant, v is tire partial specific volume of the solute particle, and p is the density of the solvent. The particle will move in the solution at a rate determined by the gravitational force and the frictional resistance. It will accelerate until the two forces balance and a terminal average velocity is attained. The instantaneous velocity will continue to fluctuate due to the random force, but a well-defined average terminal velocity is observed. The form of Equation 5.44 appropriate for this situation is ... 

The principle of corresponding states can be used to express the pressure, tan-perature, and specific volume in terms of reduced variables. Experimental observations reveal that the compressibility factor, Z Equation (2.24), for different fluids exhibits similar behavior when correlated as a function of reduced temperature, T, and reduced pressure, The reduced variables may be defined with respect to some characteristic quantity. For example, they can be defined as follows with respect to critical temperature and critical pressure ... 

The concept of free volume, Vi, and the idea that the mobility of molecules at any temperature is primarily controlled by the free volume, was brought forth by Doolittle [45] in explaining the non-Arrhenius temperature dependence of the viscosity, rj, of liquids of low molecular weight. The free volume is defined as the difference between the total specific volume v and an occupied volume, Vo. The Doolittle equation. 

In this expression, V is the polymer specific volume, and Vq is the so-called occupied volume of the polymer, which is commonly estimated by group contribution methods (20), For correlations of transport properties with free volume, the FFV defined in equation 6 is used in place of (v ) in equation 5, and the parameters A and yw are treated as empirical adjustable constants. The chapter by Laciak et al, in this book describes a new methodology for estimating permeability coefficients a priori. Their approach does not rely explicitly on correlations between transport properties and... 

The state of a system represents the condition of the system as defined by the properties. Properties are macroscopic quantities that are perceived by our senses and can be measured by instruments. A quantity is defined as the property if it depends only on the state of fhe system and independent of the process by which it has reached at the state. Some of the common thermodynamic properties are pressure, temperature, mass, volume, and energy. Properties are also classified as infensive and exfensive. Infensive properties are independent of fhe mass of fhe system and a few examples of this include pressure, temperature, specific volume, specific enthalpy, and specific entropy. Extensive properties depend on the mass of the system. All properties of a system at a given state are fixed. For a system that involves only one mode of work, fwo independent properties are essential to define the thermodynamic state of fhe system and the rest of the thermodynamic properties can be determined on the basis of fhe fwo known independent properties and using thermodynamic relations. For example, if pressure and temperature of a system are known, the state of fhe system is then defined. All other properties such as specific volume, enthalpy, internal energy, and entropy can be determined through the equation of state and thermodynamic relations. 

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