Density of phase

In order for an ensemble to represent a system in equilibrium, the density of phase must remain constant in time. This means that Liouville s equation is satisfied, which requires that g is constant with respect to the coordinates q and p, or that g = g(a), where a = a(q,p) is a constant of the motion. Constants of motion are properties of an isolated system which do not change with time. Familiar examples are energy, linear momentum and angular momentum. For constants of motion H,a = 0. Hence, if g = g a) and a is a constant of motion, then... 

Peltier coefficient for charge-transfer reaction h, V density of phase k, glcm ... 

The first or probability factor is essentially a ratio of partition functions, and represents the integrated equilibrium density of phase points on S per phase point in A. The second or trajectory-corrected frequency factor is the number of successful forward trajectories per unit time and per unit equilibrium density on S. The ratio of this to the uncorrected frequency factor 0) >s represents the number of successful forward trajectories per forward crossing. Anderson called this ratio the conversion coefficient to distinguish it from the transmission coefficent of traditional rate theory (1), which was usually defined rather carelessly and given little attention, because it could not be computed without trajectory information. 

Assuming the material density or the intrinsic averaged density of phase k to be constant (as is true for the solids and almost true for the fluid when the effect of temperature variation on the material density can be neglected), we have... 

Figure 9. Variation of density of phase Ti2S3 as a function of composition...

The essential input data are (a) the bulk chemical composition of the cement, (b) the quantitative phase composition of the cement and the chemical compositions of its individual phases, (c) the fraction of each phase that has reacted, (d) the w/c ratio, (e) the COj content of the paste and an estimate of how it is distributed among phases, and (0 the composition of each hydrated phase for the specified drying condition. If (b) is unknown, it may be estimated as described in Section 4.4, and if (c) is unknown, it may be estimated from the age as described by Parrott and Killoh (P30), or, more simply though less precisely, by using empirical equations (D12,T37). If the phase composition by volume and porosities are to be calculated, densities of phases are also required. 

Calculations based on reaction stoichiometry and densities of phases support the conclusions from experimental observations that mature pastes of composite cements are more porous than comparable pastes of Portland cements. This is indicated by the results in Table 7.3, 9.4 and 9.6. Similar calculations for 180-day-old pastes of w/s 0.45 indicate free water porosities of about 24% for a typical Portland cement, 35% for a cement with 40% slag, 35% for one with 40% pfa and 32% for one with 30% microsilica. The calculated values are in all cases somewhat higher than observed mercury porosities (F34,F41). 

The density estimate denotes the latest obtained density of phase k. In the dispersed phase p = while in the continuous phase... 

As the ensemble of systems evolves in time, each phase point moves along the trajectory (r (Z), p (Z)). Equation (1.107) states that the density of phase points appears constant when observed along the trajectory. 

Here we have added one equation—the total mass balance—which includes the density of phase one as it flows out of the system. Recall also that for a double salt MaLj, we have the following ... 

Fig.6b. Number density of phase separated areas, N, per 10 0 molecules in the monolayer.

Thus, the density of phase points behaves as an incompressible fluid. In other words, a differential volume element in phase space containing a certain number of phase points can change its shape, but not size. 

P and py are the molecular number densities of phases i and j built up from components i and j, respectively... 

Io(x) and l2(x) are modified Bessel functions of zero and second order, Bb and pe are the bulk elasticity modulus and the density of phase B, / is the capillary length (/ ac). Substituting Eqs. (12) and (14) into (8) one can obtain the general expressions for the bubble (drop) volume variation 6V ,(ico), the pressure variations inside the bubble (drop) 5P, (ici)), and inside the cell 6P (ico) at the given bP, and 6Vp . The effect of both the flow mobility and the compressibility of the medium in the capillary on the oscillating bubble or drop measurements is analysed in details in [26]. Here only the simpler case of quasi-stationary flow of incompressible media in the capillary is described. 

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