Phase-integral quantities

The phase-integral quantities in the formulas obtained in Chapter 5 can be expressed in terms of complete elliptic integrals. One thereby achieves the result that well-known properties of complete elliptic integrals, such as for instance series expansions, can be exploited for analytic studies. Furthermore, complete elliptic integrals can be evaluated very rapidly by means of standard computer programs. 

In this chapter we collect and present, without derivation, in explicit, Anal form the relevant phase-integral quantities and their partial derivatives with respect to E and Z expressed in terms of complete elliptic integrals for the first, third and fifth order of the phase-integral approximation. For the first- and third-order approximations some of the formulas were first derived by means of analytical calculations, and then all formulas were obtained by means of a computer program. In practical calculations it is most convenient to work with real quantities. For the phase-integral quantities associated with the r -equation we therefore give different formulas for the sub-barrier and the super-barrier cases. As in Chapter 6 we use instead of L2 , L2n, K2n the notations LAn+1 >, L( 2n+1 KAn+l). 

According to equation (21) the systems of the canonical ensemble are conservative. Each system moves independently of all others and the phase integral exists for each of them. Each system therefore moves on a surface of constant energy and corresponds to a microeanonical ensemble. In this sense the canonical ensemble is built up from a multitude of microeanonical ensembles. Quantities defined for the microeanonical ensemble may therefore be averaged over the canonical ensemble. The original system which is represented by the canonical ensemble however, cannot be described, even approximately, as conservative. It is customary to denote the Hamiltonian of the systems of the canonical ensemble as the Hamiltonian of the original system, which is not justified. 

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

A more general explanation of the temperature independence of the phase-transition quantities will be illustrated with ASa—although it could be illustrated equally as well with AThe integral entropy change for the phase transition is related to the partial entropy changes by Equation 21 (53). 

We emphasize that for the validity of (4.46) with the expressions (4.47a-c) for (2ra+1) the essential restriction is that d2Q2 z)/dz2 must not be too small at the top of the barrier, which means that close to its top the barrier is approximately parabolic, i.e., that the distance from the barrier to the transition points that are not associated with the barrier must be much larger than t" — t. However, when the energy is close to the top of the barrier, it is the slight deviation from parabolic shape close to the top that determines the values of the quantities K2n, n > 0, and one needs accurate values of these quantities for obtaining accurate values of in higher orders of the phase-integral approximation. 

In this chapter we shall describe the procedure for expressing the phase-integral formulas derived in Chapter 5 in terms of complete elliptic integrals. The integral in question is first expressed in terms of a Jacobian elliptic function and then in terms of complete elliptic integrals. Different elliptic functions are appropriate for different phase-integrals. For practical calculations it is most convenient to work with real quantities. For the phase-integrals associated with the r -equation it is therefore appropriate to use different formulas for the sub-barrier case and for the super-barrier case. We indicate in this chapter, where we use the notations L 2n+l L 2n+1 K< 2n+1 ) instead of the notations 2 , 2n used previously, the main... 

Phase-lnegral Quantities and Their Partial Derivatives with Respect to E and Z1 Expressed in Terms of Complete Elliptic Integrals... 

In short, one of the most significant findings in the fields of electrolytes is the view that ions may adopt a non-monotonous concentration profile at the air-water interface. Please note that this is not in contradiction to thermodynamic analysis of the surface tension isotherms. Thermodynamics does not predict a distinct concentration profile instead, it measures an integral quantity from the fluid to the gas phase. There are many... 

Related attempts had been made to calculate integral sorption heats of CO2 on all materials investigated, e.g., on D 47/2, by means of eq. (23) where the isosteric sorption heats (differential quantities) were used to calculate integral quantities over defined ranges of sorption-phase concentration, n, e.g., between its limits n = 0 and n = n. 

The expressions given in Section 35-1 for a uniform medium can be generalized in an intuitive manner provided the refractive index n(r) varies slowly over distance equal to the wavelength of light. Thus the accumulated phase of Eq. (35-3) becomes the integrated quantity... 

The quantities of interest are (i) n, moles of adsorbate (ii) m, mass of adsorbent (iii) V, volume (iv) p, pressure (v) T, absolute temperature (vi) R, molar ideal gas constant (vii) A, surface area of the adsorbent (viii) Q heat (ix) U, internal energy (x) H, enthalpy (xi) 5, entropy and (xii) G, Gibbs free energy. Superscripts refer to differential quantities (d) experimentally measured quantities (exp) integral quantities (int) gas phase (g), adsorbed phase (s) and solid adsorbent (sol) quantities standard state quantities (°). Subscript (a) refers to adsorption phenomena (e.g. AaH) [13, 91]. 

The preceding equations describe the thermodynamic behaviour of a single phase. In an unconstrained equilibrium between two phases each component has the same chemical potential and the same activity in each phase and the integral quantities are linear functions of the composition in a two-phase region. In the diagrams, the functions are drawn with dashed lines in these regions. 

Table Ilia. Integral quantities for the liquid phase at 1273 K. ...

Fig. 2. Integral quantities of the liquid phase at Fig. 3. Activities in the liquid phase at T=1600 K. T=1600 K.

Fig. 4. Integral quantities of the stable phases at Fig. 5. Activities in the stable phases at T=573 K. T=573 K.

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