Phase rule isobaric

It is possible to have a four-phase tie-line in a binary system, but this could only be at a unique temperature and pressure, just like a triple point in a unary system. Binary sections are not usually drawn for such unique conditions. That is, when we chose our pressure for our T-X section in 17.3.2 (the isobaric phase rule), it would be extremely unlikely for this choice to be just the pressure needed for four-phase equilibrium, and so three-phase tie-lines are the norm in binary sections. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

In the preceding paragraphs we have seen a few examples of phase equilibria occurring in binary systems represented by means of 2D diagrams built, at constant pressure (isobaric), on temperature/composition axes for which the phase rule... 

The steps for constructing and interpreting an isothermal, isobaric thermodynamic model for a natural water system are quite simple in principle. The components to be incorporated are identified, and the phases to be included are specified. The components and phases selected "model the real system and must be consistent with pertinent thermodynamic restraints—e.g., the Gibbs phase rule and identification of the maximum number of unknown activities with the number of independent relationships which describe the system (equilibrium constant for each reaction, stoichiometric conditions, electroneutrality condition in the solution phase). With the phase-composition requirements identified, and with adequate thermodynamic data (free energies, equilibrium con-... 

Cooling the system is continued until the temperature of Point 2, where the hydrate phase (vertical area that begins at Point 7) forms from the vapor (Point 8) and liquid (Point 6). At Point 2 three phases (Lw-H-V) coexist for two components, so Gibbs Phase Rule (F = 2 — 3+2) indicates that only the isobaric pressure of the entire diagram is necessary to obtain the temperature and the concentrations of the three phases (Fw, H, and V) in equilibrium. 

This means that to fix all the properties of both kinds of crystals, we need only choose the temperature (pressure being already fixed at 1 bar). However, when the first drop of liquid forms, p = 3 (diopside crystals, anorthite crystals, and liquid), and / = 0. Another word for / = 0 is invariant. When p = 3 on an isobaric plane, we have no choice as to T, P, or the compositions of the phases -they are all fixed. This explains why all mixtures begin to melt at the same temperature, and why the liquid formed is always the same composition no matter what the proportions of the two kinds of crystals. No other arrangement would satisfy the phase rule. 

The entropy, Spontaneous vs non-spontaneous, Reversible and irreversible processes, Calculation of entropy changes (Isothermal, isobaric, isochoric, adiabatic), Phase changes at equilibrium, Trouton s rule, Calculation for irreversible processes... 

The specific heat of Si3N4 ceramics is in the temperature range 293 up to 1200 K [Cp (293 K) = 0.67 KJ (K kg)-1] nearly independent of the composition of the additives. The isobaric specific heat values agree well with the isochoric specific heat calculated by Debye s theory. Also the Dulong Petit s rule can applied as an approximation of the Cv values [25 J(K mol)-1] at temperatures >1100 K [371]. From the Cp values at around 100 K the amount of the amorphous grain boundary phase can be calculated [371]. 

The phase diagrams of two-component systems are represented in the two-dimensional space, where the composition is shown on the x axis (in molar or in mass fractions) in agreement with the lever rule, and the temperature is given on the y axis (in °C or in Kelvin). They are the so-called isobaric diagrams, since the constant pressure, mostly the atmospheric one, is assumed. The Gibbs phase law attains thus the form... 

A straightforward, but tedious, route to obtain information of vapor-liquid and liquid-liquid coexistence lines for polymeric fluids is to perform multiple simulations in either the canonical or the isobaric-isothermal ensemble and to measure the chemical potential of all species. The simulation volumes or external pressures (and for multicomponent systems also the compositions) are then systematically changed to find the conditions that satisfy Gibbs phase coexistence rule. Since calculations of the chemical potentials are required, these techniques are often referred to as NVT- or NPT- methods. For the special case of polymeric fluids, these methods can be used very advantageously in combination with the incremental potential algorithm. Thus, phase equilibria can be obtained under conditions and for chain lengths where chemical potentials cannot be reliably obtained with unbiased or biased insertion methods, but can still be estimated using the incremental chemical potential ansatz [47-50]. 

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