Clausius - Clapeyron relation

In some regions, the equilibrium pressure of coexistence of gas hydrate with the corresponding gaseous state follows a Clausius-Clapeyron relation ... 

Neutron diffraction studies under pressure [84] on the 70/30 composition have revealed that transitions in this copolymer are displaced towards higher temperature with increasing pressure, as can be seen in the phase diagram of Fig. 11. In addition, it is worth noting the non-linear increase of the Curie temperature with pressure. By considering the Clausius-Clapeyron relation dTc/dP = TCAVC/Ahc, this effect can be related to a decrease in the volume... 

Villard measured hydrates of Ar, and proposed that N2 and O2 form hydrates first to use heat of formation data to get the water/gas ratio deForcrand and Thomas sought double (W/H2S or H2Se) hydrates found mixed (other than IpSx) hydrates of numerous halohydrocarbons mixed with C2H2, CO2, (pHg de Forcrand first used Clausius-Clapeyron relation for AH and compositions tabulated 15 hydrate conditions Scheffer and Meyer refined Clausius-Clapeyron technique de Forcrand measured hydrates of krypton and xenon... 

A commonly used expression for this number is Sh = 2.0 - - 0.55Re/ Scp where Rcp = d u —u /i/ is the particle Reynolds number and Scp is the fuel vapor Schmidt number. In this definition v is the carrier phase kinematic viscosity. In Eq. (8.2), one important parameter is the Spalding number Bm = Ypx f)/(1 — Yfx) where Ypx is the fuel mass fraction at the droplet surface, calculated from the fuel vapor partial pressure at the interface ppx which is evaluated from the Clausius-Clapeyron relation ... 

Considering that dissociation occurs upon volatilization, the temperatures can be correlated extremely well on a In P vs (1/rd.voi) plot, where P is the total system pressure and T a.voi decomposition temperature, as the case dictates. Such a plot is shown in Fig. 11. Since the Clausius-Clapeyron relation for vapor pressure of pure substances shows an exponential dependence on temperature, TVoi was considered a pseudo-boiling point at the respective system pressure. For a substance that vaporizes congruently to its gaseous state, the slope of lines on a In P vs (l/Tvoi) plot represents the enthalpy of vaporization. Indeed, the enthalpy of vaporization calculated from the slope on a In P vs (l/TVoi) plot for the B-O2 system (360 kJ/mol) agrees exactly with the value calculated by using... 

The vapor pressure of any substance increases nonlinearly with temperature according to the Clausius-Clapeyron relation. 

Temperature The relation between aqueous solubility and temperature can be defined by a relation analogous to the Clausius-Clapeyron relation derived for vapor pressure... 

The temperature is tentatively lowered by a few degrees. The loss of enthalpy of the liquid then serves for vaporization. The volume released is replaced by vapour, as long as there is still liquid and the quantity of vaporized liquid is sufficient. Otherwise this quantity is the upper limit. As a consequence we obtain a new value for the pressure. By iteration the temperature is subsequently modified until the values for pressure and temperature lie on the vapour pressure curve (vid. Fig. 10.4). The latter can be determined from approximate equations [13] or the Clausius-Clapeyron relation [19]. The connection between temperature and pressure is ensured by the equation of state for gases. 

Fig. 16. a) schematic diagram of the area for entropy change estimation from the Clausius-Clapeyron equation, from a M vs. H plot of a magnetic first-order phase transition system, and b) magnetic entropy change versus temperature, estimated from the Maxwell relation (full symbols) and corresponding entropy change estimated from the Clausius-Clapeyron relation (open symbols). 

In Fig. 1, it is assumed that the LLPT line has a negative slope, (d7/dP)LLPT < 0, but this is not always the case. The slope of the LLPT line depends on the entropy and volume differences between LDL and HDL. Specifically, at any given point (71 P) lying on the coexistence line, the Clausius-Clapeyron relation specifies that... 

The suggestion that LLPT should appear among elements and compounds at constant chemical composition first arose from the analysis of the unusual melting behavior recorded for certain solids at high pressure. The melting relation dTm/dP is determined by the Clausius Clapeyron relation ... 

Figure 5.12 Estimated normal boiling point temperatures, of [C Cjim][Nty ionic liquids as a function of alkyl side chain length, n. O, 3" using the Eotvos equation with the data under discussion [26] , using the Guggenheim equation with the data under discussion [26] A, using the Guggenheim equation with data from [1] , using the Clausius-Clapeyron relation with experimental vapor pressure values from [33-35].

In general, a weight loss of up to 20% observed as a result of decomposition for all the MAX phases can be attributed to the release of gaseous Al by sublimation during the decomposition process because the vapour pressures of the A elements exceed the ambient pressure of the furnace (i. e. < 5 x 10 torr) at > 1500°C. Since the vapor pressure of a substance increases non-linearly with temperature according to the Clausius-Clapeyron relation [25], the volatility of A elements will increase with any incremental increase in temperature. 

For a first-order transition from a phase a to a phase) , the continuity of the free energy function and the equilibrium condition result in the well-known Clausius-Clapeyron relation (equation 8)... 

Up to this point, the theoretical approaches treated the origin of freezing and melting in various confined geometries as completely different. In this chapter we show that a unified thermodynamic approach based on Clausius-Clapeyron relations for the coexistence of solid and liquid phases jointly, with equations relating the vapor pressures above bulk and confined state of substance, can explain the shift of the temperature of triple point in various confined systems and define the key physicochemical parameters determining this shift. 

Since in all the above-mentioned systems the phase transition occurs with the presence of the vapor phase, the problem of finding the melting/freezing point reduces to the determination of the triple point for each of the cases considered. The general approach to treating this problem is based on the Clausius-Clapeyron relations, describing the solid-vapor and liquid-vapor lines of equilibrium on the phase diagrams in the pressure versus temperature coordinates ... 

In this chapter we have proposed a unified thermodynamic approach to the analysis of melting/freezrng phenomena in confined systems. The approach is based on the Clausius-Clapeyron relations for coexistence of solid and liquid phases jointly with equations relating the vapor pressures above bulk (with plane interfaces) and confined states of a substance. For illustration we have applied our analysis to three types of confinement plane interfaces, small particles and pores. The analysis for other types of confinement like free, wetting and adsorption films, emulsions, and so on is straightforward. Notably, the logic of derivation allows us to better understand the influence and physical content of the parameters involved in the key equations. 

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