Clapeyron relation

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

Near the critical region, the property ratio can be replaced by T(dP/dT)sat via the Clapeyron relation since both hfg and tyg are approaching zero. Here (dP/dT)sat is the slope of the vapor-pressure curve and has units of Pa/K. For multicomponent fluids, Eq. (23-93) is evaluated for each major component (> 10 % wt), and the largest single-component venting requirement is used. Refer to CCPS Guidelines (1998) for more complex schemes. 

The average dT/dt is typically an arithmetic average between the value at set pressure and the value at peak allowed pressure. The properties Cp, hfg, i, either can be evaluated at the set conditions or can be taken as the average values between the set condition and the peak allowed pressure condition. Alternatively, the term h/g/t)/g in Eq. (23-95) can be replaced by T(dP/dT)tat via the Clapeyron relation. This holds reasonably well for a multicomponent system of near constant volatility. Such an application permits direct use of the experimental pressure-temperature data obtained from a closed-system runaway VSP2 test. This form of Eq. (23-95) has been used to demonstrate the advantageous reduction in both vent rate and vent area with allowable overpressure (Leung, 1986a). 

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... 

Equation (1.210) means that dP dx] = 0, and at a specified temperature and the total vapor pressure of a binary liquid mixture is a minimum or a maximum at the composition of the azeotropic mixture. A similar analysis of constant pressure systems indicates that, at a specified pressure and temperature of a binary liquid, mixture is a minimum or a maximum at the composition of the azeotropic mixture. If we simultaneously vary the temperature and pressure for an azeotropic mixture, we have the Clapeyron relation... 

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 sequence of calculations using the above equations is as follows the properties of the saturated vapor were calculated first using (3) and (4) the Clapeyron relation (5) was then used to establish the properties for the saturated liquid with values of Ap ap calculated from solutions of (1) and (2) and dPjdT from (2). The saturated liquid line as determined was then used as a reference for calculating the entropy and enthalpy for any temperature below critical and densities greater than the saturated liquid using (6) and (7). Calculations for all remaining coordinates of temperature and density were performed using (3) and (4). 

In view of the various shortcomings inherent in the DSC method, use of the Clapeyron relation is more appropriate for the estimation of (A5ni)/> ... 

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].

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