Clausius equation, phase diagrams

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

If you plot the temperature and vapor pressure data given in Table 1, you reconstruct the liquid-vapor equilibrium line in the phase diagram of that liquid (Fig. 139). The equation of this line, and you might remember this from your freshman chemistry course, is the Clausius -Clapyron equation ... 

Phase Diagrams Construct phase diagram from Tfu, and Tvap measurements Phase transitions Identification of phase boundaries Comparison with Clausius-Clapeyron equation... 

Suppose we have a solid-liquid coexistence point T, p) for pore fluid on the bulk phase diagram. Though the bulk pressure is at p, the fluid in pore is supposed to have different pressure because of the pore-wall potential and the capillary effect. Not for the bulk pressure but for this pressure felt by fluid in pore, p, the Clausius-Clapeyron equation for the bulk is assumed to hold. 

The shapes of the phase boundaries in a phase diagram can be determined from thermodynamics using the Clapeyron equation. If one of the phases is a vapor, then the Clausius-Clapeyron approximation can be used. 

Clausius-Clapeyron equation (358) boiling point (359) melting point (360) phase diagram (360) triple point (360) critical point (361)... 

The vaporization line in the phase diagram is a plot of the Clapeyron equation or the Clausius-Clapeyron equation. Notice, however, that this line ends at a particular pressure and temperature, as shown in Figure 6.5. It is the only line that doesn t have an arrow on its end to indicate that it continues. That s because beyond a certain point, the liquid phase and the gas phase become indistinguishable. This point is called the critical point of the substance. The pressure and temperature at that point are called the critical pressure pQ and critical temperature Tc- For H2O, pQ and Tq are 218 atm and 374°C. Above that temperature, no pressure can force the H2O molecules into a definite liquid state. If the H2O in the system exerts a pressure higher than pQ, then it cannot exist as a definite liquid or gas. (It can exist as a solid if the temperature is low enough.) The state of the H2O is called supercritical. Supercritical phases are important in some industrial and scientific processes. In particular, there is a technique called... 

Single-component systems are useful for illustrating some of the concepts of equilibrium. Using the concept that the chemical potential of two phases of the same component must be the same if they are to be in equilibrium in the same system, we were able to use thermodynamics to determine first the Clapeyron and then the Clausius-Clapeyron equation. Plots of the pressure and temperature conditions for phase equilibria are the most common form of phase diagram. We use the Gibbs phase rule to determine how many conditions we need to know in order to specify the exact state of our system. 

This result is the Clausius-Clapeyron equation and describes the shapes of the phase boundary lines on a pressure temperature (PT) phase diagram. In particular, this expression is useful for calculating the vapor pressures of different materials (for either evaporation or sublimation). 

The pressure/temperature phase diagram for water shows that increasing pressure can melt ice. A commonly stated example of this effect is that the pressure on the blade of an ice skate melts a layer of ice beneath the skate, allowing the skater to glide smoothly. Do you think this idea is true Use the Clausius-Clapeyron equation to investigate. What other factors could contribute to smooth skating ... 

Clausius-Clapeyron equation This equation describes the shape of the phase boundary on the phase diagram. 

The Clausius and Clausius-Clapeyron equations govern the curves in phase diagrams. 

The criteria for equilibria involving solid phases are exactly those given in 7.3.5 for any phase-equilibrium situation phases in equilibrium have the same temperatures, pressures, and fugacities. Moreover, pure-component solid-fluid equilibria obey the Clapeyron equation (8.2.27). This means the latent heat of melting is proportional to the slope of the melting curve on a PT diagram and the latent heat of sublimation is proportional to the slope of the sublimation curve. In the case of solid-gas equilibria, the Clausius-Clapeyron equation (8.2.30) often provides a reliable relation between temperature and sublimation pressures, analogous to that for vapor-liquid equilibria. 

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). 

If the system consists of two phases - water and water vapor, for example (i.e., boiling water) - F = 3 —2 = 1. Only one intensive variable can be freely selected in the case of equilibrium between two phases. If temperature is thus selected, the pressure (i.e., the vapor pressure at that selected temperature) is fixed. The connection between the two variables is a curve in the p-T diagram (curve 3 in Figure 3.6). The slope of the equilibrium curve between pressure and temperature is defined in the Clausius-Clapeyron equation ... 

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