Melting pressure curve

The solid-liquid phase equilibrium of a pure substance is described in the state diagram (see Fig. 1-15) by the melting pressure curve p T). This curve is formed by the connection of state points, where the liquid and solid phase coexist at phase equilibrium. For crystallization processes which start from the melting point, a knowledge of this curve is important. 

The melting pressure curve may be given based on the general form of the Clausius-Clapeyron equation in a similar manner to the vapor pressure and sublimation curves (Eq. (1-71)) 

Depending on the sign, two cases for the difference of A P have to be distinguished for the course of the melting pressure curve  

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The almost vertical curve in the illustration (Fig. 11.10), called the melting (pressure) curve, separates the regions at which solid or liquid condensate is stable. The slope of this curve has been greatly exaggerated for the sake of clarity. 

The point at which the sublimatiOTi, boiling, and melting pressure curves all converge is called the triple point because at the conditions of temperature and pressure there, the substance is simultaneously in a solid, liquid, and gaseous state. At the triple point both pressure and temperature are characteristic properties of a pure substance. The triple point of water, for example, is at 273.16 K and 611 Pa. Only at this exact temperature and this exact pressure are ice, liquid water, and water vapor in equilibrium with each other. This triple point is used to define the unit called Kelvin (compare Sect. 3.8). If the pressure at the triple point lies noticeably above 100 kPa, the liquid state cannot exist no matter what the temperature, and only sublimation can be observed. An example of this is carbon dioxide (217 K, 518 kPa), which, when exposed to air, changes directly from solid to gas (so it is called dry ice ). 

A linear approach for both the temperature and pressure dependency of the chemical potential is sufficient for calculating the melting pressure curve (compare Sect. 5.4). For the process... 

For some substances the sign of AF changes with increasing pressure, (for example with rubidium, caesium and graphite) the sign changes from plus to minus. A maximum temperature for the melting pressure curve then is when A F = 0 or... 

Different approximations for the melting pressure curve are discussed in [1.88]. 

At very high pressures, the pressure dependence of the melting point of water is extremely complicated, as a lot of different crystal structures occur (19]. The lowest melting point observed is i , = —23 at 2100 bar, whereas very high melting points like = 175 C at 36500 bar are also possible. Correlations for the melting pressure curves of the different modifications of ice are given in (20j. 

The p T) phase diagram of carbon dioxide is presented in Fig. la [21]. On the vapor pressure curve (Ig) a liquid (1) and a gaseous (g) phase coexist. With increasing pressure the vapor pressure curve rises and ends at the critical point CP ((304.1282 0.015) K, (7.3773 0.0030) MPa, (0.4676 0.0006) g cm [22]), where the two coexisting phases (1 and g) become identical. With decreasing pressures it ends at the triple point Tr (216.58 K, 0.5185 MPa [22]), where solid, liquid, and gaseous carbon dioxide are in equilibrium. The sublimation pressure curve (gs) and the melting pressure curve (Is) are not important within the scope of the present chapter. 

The reason for the constancy and sharpness of the melting j)oint of a pure crystalline solid can be appreciated upon reference to Fig. 7,10, 1, in which (a) is the vapour pressure curve of the solid and (6) that of the liquid form of the substance. Let us imagine a vessel, maintained at constant temperature, completely filled with a mixture of the above liquid and solid. The molecules of the solid can only pass into the liquid and the molecules of the liquid only into the solid. We may visualise two competitive processes taking place (i) the solid attempting to evaporate but it can only pass into the liquid, and (ii) the liquid attempting to distil but it can only pass into the solid. If process (i) is faster, the solid will melt, whereas if process (ii) proceeds with greater speed the... 

It is a well-known fact that substances like water and acetic acid can be cooled below the freezing point in this condition they are said to be supercooled (compare supersaturated solution). Such supercooled substances have vapour pressures which change in a normal manner with temperature the vapour pressure curve is represented by the dotted line ML —a continuation of ML. The curve ML lies above the vapour pressure curve of the solid and it is apparent that the vapour pressure of the supersaturated liquid is greater than that of the solid. The supercooled liquid is in a condition of metastabUity. As soon as crystallisation sets in, the temperature rises to the true freezing or melting point. It will be observed that no dotted continuation of the vapour pressure curve of the solid is shown this would mean a suspended transformation in the change from the solid to the liquid state. Such a change has not been observed nor is it theoretically possible. 

The horizontal isopiestic cuts the vapour-pressure curve of the solid in the first case, that of the liquid in the second. Melting can be brought about in case (1) by an increased pressure. 

The melting-point (T,p) curve (unlike a vapour-pressure curve of a liquid) does not end abruptly at a critical point (Ar = 0, L = 0) it is an endless curve, probably forming a closed loop ABCD, unless it intersects some other curve or the axes of co-ordinates. At high pressures it bends round towards the p axis, and according to Tammann, takes the shape indicated by the following considerations. It is known from experiment that (for substances of the wax-type) the melting-point increases with rise of pressure,... 

Kirchhoffs investigation does not show that the sublimation and evaporation curves meet each other at the temperature at which solid and liquid are in equilibrium with vapour it proves that they are inclined at an angle, but the further fact that they intersect requires separate proof, which was inferred by James Thomson, and experimentally demonstrated by Ferche (1891) in the case of benzene the point of intersection, calculated from the vapour-pressure curves, was 5 405° C, whereas the melting-point was 5 42° C. 

Even if nowadays, the MCT may be considered a primary thermometer only on a narrow temperature range, it is considered the best dissemination standard in the millikelvin range [56-59], In fact, the 3He melting pressure is a good thermometric property because of its sensitivity over three decades of temperature with a resolution A T/T up to 10 5 [56], The good repeatability, the insensitivity to magnetic fields up to 0.5 T [60] and the presence of temperature-fixed points allow for the control of possible shifts in the calibration curve of the pressure transducer. The usefulness of these fixed points is evident, considering that the ITS-90 is based just on the definition of fixed points. 

Pure CIF3O2 is colorless as a gas or liquid and white as a solid. Some of its measured (68) physical properties are summarized in Table XX. Near its melting point the vapor pressure above liquid CIF3O2 was found to be reproducibly lower than expected from the vapor pressure curve given in Table XX. This indicates that close to the melting point some ordering effect occurs in the liquid. 

In a very similar way as discussed above for estimating pi from boiling point data, one can treat the vapor pressure curve below the melting point. Again we use the Clausius-Clapeyron equation ... 

The condition for valid results in this work is the fact that the system is in the gel-state that is, it is in a state of inner equilibrium, and not in the glass state. Measurements of Hellwege, Knappe, and Lehmann (5) on pure PVC show the expected discontinuity in the compressibility vs. pressure curve at the melting point transition in the glass state. It is inferred from this study that the glass temperature for pure PVC and for the plasticized, gel-type PVC does not rise above 110° C. at pressures of 200 atm. (Figure 7). Our study showed that the system is always in a state of inner equilibrium. 

By extrapolation of the vapour pressure curves (fig. 3) Horiba showed the melting point to be 817° to 818° C. at the corresponding pressure of 35-8 atm. Johnston 1 deduced the following boiling temperatures at low pressures—... 

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