Phase transitions Clausius-Clapeyron equation

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

Since surface pressure is a free energy term, the energies and entropies of first-order phase transitions in the monolayer state may be calculated from the temperature dependence of the ir-A curve using the two-dimensional analog of the Clausius-Clapeyron equation (59), where AH is the molar enthalpy change at temperature T and AA is the net change in molar area ... 

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

Aeeording to the Clausius-Clapeyron equation the transition temperature at the solid —> liquid phase boundary of a binary mixture shifts by (5Z) upon a relative eoneentration ehange (dc) between the two eonstituents ... 

This expression reduces to the classical Clausius-Clapeyron equation when the difference in compressibility, thermal expansion and heat capacity vanish as is observed for most phase transitions in lipids [80]. 

The Clausius-Clapeyron equation serves the same purpose, but it is not exact its derivation involves approximations, in particular the assumptions that the perfect gas law holds and that the volume of condensed phases can be neglected in comparison to the volume of the gaseous phase. It applies only to phase transitions between the gaseous state and condensed phases. 

This is the Clausius-Clapeyron equation. From it, the slope (dpjdT) of the phase boundary and the observed volume difference between the two phases, the entropy of the transition and hence its latent heat can be found. These quantities, evaluated at the various triple points, are shown for ordinary water in table 3.1. 

The vapor pressure equation for the alpha phase is derived by evaluating free energy functions for the solid and the gas at 25 K intervals from 1000 to 1750 K and the transition temperature. For the liquid phase, values are evaluated at 50 K intervals from 1800 to 3600 K and the melting point. For the beta phase, values were evaluated at the transition and melting point temperatures and fitted to the Clausius-Clapeyron equation (Table 15). 

In the region of the alpha-beta transition at 270 K, the measurements of Gerstein et al. (1964) show a considerable scatter and no firm evidence of a transition, possibly because the samples consisted of a mixture of both phases and the transition would be smeared out. Based on a measurement of the volume difference between the two phases and the slope of the pressure curve, Bucher et al. (1970) used the Clausius-Clapeyron equation to calculate an... 

If the characteristic time of relaxation tends to zero, the transition occurs instantaneously in a sharply defined front. We denote the velocity of the front by U and the velocity of the mixture by u. The state of the mixture is characterized by pressure p, temperature T and liquid mass fraction fp. The gas and the vapour are considered perfect so they obey Dalton s law. For simplicity we shall from now on refer to the supersaturated vapour state and to the state of liquid-vapour equilibrium as state 1 and state 2, respectively. It is dear there exists no liquid phase in state 1, fp = 0, and the vapour mass fraction f is equal to the total mass fraction of the condensable component of the mixture. The vapour pressure p 2 state 2 has to satisfy the Clausius-Clapeyron equation (2). The laws of conservation of mass, momentum and energy are, applied to the condensation discontinuity ... 

The enthalpy change of a substance associated with the isobaric transition from one phase to another at equilibrium may be described analogously to the Clausius-Clapeyron equation as... 

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). 

The Clausius-Clapeyron equation is very useful in considering gas-phase equilibria. For example, it helps predict equilibrium pressures at differing temperatures. Or it can predict what temperature is necessary to generate a particular pressure. Or pressure/temperature data can be used to determine the change in enthalpy for the phase transition. 

State whether or not the Clausius-Clapeyron equation is strictly applicable to the following phase transitions. 

Reusable hot-packs sometimes use the precipitation of supersaturated sodium acetate or calcium acetate to give off heat of crystallization to warm a person. Can the conditions of this phase transition be understood in terms of the Clapeyron or the Clausius-Clapeyron equation Why or why not ... 

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