Phase Clausius-Clapeyron equation

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

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 ... 

The Clausius-Clapeyron equation implies that if we plot the natural log of the pressure of the gas phase versus inverse temperature, the slope of the resulting line is the heat of vaporization divided by the gas constant (R). A plot of In P (vapor pressure of water) versus inverse temperature is given in Figure 3. The calculated heat of vaporization (determined by multiplying the slope by R) is 10,400 cal/mol. The important aspect of Eq. (10) with regard to moisture sorption is the fact that increasing the temperature also increases the vapor pressure. 

It is possible, however, to simplify the calculation of the energy transfer by assuming that the vapor phase is always a saturated vapor. O Connor (Ol) has shown that the rate of approach of a superheated vapor to saturated conditions is extremely rapid when the superheated vapor is in direct contact with its liquid phase. If the vapor phase is assumed to be saturated, the temperature of the phase can be calculated from an integrated form of the Clausius-Clapeyron equation instead of from the vapor-phase energy-transfer equation. 

If the vapor-phase temperature is to be evaluated from the Clausius-Clapeyron equation, the pressure in the two-phase tubular contactor must be known at each axial position. This need once again illustrates the necessity of obtaining an understanding of the hydrodynamics of two-phase systems in order to carry out the design of heat-transfer contactors. 

Equation 3 is analogous to the Clausius-Clapeyron equation for equilibrium of a substance in the vapor and condensed phases (4). 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

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,... 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

Recall 7F o = 1 for a pure fuel condensed phase, and Ff(0) is not known. For a liquid fuel, Ff(0) is found from the Clausius-Clapeyron equation provided we know 7 (0). 

Having qualitatively discussed the way a pressure cooker facilitates rapid cooking, we now turn to a quantitative discussion. The Clapeyron equation, Equation (5.1), would lead us to suppose that dp oc dT, but the liquid-gas phase boundary in Figure 5.12 is clearly curved, implying deviations from the equation. Therefore, we require a new version of the Clapeyron equation, adapted to cope with the large volume change of a gas. To this end, we introduce the Clausius-Clapeyron equation ... 

We need to understand that the Clausius-Clapeyron equation is really just a special case of the Clapeyron equation, and relates to phase changes in which one of the phases is a gas. 

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 ... 

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

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

Clapeyron and Clausius-Clapeyron Equations for Phase Boundaries... 

The vapor pressure, pv, is the pressure exerted by fluids and solids at equilibrium with their own vapor phase. The vapor pressure is a strong function of T, as expressed in the Clausius-Clapeyron equation [1] ... 

In Fig. 4 the experimental isobaric volume temperature curve of the 1-l.c. 4-hexyloxybenzoic acid 4 -hexyloxyphenylester is shown, which possesses a nematic phase 37,38). Two phase transformations are indicated by the jumps of the V—T curve the isotropic to nematic and, at lower temperatures, the nematic to crystalline transformation. As well known, both transformations are of first order and obey the Clausius-Clapeyron equation. 

To check the phase transformation isotropic -> nematic, the validity of the Clausius Clapeyron equation is examined. It has been shown 38), that within the experimental error the results fulfill Eq. 1 in analogy to the low molar mass l.c. The phase transformation isotropic to l.c. is therefore of first order with two coexisting phases at the transformation point. Optical measurements on the polymers confirm these thermodynamical measurements (refer to 2.3.1.3). 

Table 4.10 shows the literature values for hydrate numbers, all obtained using de Forcrand s method of enthalpy differences around the ice point. However, Handa s values for the enthalpy differences were determined calorimetrically, while the other values listed were determined using phase equilibrium data and the Clausius-Clapeyron equation. The agreement appears to be very good for simple hydrates. Note also that hydrate filling is strongly dependent on... 

The Clausius-Clapeyron equation" is an integrated version of the Clapeyron equation that applies to equilibrium between an ideal gas vapor phase and a condensed phase, with the conditions that the volume of the... 

The fundamental relationship that allows the determination of the equilibrium vapor pressure, P, of a pure condensed phase as a function of temperature is the Clausius-Clapeyron equation... 

Gas and condensed phase equilibrium the Clausius-Clapeyron equation... 

If the gas phase activity of the host is controlled by the presence of a pure condensed phase, solid or liquid, the equilibrium between host and guest in a stoichiometric clathrate can be described in terms of the gas phase pressure of the guest. This is, in effect, a vapor pressure for the guest. At higher pressures the guest will condense to form clathrate, and at lower pressures the clathrate will decompose. Temperature variation of this pressure will follow the Clapeyron equation which, with the usual assumptions (ideal gas behavior of the vapor and negligible volume of the condensed phase), reduces to the Clausius-Clapeyron equation ... 

---