Clapeyron-Clausius Law

By increasing the temperature, the vapor pressure of the reaction mass may increase. The resulting pressure can be estimated by the Clausius-Clapeyron law, which links the pressure to the temperature and the latent enthalpy of evaporation A Hv ... 

By imposing thermodynamic equilibrium (with the Clausius-Clapeyron law whose validity holds far away from the critical point) at the droplet surface, the conservation of mass and enthalpy (considering that changes in the droplet temperature must be related to the latent heat of evaporation) yields the following expressions ... 

During the vaporization process, it is striking that the enthalpy increases considerably, sometimes much more an the Clausius-Clapeyron law leads us to expect. Vaporization will naturally become more difficult when the pressure is increased, but there is no explanation for an increase by a factor of 5 or 6. Therefore data of samples 14, 24, and 25 seem to be anomalous. 

A distinct approach for describing the evaporation of volatile thin films can also be employed for calculating the evaporation flux of a thin droplet (or a thin disc) [33]. The difference between thin film and thin droplet is that the height (i.e., thickness) of a droplet changes with the distance to the center, while the thickness of thin film is uniform. In this approach, the influence of the gas phase on the evaporative flux is neglected. Thus, the liquid phase and vapor phase are decoupled in the calculation. Such a model is referred to as a nonequilibrium one-sided (NEOS) model while the abovementioned model developed by Deegan is referred to as the lens model. Based on the Clausius-Clapeyron law [34], which is used to relate the temperature and the pressure, the boundary condition at the liquid-gas interface is given as [19]... 

Fundamental Property Relation. The fundamental property relation, which embodies the first and second laws of thermodynamics, can be expressed as a semiempifical equation containing physical parameters and one or more constants of integration. AH of these may be adjusted to fit experimental data. The Clausius-Clapeyron equation is an example of this type of relation (1—3). 

Nomograph defined. This method assumes the application of the Clausius-Clapeyron equation, Henry s law, and... 

The introduction of the perfect gas law to the Clausius-Clapeyron equation (Equation (6.14)) allows us to obtain a more direct approximation to p p(T) in the saturation region. We use the following ... 

There is another important law that follows from the classical theory of capillarity. This law was formulated by J. Thomson [16], and was based on a Clausius-Clapeyron equation and Gibbs theory, formulating the dependence of the melting point of solids on their size. The first known analytical equation by Rie [17], and Batchelor and Foster [18] (cited according to Refs. [19,20]) is... 

Raoult s law works for small polymers as well as small molecules. Determination of M is based for both ebulliometry (boiling point elevation) and cryometry (freezing point lowering) on the Clausius-Clapeyron equation ... 

Critical Tables (7) give values of vapor pressure of 5.0 and 7.5% NaCl solutions over the range of 0° to 110° C. From these data the BPE for a 7.0% solution (50% recovery) at 1 atm. is readily calculated to be 2.34° F. From the ideal solution law (which should apply well to water in dilute solutions) and the Clausius-Clapeyron equation we get... 

Another useful equation is the Clausius-Clapeyron equation. It states that, provided the ideal gas law holds and the enthalpy of vaporization, Aft, is independent of T (which is a reasonable assumption for a small temperature range), the slope of the vapor pressure curve is given by... 

Derive the Clausius-Clapeyron equation [Eq. (44)] from Eq. (40) by neglecting the volume of the condensed phase and using the ideal gas law for the vapor. 

Chapter 6 dealt with the application of vacuum technology in three areas of the chemical sciences. The first was concerned with its use in chemical technology, particularly in purification/separation operations such as distillation and evaporation. For distillation, the use of the Clapeyron and Clausius-Clapeyron equations was demonstrated (Examples 6.1 and 6.2) whilst Raoult s and Henry s laws were stated and applied (Examples 6.3, 6.4). The removal of water (drying) is an important but poorly understood operation. Aspects of this were discussed in Examples 6.5-6.7. Condensers, particularly in conjunction with vacuum pumps, are indispensable in applications such as distillation and drying. Simple treatment of condenser theory was stated and applied in Examples 6.7-6.9. 

If the vapor phase in VLE is ideal and the liquid molar volumes are negligible (assumptions inherent in Raoult s law), then the Clausius/Clapeyron equation applies (see Ex. 6.5) ... 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

The requirement of thermodynamic reversibility also applies to the chromatographic method, but in this case it is necessary to work at very low surface coverage (at zero coverage ) in the Henry s law region. Values of the specific retention volume, Vs, determined at different temperatures are inserted in the Clausius-Clapeyron equation in place of the equilibrium pressures to obtain A h. Provided that a number of conditions are observed, the method is capable of providing a fairly easy and rapid assessment of the adsorbent—adsorbate interaction energy. 

The emphasis on RaOult s law and the liquid state raises difficulties for solutes which are above their critical temperatures and for which A// , p , and F appear to have little meaning. From an empirical point of view it appears that the extrapolation of the Clausius-Clapeyron equation to super-critical temperatures gives usable values of p (see Hildebrand and Scott, loc. ciL, chap. 14). 

Equation (22-19) is useful particularly for pairs of chemically similar liquids. If Raoult s law holds, relative volatility is equal to pjpi- Therefore, it is possible to plot liquid-vapor composition diagrams for closely similar liquids, such as benzene-toluene, without further ado. Note that the value of iJFis not strictly constant over the whole composition range, even for such a mixture, because pi and P2 do not necessarily vary similarly with temperature (Clausius-Clapeyron equation). 

Several points are to be noted, (i) First, the obvious analogy to the Clausius-Clapeyron equation, (ii) According to the Third Law, the entropy of any material must reach a minimum value with zero slope hence, as T 0, d7ic/dT)p 0 as well (cf. Fig. 5.9.1). (iii) Experimentally it is found that (dHc/dT)p <0 therefore, in a magnetic field the superconducting state has a lower entropy than does... 

This rule is very similar to the general law of Wrewsky, as would be expected from the Clausius-Clapeyron equation. The difference between the two rules is that Wrewsky s is based on a knowledge of the latent heats of evaporation of the components from the azeotropic solution, while the present rule is expressed in terms of more readily available quantities, the latent heats of evaporation of the pure liquids. 

Go back to the temperature-mole fraction diagram for the isopropyl alcohol-isobutyl alcohol system (Fig. 175). The composition of the vapor is always different from that of the liquid, and we can separate the two compounds. If the composition of the vapor is the same as that of the liquid, that separation is hopeless. Since we ve used the notions of an ideal gas in deriving our equations for the liquid and vapor compositions (Clausius-Clapeyron, Dalton, and Raoult), this azeotropic behavior is said to result from deviation from ideality, specifically deviations from Raoult s law. Although you might invoke certain interactive forces in explaining nonideal behavior, you cannot predict azeotrope formation a priori. Very similar materials form azeotropes (ethanol-water). Very different materials form azeotropes (toluene-water). And they can be either minimum-boiling azeotropes or maximum-boiling azeotropes. 

Univariant Systems.—Equilibrium between liquid and vapour. Vaporisation curve. Upper limit of vaporisation curve. Theorems of van t Hoff and of Le Chatelier. The Clausius-Clapeyron equation. Presence of complex molecules. Equilibrium between solid and vapour. Sublimation curve. Equilibrium between solid and liquid. Curve of fusion. Equilibrium between solid, liquid, and vapour. The triple point. Complexity of the solid state. Theory of allotropy. Bivariant systems. Changes at the triple point. Polymorphism. Triple point Sj—Sg— V. Transition point. Transition curve. Enantiotropy and monotropy. Enantiotropy combined with monotropy. Suspended transformation. Metastable equilibria. Pressure-temperature relations between stable and metastable forms. Velocity of transformation of metastable systems. Metastability in metals produced by mechanical stress. Law of successive reactions. 

Here A is the ordinary heat of vaporization per mol, U the heat development associated with the condensation without dbing external work p and i are the pressure and concentration of the saturated vapour. The two equations are fundamentally identical they result from the Clausius-Clapeyron equation by introducing the laws of the ideal gaseous state and limiting the application to small pressures. 

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. 

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