Volume Clausius-Clapeyron equation

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... 

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 ... 

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... 

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. 

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... 

For pressures below one atmosphere, a number of simplifications can be made. The volume change on formation of vapor can be approximated reasonably by the volume of the vapor. The vapor is assumed to behave like an ideal gas. In particular, AV = VM = RT/ P. Substituting this value for AV into Equation (1) yields 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 ... 

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. 

Equation (9) is valid for evaporation and sublimation processes, but not valid for transitions between solids or for the melting of solids. Clausius-Clapeyron equation is an approximate equation because the volume of the liquid has been neglected and ideal behaviour of the vapour is also taken into account. 

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. 

If we use the ideal gas equation to approximate the volume of vapor as Fvap = RT/P, we obtain the Clausius-Clapeyron equation... 

Calculate the number n of moles of HCl in the solution dispensed. Give S and 5 for the initial and final volumes, and give a limit of error (95 percent confidence) for n. The heat of vaporization of a liquid may be obtained from the approximate integrated form of the Clausius-Clapeyron equation. 

As in the development of tlie Clausius/Clapeyron equation (Example 6.5), if for low pressures one assumes tliat tlie gas pliase is ideal and tliat tlie adsorbate is of negligible volume... 

This expression is sometimes referred to as the Clausius-Clapeyron equation, for it was first derived by R. Clausius (1850), in the course of a comprehensive discussion of the Clapeyron equation. Although the Clausius-Clapeyron equation is approximate, for it neglects the volume of the liquid and supposes ideal behavior of the vapor, it has the advantage of great simplicity. In the calculation of dp/dT (or dT/dP) from a knowledge of the heat of vaporization, or vice versa, it is not necessary to use the volumes of the liquid and vapor, as is the case in connection with equations (27.9) and (27.10). However, as may be expected, the results are less accurate than those derived from the latter expressions. 

T and pressure P. It should be noted that equation (33.26) is the exact form of the Clausius-Clapeyron equation (27.12). If the vapor is assumed to be leal, so that the fugacity may be replaced by the vapor pressure, and the total pressuic is taken as equal to the equilibrium pressure, the two equations become identical. In this simplification the assumption is made that the activity of the liquid or solid does not vary with pressure this is exactly equivalent to the approximation used in deriving the Clausius-Clapeyron equation, that the volume of the liquid or solid is negligible. 

It is well known that the volume of a vapour, V(g), is much larger than that of the equivalent amount of a liquid. As a result, it is reasonable that the V(g) of vapour volume represents the volume change of AVm. Further, the vapour can be considered a perfect gas. Based on the state equation of perfect gases for a mole gas, the Clausius-Clapeyron equation can be rewritten as... 

The Clausius-Clapeyron equation relates pressure with temperature, enthalpy, and volume, and has been used to develop semi-theoretical expressions of vapor pressure ( ). Many properties, including viscosity, can be related to an energy barrier, free volume and temperature. The attempt here is to express viscosity in the form of the Clausius-Clapeyron equation. 

Salmeterol xinafoate is known to exist in two polymorphic forms, forms I and II. Form I is stable, and form II is the metastable polymorph at ambient temperature. The enthalpies of solution (AHso ) of forms I and II determined from van t Hoff solubility-temperature plots are 32.1 and 27.6 kJ/mol, respectively, and the transition temperature obtained by linear extrapolation of the van t Hoff plots is 99 °C. The enthalpy of polymorphic conversion (AHii i) calculated from the plots of log solubility ratio of polymorphs versus the reciprocal of absolute temperature is negative (—4.55 kJ/mol) (5). However, the change in molar volume (AFu i) due to the conversion is positive. Therefore, according to the Clausius-Clapeyron equation,... 

To integrate the equation at low pressure, we assume the vapor volume to be given by the ideal-gas equation the liquid volume to be negligible and the heat of vaporization to be unchanging with T. We obtain from Equation (4.453) the Clausius-Clapeyron equation. 

In this equation, known as the Clausius-Clapeyron equation, represents the heat absorbed, per gram, in the transformation of one phase into the other, and Vi are the specific volumes of the two phases, and T is the absolute temperature at which the change occurs. The above equation enables one to calculate only the slope of the curve at a given point, not the actual values of the pressure. It is possible, however, to derive an expression by means of which the individual points on the vapour-pressure curve can be calculated approximately. 

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. 

Clausius-Clapeyron equation - An approximation to the Clapeyron equation applicable to liquid-gas and solid-gas equilibrium, in which one assumes an ideal gas with volume much greater than the condensed phase volume. For the liquid-gas case, it takes the form d(lnp)/dT = A HIRV- where R is the molar gas constant and A H is the molar enthalpy of vaporization. For the solid-gas case, A H is replaced by the molar enthalpy of sublimation, A H. 

Clausius-Clapeyron Equation. Derived from equation 1, the Clapeyron equation is a fundamental relationship between the latent heat accompanying a phase change and pressure—volume—temperature (PVT) data for the system (1) ... 

The transition point, like the melting point, is affected by pressure. Depending on the relative values of the specific volumes of the two polymorphs, an increase in pressure can either raise or lower the transition temperature. However, since this difference in specific volumes is ordinarily very small, the Clausius—Clapeyron equation predicts that the magnitude of dT/dP will not be great. 

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. 

Equation (6.61) is the starting point for the Clausius - Clapeyron equation. The equation states that if in the limit the entropies for both phases become equal, also the volumes of the phases must become equal, or the freedom to change the pressure will be lost, dp = 0. Otherwise, the equation does not hold longer. In the critical point, the entropies for both phases become equal. Therefore, also the volumes become equal. Of course, this statement holds also vice versa. 

The ideal gas law characterizes the relationship between pressure, temperature, and volume for gases. Both the Clausius-Clapeyron and Antoine equations characterize the vapor-liquid equilibrium of pure components and mixtures. At atmospheric pressure and ambient temperature, water is a liquid but an equilibrium exists with its vapor phase concentration—its vapor pressure. The vapor pressure is a function of temperature. The formula for the Clausius-Clapeyron equation is ... 

The heat of sorption is the difference in specific heat content or enthalpy between the bound moisture and that freely attached at the same temperature and total pressure. This enthalpy difference is normally derived from a form of the Clausius-Clapeyron equation on the assumption that the moisture vapor phase acts like an ideal gas and the molal volume of the condensed phase is negligible compared with that of the vapor. These considerations lead to the expression... 

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