Heat Clausius-Clapeyron equation

Enthalpy of Vaporization The enthalpy (heat) of vaporization AHv is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liqmd of a pure component i.e., at a temperature (below the critical temperature) anci corresponding vapor pressure. AHy is related to vapor pressure by the thermodynamically exact 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 ... 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

As already mentioned, the system ofEqs. (8.1-8.5) is supplemented by the Clausius-Clapeyron equation, as well as by the correlation that determines the dependence of enthalpy on temperature and describes the thermohydrodynamical characteristics of flow in a heated capillary. It is advantageous to analyze parameters of such flow to transform the system of governing equations to the form that is convenient for significant simplification of the problem. 

This latent heat of evaporation, Le, also appears in the fundamental description of the dependence of the vapor pressure of water, p, on temperature, T - the Clausius-Clapeyron equation ... 

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. 

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. 

Use the Clausius-Clapeyron equation to solve for the molar heat of vaporization of isopropyl alcohol, A7/vap. 

Any one of Equations (8.14), (8.15), or (8.16) is known as the Clausius-Clapeyron equation and can be used either to obtain AH from known values of the vapor pressure as a function of temperature or to predict vapor pressures of a hquid (or a solid) when the heat of vaporization (or sublimation) and one vapor pressure are known. The same equations also represent the variation in the boiling point of a liquid with changing pressure. 

What would be the form of the integrated Clausius-Clapeyron equation if the heat capacity of the vapor were given by the equation... 

Heats of adsorption are usually determined in two ways either by direct calorimetric determination at a chosen temperature, or by calculating the isosteric heats from adsorption isotherms measured at different temperatures and using the Clausius-Clapeyron equation. Thus, isosteric heats of adsorption are calculated from the... 

In chemisorption where severe surface perturbations can occur, the Clausius-Clapeyron equation cannot be applied, since equilibrium pressures are low and often unobtainable. Nonetheless, a differential heat analogous to the isosteric heat can be obtained from heats of immersion without recourse to pressure data where the amounts adsorbed prior to immersion can be measured gravimetrically (Sec. VII,A). 

The Clausius-Clapeyron equation is an exact thermodynamic relationship between the slope of the vapor pressure curve and the molal heat of vaporization ... 

The Clausius-Clapeyron equation makes it possible to calculate the heat of vaporization of a liquid by measuring its vapor pressure at several temperatures and then plotting the results to obtain the slope of the line. Alternatively, once the heat of vaporization and the vapor pressure at one temperature are known, the vapor pressure of the liquid at any other temperature can be calculated, as shown in Worked Example 10.5. 

The heat of vaporization, AHvap, of a liquid can be obtained either graphically from the slope of a plot of In Pvap versus 1 /T, or algebraically from the Clausius-Clapeyron equation. As derived in Worked Example 10.5,... 

Accuracy of the Clausius-Clapeyron Equation for Hydrate Heat of Dissociation to Vapor and Water... 

The most recent confirmation of the validity of the Clausius-Clapeyron equation for hydrates was by Handa (1986a,b), who measured the heat of dissociation (via calorimetry) of the normal paraffins that form simple hydrates. Table 4.8 shows Handa s values for hydrate dissociation enthalpy compared to those calculated with the Clausius-Clapeyron equation by Sloan and Fleyfel (1992). The agreement appears to be very good for simple hydrates. 

To a fair engineering approximation AH is not only a function of the hydrogen bonds in the crystal, but also a function of cavity occupation. Because the Clausius-Clapeyron equation determines the heat of hydrate formation by the slopes of plots of In P versus 1/T, one may easily determine relationships between heats of dissociation. 

Isosteric heats were obtained using the fitted curves with the aid of the Clausius-Clapeyron equation ... 

The integration of Equation (11.22) to determine the equilibrium constant as a function of the temperature or to determine its value at one temperature with the knowledge of its value at another temperature is very similar to the integration of the Clausius-Clapeyron equation as discussed in Section 10.2. The quantity AHB must be known as a function of the temperature. This in turn may be determined from the change in the heat capacity for the change of state represented by the balanced chemical equation with the condition that all substances involved are in their standard states. 

The vapor pressure of water is 634mmHg at 95°C and 1074mmHg at 110°C. Estimate the standard heat of vaporization of water using the Clapeyron equation and the Clausius-Clapeyron equation. 

Equation (9) is sometimes known as Clausius-Clapeyron equation and is generally spoken to as first latent heat equation. It was first derived by Clausius (1850) on the thermodynamic basis of Clapeyron equation. 

The heat duties associated with these fractions can be estimated by the Clausius-Clapeyron equation (20)... 

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... 

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