Table Clausius-Clapeyron equations

In using the Clausius-Clapeyron equation, the units of AH and R must be consistent. If AH is expressed in joules, then R must be expressed in joules per mole per kehrin. Recall (Table 5.1, page 107) that... 

A From Table 13-1 we know that AHvap = 38.0 kJ / mol for methyl alcohol. We now can use the Clausius-Clapeyron equation to determine the vapor pressure at 25.0° C = 298.2 K. 

Table III shows that the experimental and predicted evaporation rates are in good agreement at all beam intensities. There is some inconsistency at the highest power levels. It was difficult to maintain the droplet in the center of the laser beam at the highest power level, and the measured evaporation rate is somewhat low as a result of that problem. Additional computations demonstrate that the predicted evaporation rate is quite sensitive to the choice of the imaginary component of N, so the results suggest that this evaporation method is suitable for the determination of the complex refractive index of weakly absorbing liquids. For strong absorbers, the linearizations of the Clausius-Clapeyron equation and of the radiation energy loss term in the interfacial boundary condition may not be valid. In this event, a numerical solution of the governing equations is required. The structure of the source function, however, makes this a rather tedious task.

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... 

As indicated by the plots in Figure 10.13a, the vapor pressure of a liquid rises with temperature in a nonlinear way. A linear relationship is found, however, when the logarithm of the vapor pressure, In Pvap, is plotted against the inverse of the Kelvin temperature, 1 /T. Table 10.8 gives the appropriate data for water, and Figure 10.13b shows the plot. As noted in Section 9.2, a linear graph is characteristic of mathematical equations of the form y = mx + b. In the present instance, y = lnPvap, x = 1/T, m is the slope of the line (- AHvap/R), and b is the y-intercept (a constant, C). Thus, the data fit an expression known as the Clausius-Clapeyron equation. ... 

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. 

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... 

Some measurements with the differential thermal analyzer are summarized in Table I. The data are condensed into the two constants of the Clausius-Clapeyron equation ... 

Data were also obtained by this method for the solid states for the methyl ester of 2,4-D, the n-propyl ester of 2,4,5-T, and the butyl ester (liquid) of 2,4-D. The results are shown in Table III. These data were fitted by the least squares method to the Clausius-Clapeyron equations given in footnotes to Table III. These equations were used to estimate the vapor pressures at several temperatures, including the melting point. In Table IV, these are compared with estimates from other sources. Jensen s unpublished data with the Knudsen method compare favorably with those reported in this work, but the published values obtained by other methods are larger. 

The deviations observed between extrapolated estimates from GLC data, and direct measurements with the effusion measurements appear to be too large to be accounted for by extrapolation uncertainties. The best estimate can probably be obtained by fitting the combined data to the Clausius-Clapeyron equation (footnote b of Table IV). The obvious implication is that where possible, extrapolation of pesticide vapor pressures obtained at elevated temperatures be converted to interpolation by including a direct measurement at room temperature. In terms of the work described here, vapor pressure measurements requiring the DTA should be supplemented with Knudsen cell measurements. This would require a temperature at which the vapor pressure was 10 3 mm. or less. 

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... 

After freezing, the time to sublimate the solvent is given by the drying expressions in Tables 8.3 and 8.4, where the enthalpy of vaporization for drying is replaced by the enthalpy of sublimation. The enthalpy of sublimation is often equal to the sum of the heats of fusion and vaporization [16]. The enthalpy of sublimatian is also substituted for the enthalpy of vaporization in the Clausius Clapeyron equation (8.9) required for the calculation of the solvent partial pressure. The same rate determining steps of boundaiy layer mass transfer and heat transfer as well as pore diffusion and porous heat conduction are applicable in sublimation. 

Estimate the vapor pressure of acetone (mm Hg) at 50 C (a) from data in Perry s Chemical Engineers Handbook and the Clausius-Clapeyron equation, (b) from the Cox chart (Figure 6.1-4), and (c) from the Antoine equation using parameters from Table R4. 

If you plot the temperature and vapor pressure data given in Table 1, you reconstruct the liquid-vapor equilibrium line in the phase diagram of that liquid (Fig. 174). The equation of this line, and you might remember this from your freshman chemistry course, is the Clausius-Clapeyron equation ... 

The P° s of Dalton-Raoult are vapor pressures taken at fixed temperatures. They are the p s in the Clausius-Clapeyron equation found with a variation in temperature. You don t beheve me Pick a vapor pressure and temperature pair from Table 1 for either liquid, and let these bep and T° (and don t forget to use K, not °C). Now what happens when the unknown temperature (T) is the same as T° The (UT-VT° becomes zero, the entire exponent becomes zero, p is multiplied by 1 (anything to the power zero is 1, eh ), and so on p=p°. 

The isosteric heats of adsorption have been calculated from isotherms by the use of Clausius-Clapeyron equation. The detailed results 5) show that in all the cases measured physical adsorption is taking place. In this paper the heats given in Table I correspond to half-surface coverage. 

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. 

In this equation, R is the ideal gas constant, 6o is a constant that varies from one substance to another, and Ai/vap is the heat of vaporization of the substance of interest, assumed to be constant over the temperature range studied. In an experiment to determine the heat of vaporization of carbon tetrachloride, nine runs were performed. Their results are reproduced in Table 5A.5 (Simoni, 1998). If the Clausius-Clapeyron equation holds under these conditions. In p p can be described by a linear function of 1/T, from whose slope we can determine the heat of vaporization of carbon tetrachloride. 

Comprehensive tables of vapor-pressure data of common liquids, such as water, common refrigerants, and others, may be found in Refs. [2,3]. For most liquids, the vapor-pressure data are obtained at a few discrete temperatures, and it might frequently be necessary to interpolate between or extrapolate beyond these measurement points. At a constant pressure, the Clausius-Clapeyron equation relates the slope of the vapor pressure-temperature curve to the latent heat of vaporization through the relation... 

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). 

No validated QSARs are available to predict directly from chemical structure, but there are several methods for calculating p based on derivations of the Clausius-Clapeyron equation (Table 4.4). 

The enthalpies of vaporization can be evaluated with the Clausius-Clapeyron equation (see Section 2.5.3), using the vapor pressure equations given in Table 13.4. The vapor phase is treated as an ideal gas, this means the vapor phase reality is neglected. 

In some cases, in order to apply the theory, it is necessary to extrapolate the vapour pressure of the liquefied gas beyond the critical point. For example, suppose that it is desired to estimate the ideal solubility of methane at a temperature of 25 C, which is far above critical. If the observed vapour pressures are extrapolated by means of the Clausius-Clapeyron equation, the estimated value of p at 25 is found to be 289 atm— but of course this does not correspond to a stable state of gas-liquid ecjuUibrium. The ideal solubility of methane at 25 is therefore 1/289=0.0035. Some of the observed solubilities, as quoted by Hildebrand and Scott, are given in the table. 

When we apply Eq. 12.1.14 to the vaporization process A(l) A(g) of pure A, it resembles the Clausius-Clapeyron equation for the same process (Eq. 8.4.15 on page 219). These equations are not exactly equivalent, however, as the comparison in Table 12.1 shows. 

Table 12.7 Pressure compensation factors from Clausius-Clapeyron Equation...

Theoretically, the Rombusch equation can be used to calculate the thermodynamic properties in the liquid phase. But this equation is used in Ref. [0.46] and then in [0.11] only to calculate the tables for the values v, h, s of superheated and saturated vapors of Freon-21 at temperatures from 213 to 473 K and pressures up to 5 MPa. To determine the thermodynamic properties of the saturated liquids in Ref. [0.46], as is the case in the majority of other works, an autonomous system of equations including the Clausius-Clapeyron equation... 

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