Clausius-Clapeyron equation and

Since 7 , is not an experimentally measurable quantity, it is useful to insert the solution for Ts (from the Clausius-Clapeyron equation) and solve for W h as an explicit function of RH0 and RHC. VanCampen et al. showed (using sample algebraic approximations and conversion factors) that substituting for Ts in Eq. (35) gives the useful solution... 

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... 

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. 

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.

Clausius-Clapeyron equation, and to obtain a theoretical expression for a vapor bubble growing in a superheated liquid. The equation (F5, F6) is a second-order differential equation which is so complex as to be of limited usefulness without serious modification. Fortunately, the equation becomes enormously simpler if the inertia of the liquid can be ignored during bubble growth. Forster and Zuber give a careful discussion of the physical requirements for neglecting inertia of the liquid. These are that either the bubble must be very small or the temperature of the bubble... 

Solution The enthalpy of vaporization (33.05 kj-mol1) corresponds to (3.305 X 104 J-moE1) 25°C corresponds to T2 = 298 K, and 57.8°C corresponds to T, = 332.0 K. We substitute these values into the Clausius-Clapeyron equation and obtain... 

Substituting this formula into the Clausius-Clapeyron equation and integrating by parts... 

Discuss similarities and differences between Eq. (77) and the Clausius-Clapeyron equation and the phenomena underlying each of these. 

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. 

Estimate the vapor pressure of a pure substance at a specified temperature or the boiling point at a specified pressure using (a) the Antoine equation, (b) the Cox chart, (c) the Clausius-Clapeyron equation and known vapor pressures at two specified temperatures, or... 

Calculate the latent heat of vaporization and the parameter B in the Clausius-Clapeyron equation and then estimate p at 42.2°C using this equation. 

To show you how to use the Clausius-Clapeyron equation, and to show you how well the equation fits over small temperature ranges. The calculated boiling point pressure for isobutyl alcohol (760,04 torr) is not very different from the normal boiling point pressure of 760.00 torr (0.005%). 

In this paper, recent work on the viscosity of mold flux compositions is reviewed, and a relation to describe the temperature dependence of viscosity is discussed. This relation is based on the Clausius-Clapeyron Equation and was originally developed by Kirchoff and Rankine to describe the temperature dependence of vapor pressure. 

Many families of compounds are characterized by the fact that their vapor pressures may be approximated by the Clausius-Clapeyron equation, and by the fact that their latent heats of vaporization are approximately equal. The logarithm of the vapor pressures of the members of such families of compounds fall on parallel lines when plotted against the reciprocal of the absolute temperature. For any two members / and b of such a mixture, it is readily shown that a, is independent of temperature. 

Many procedures have been developed to predict vapor pressure, and the predictive error of eleven different methods have been evaluated using a series of PCBs. A number of approaches use a set of known vapor pressures to develop a correlation with molecular properties that can be used for predictions of unknowns. For example, the free energy of vaporization has been correlated with molecular surface area to predict vapor pressures of PCB congeners. More direct approaches are based on the Clausius-Clapeyron equation, and vapor pressures can be predicted quite effectively for some series of compounds using only boiling points along with melting points for compounds that are sohds at ambient temperatures. 

Values of the enthalpy of adsorption, determined either from the variation of adsorption with temperature (isosteric enthalpy of adsorption) or by direct calorimetric measurements, provide a valuable insight into the mechanism of adsorption. When taken together wifri the data from adsorption isotherms, they provide information which could not be extracted from either set of data alone. Heats of adsorption and other thermodynamic parameters can be obtmned either by direct calorimetric determination, -AH =j n (where ria = adsorbed amount), or by using the Clausius-Clapeyron equation and the data from isosteric measurements. However, the faacX tiiat adsorption is often irreversible in the presence of micropores, fr equently makes estimates of adsorption heats obtained from isosteres very unreliable. 

The freezing point depression follows from the lowering of vapor pressure. From the Clausius-Clapeyron equation and Raoulfs law, it follows enthalpy of... 

The value of Q ff is thus defined for a particular coverage Fi. The r.h.s. of Eq. (19) is artalogous to the Clausius-Clapeyron equation and contains a directive for the experimentahst to measure the differential heat of adsorption as a function of coverage [69Traj. The isosteric heat of adsorption is defined very similarly and differs from the differential heat only by kT (about 50 meV at 600 K), = Qjjg. + kT. 

The vapor pressure equation for the beta phase was obtained by evaluating the free energy functions of the solid and the gas at 10 K intervals from 1000 to 1130 K and the transition temperature for the gamma phase, the extreme values were fitted to the Clausius-Clapeyron equation and for the liquid phase, at 50 K intervals from 1200 to 3700 K and the melting point (Table 27). 

Since the first two series were only given in the form of the Clausius-Clapeyron equation and the third series were given in complete detail then all were evaluated separately. 

Estimate the enthalpy of vaporization at = 5 C using the Clausius-Clapeyron equation and interpolate the vapor pressure at = 5 C with the Hoffmann-Florin equation. 

In Chapter 6, we introduced some important concepts that we can apply to systems at equilibrium. The Clapeyron equation, the Clausius-Clapeyron equation, and the Gibbs phase rule are tools that are used to understand the establishment and changes of systems at equilibrium. However, so far we have considered only systems that have a single chemical component. This is very limiting, because most chemical systems of interest have more than one chemical component. They are multiple-component systems. 

Trens et al. [11], have correlated the intersection of the desorption branch with the adsorption branch at the low pressure (referred to as the reversible pore filling or rpf ) with thermodynamic properties. Specifically, it seems to follow the Clausius-Clapeyron equation and follows that relationship expected from corresponding states relationship. This indicates that the rpf is characteristic of a first-order gas-liquid transition. The enthalpy of this transition is somewhat higher than the hquid-gas transition in llie bulk, which should not be surprising since the interaction of the solid with the adsorbate should supply an extra energy. 

From these equations one can then estimate the molar adsorption potential, eq> its distribution function. The isosteric differential heat of adsorption, q- o, can be derived from the Clausius-Clapeyron equation and the modified Freundlich equation ... 

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