Clapeyron equation and

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

This equation is called the Clapeyron equation and can be applied to any two phases in equilibrium, e.g., solid and liquid, liquid and vapor, solid and vapor or two crystalline forms of the same solid. Thus for the equilibrium... 

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... 

We need to understand that the Clausius-Clapeyron equation is really just a special case of the Clapeyron equation, and relates to phase changes in which one of the phases is a gas. 

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.

Bosworth and Rideal (98) also measured the rate of change of the C.P.D. of an oxygenated W filament which was maintained at temperatures varying from 1270° to 1930° K. Again the experimental conditions permitted the heat of evaporation to be calculated by the Clapeyron equation and the value recorded at low coverage—about 150 kcal./mole—is in satisfactory agreement with the measured calorimetric heat of adsorption of O2 on W (24). 

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... 

Kistiakowsky (1923) utilized the Clapeyron equation and the ideal gas law to derive an expression to estimate each individual compound s in which the... 

In accordance with the Clapeyron equation and Le Chatelier s principle, the more highly ordered (low-entropy) phases tend to lie further to the left (at lower 7), whereas the higher-density phases tend to lie further upward (at higher 7). The mnemonic (7.32) allows us to anticipate the relative densities of adjacent phases. From the slope, for example, of the ice II-ice III coexistence line (which tilts forward to cover ice III), we can expect that ice II is denser than ice III (pn > pm). Similarly, from the forward slopes of the liquid coexistence lines with the high-pressure ices II, V, and VI, we can expect that cubes of ice II, ice V, and ice VI would all sink in a glass of water, whereas ice I floats (in accord with the backward tilt of its phase boundary). Many such inferences can be drawn from the slopes of the various phase boundaries in Fig. 7.3, all consistent with the measured phase densities Pphase (in gL 1), namely,... 

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... 

This equation is the well-known Clapeyron equation, and expresses the pressure of the two-phase equilibrium system as a function of the temperature. Alternatively, we could obtain dp/dT or dp/dP. When three phases are present, solution of the three equations gives the result that dT, dP, and dp are all zero. Thus, we find that the temperature, pressure, and chemical potential are all fixed at a triple point of a one-component system. 

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. 

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

Problem 6 Give the thermodynamic derivation of Clapeyron equation and Clausius-Clapeyron equation. Discuss their applications also. 

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. 

By applying the Clapeyron equation and Trouton s rule to an ideal binary mixture, Rose derived the relation... 

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. 

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