Clausius-Clapeyron equation, application

Nomograph defined. This method assumes the application of the Clausius-Clapeyron equation, Henry s law, and... 

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

Chapter 6 dealt with the application of vacuum technology in three areas of the chemical sciences. The first was concerned with its use in chemical technology, particularly in purification/separation operations such as distillation and evaporation. For distillation, the use of the Clapeyron and Clausius-Clapeyron equations was demonstrated (Examples 6.1 and 6.2) whilst Raoult s and Henry s laws were stated and applied (Examples 6.3, 6.4). The removal of water (drying) is an important but poorly understood operation. Aspects of this were discussed in Examples 6.5-6.7. Condensers, particularly in conjunction with vacuum pumps, are indispensable in applications such as distillation and drying. Simple treatment of condenser theory was stated and applied in Examples 6.7-6.9. 

The techniques described in the foregoing do not easily provide information about the chemisorption bond strength. The Clausius-Clapeyron equation is not applicable in the range of irreversible adsorption. Only by measurements of the desorption rates during the thermal desorption processes at two slightly different temperatures can the activation energy of desorption be estimated. This method has been used by Kubokawa (55). Desorption rates can be calculated from the evolution curves obtained during isothermal desorption as shown, for example, by Czanderna (54). 

Although we shall not be concerned experimentally with measuring heats of adsorption, it is appropriate to comment that Mi for the physical adsorption of a gas is always negative, since the process of adsorption results in a decrease in entropy. The isosteric heat of adsorption (the heat of adsorption at constant coverage 6) can be obtained by application of the Clausius-Clapeyron equation if isotherms are determined at several different temperatures the thermodynamics of adsorption have been fully discussed by Hill. ... 

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. 

A further example of the applicability of the Clausius-Clapeyron equation is in the assessment of risk associated with the handling of hazardous dmgs, particularly by personnel who are potentially exposed to cytostatic... 

The Clausius-Clapeyron equation is not limited to ordinary phase changes, but is applicable to any system whose state can be described in terms of the variables T, p, When applied to the present problem we obtain a formula due to Ehrenfest.f... 

Sarasohn, I. M. Application of the Clausius-Clapeyron Equation to DTA. DuPont Thermogram 2, 1 (1965). 

Here A is the ordinary heat of vaporization per mol, U the heat development associated with the condensation without dbing external work p and i are the pressure and concentration of the saturated vapour. The two equations are fundamentally identical they result from the Clausius-Clapeyron equation by introducing the laws of the ideal gaseous state and limiting the application to small pressures. 

Heats of snrface reactions are directly obtainable from simple LEED observations. The nsnal application is to measure the enthalpy of adsorption of reversibly bound adsorbates. When the adsorbate produces a characteristic LEED pattern with extra beams, the mere existence of these beams, and not detailed intensity analysis, informs one of the presence of the characteristic adsorbed structure on the surface. At a given temperature there is a pressure at which this surface structure is just maintained, and the rates of evaporation and condensation into the structure are equal. Measurements of this pressure p as a function of absolute temperature T give the isosteric enthalpy of adsorption AH by application of the Clausius-Clapeyron equation for constant coverage... 

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. 

Application of the Clausius-Clapeyron, Equation (11.5) of the text, predicts that the more the vapor pressure rises over a temperature range, the smaller the heat of vaporization will be. Considering the equation below,... 

The Clausius-Clapeyron equation, one of the most famous in physical chemistry, is most applicable for this discussion. The equation states that the partial differential with respect to absolute temperature of the logarithm of a pure liquid s vapor pressure is inversely related to the liquid s absolute temperature. We again consider the liquid TCE and state the Clausius-Clapeyron equation mathematically (29) ... 

Because the calorimetric methods of measurement of enthalpy of vapor formation are very difficult, the indirect mefliods are used, especially for less volatile substances. The application of generalized expression of the first and second laws of thermodynamics to the heterogeneous equilibrium between a condensed phase in isobaric- thermal conditions is given in the Clausius-Clapeyron equation that relates enthalpy of a vapor formation at the vapor pressure, P, and temperature, T. For one component system, the Clausius-Clapeyron equation has the form ... 

Application of Eq. (1-49) to a one component system at phase equilibrium with a vapor phase (II) and a liquid phase (I) gives the Clausius-Clapeyron equation (see also Chapter 1.4.3.1). With x, = 1 and y, = 1 for both phases I and II, it follows that... 

The application of the Clausius-Clapeyron equation should be restricted to a certain temperature range. The first term dP/dT can be considered as quite exact, as a reliable vapor pressure equation is a necessary requirement for process simulation. The limitation is that the common vapor pressure equations like Wagner and Antoine badly extrapolate to low temperatures. Furthermore, even if measured data at low temperatures are available, the relative errors are quite high. As a rule of thumb, it is recommended not to use the Clausius-Clapeyron equation... 

With these two restrictions, it can be assumed that an error of 1-2% coming from lA is usually obtained for the calculated enthalpy of vaporization. The values outside the application range of the Clausius-Clapeyron equation can be estimated by fitting Eqs. (3.61) or (3.63) to data generated with the Qausius-Clapeyron equation and extrapolating it. 

His essential contributions to Chemical Kinetics, besides the part previously cited in the first part of this chapter, culminated in the discovery of the relation between the rate constant and the equilibrium constant (Van t Hoff, 1884). He interpreted the Chemical Equilibrium as the balance between opvposite reactions so he related equilibrium constant to the ratio of the rate constants of the direct and reverse reaction. From an application of Clausius-Clapeyron equation Van t Hoff found the dependence of the equilibrium constant K from the absolute temperature T ... 

The Isosteric Heat of Adsorption. The value of the heat of adsorption is abnormally high in micropores. Sing and Ramakrisbna (173) found that by careful choice of adsorbates and application of the method of analysis, it is possible to distinguish between capillary adsorption and adsorption on high-energy sites. They showed that from p/po 0.01 to 0.2 the isosteric heat of adsorption of nitrogen on silica free of mesopores remains essentially constant at 2.0 kcal mole. On mesoporous gel it drops from 2.3 to 2.0 and on microporous gel from 2.7 to 2.0. The isosteric heat q t is calculated from the adsorption isotherm by the Clausius-Clapeyron equation. 

As an example of the application of the Clausius-Clapeyron equation, the vapor-pressure data for ethanol shown in Figure 11.24 are graphed as In P versus 1/T in Figure 11.25 . The data lie on a straight line with a negative slope. We can use the slope of the line to determine AH ap for ethanol. We can also extrapolate the line to obtain values for the vapor pressure of ethanol at temperatures above and below the temperature range for which we have data. 

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