Clapeyron equation surface

Referring to Figure 8, temperature Tc is the chamber temperature and Ts is the surface temperature at the salt solution/vapor interface. The temperature of the chamber is well defined and is an experimental variable, whereas Ts must be higher than Tc due to condensation of vapor on the saturated solution surface. We can determine Ts by applying the Clausius-Clapeyron equation to the problem. Assume that the vapor pressures of the surface and chamber are equal (no pressure gradients), indicating that the temperature must be raised at the surface (to adjust the vapor pressure lowering of the saturated solution) to Pc (at Tc) = Ps (at Tc). However, there is a difference in relative humidity between the surface and the chamber, where RHC is the relative humidity in the chamber and RH0 is the relative humidity of the saturated salt solution, and we obtain... 

The vapor pressure of the liquid at the surface Pg can be evaluated from an integrated from of the Clausius-Clapeyron equation if the surface temperature Ts is known. 

Since surface pressure is a free energy term, the energies and entropies of first-order phase transitions in the monolayer state may be calculated from the temperature dependence of the ir-A curve using the two-dimensional analog of the Clausius-Clapeyron equation (59), where AH is the molar enthalpy change at temperature T and AA is the net change in molar area ... 

If the surface temperature does not differ greatly from the surrounding temperature, the highly nonlinear surface boundary condition may be simplified by linearizing the expression for the radiation flux and the Clausius-Clapeyron equation to yield the approximation... 

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 temperature variation of ttv may be analyzed by a relationship analogous to the Clapeyron equation to yield the two-dimensional equivalent to the heat of vaporization. The numerical values obtained for this quantity more nearly resemble the bulk values for hydrocarbons than those for polar molecules. This suggests that most of the change in the surface transition involves the hydrocarbon tail of the molecule rather than the polar head. 

The requirement of thermodynamic reversibility also applies to the chromatographic method, but in this case it is necessary to work at very low surface coverage (at zero coverage ) in the Henry s law region. Values of the specific retention volume, Vs, determined at different temperatures are inserted in the Clausius-Clapeyron equation in place of the equilibrium pressures to obtain A h. Provided that a number of conditions are observed, the method is capable of providing a fairly easy and rapid assessment of the adsorbent—adsorbate interaction energy. 

The partial pressure of a pure liquid at the svuface varies as a function of the surface temperature, Tg, according to the Clausius Clapeyron equation [16] ... 

Initial conditions for the dependent variables are assumed to be arbitrary. By further assuming that the fuel vapor is saturated at the droplet surface, the fuel vapor weight fraction, Wt)s, and the temperature at the droplet surface, Ts, are related by the Clausius-Clapeyron equation... 

Equations 22-29 for the gas-phase profiles, temperature gradient, flame position, and vaporization rate depend only on the temperature at the surface of the droplet, Tg. (The Clausius-Clapeyron equation relates to Tg.) An iterative numerical procedure for satisfying the continuity of Tg at the liquid/gas interface is described in the Appendix. 

As a useful thermodynamic property, the isosteric heat of adsorption has been generally applied to characterize the adsorbent surface. The isosteric heat of adsorption is evaluated simply by applying the Clausius-Clapeyron equation if one has a good set of adsorption equilibrium ta obtained at several temperatures. 

The Clapeyron equation, derived in Sec. 6.4 for the latent heat of phase transition of pure chemical species, is also applicable to pnre-gas adsorption eqnilibrinm. Here, however, the two-phase equilibrium pressure depends not only on temperature, but on surface coverage or the amount adsorbed. Thus the analogous equation for adsorption is written... 

The classical Clapeyron equation adequately predicts the features of first-order phase transitions, and this has been established for a number of examples of first-order transitions effected by the deliberate variation of temperature or pressure. Second- or higher-order transitions are not readily explained by classical thermodynamics. Unlike the case of first-order transitions, where the free-energy surfaces of the two phases... 

In the previous section it was observed that the Langmuir postulates of sites of equal activity and no interaction between occupied and bare sites were responsible for nonagreement with experimental data. It might be surmised that these assumptions correspond to a constant heat of -ad-sm-pt-ion—Indeed.-it-is-p.QssibIe to derive the Langmuir isotherm by assuming that is independent of d. The heat of adsorption can be evaluated from adsorption-equilibrium data. First the Clausius-Clapeyron equation is applied to the two-phase system of gas and adsorbed component on the surface ... 

For the Clausius-Clapeyron equation to be valid the process must be univariant. This means that Eq. (9-10) can be applied only for constant concentration of adsorbate on the surface, that is, constant 6. If adsorption-equilibrium data are available at different temperatures, the slopes of p-vs-T curves at constant 6 may be used with Eq. (9-10) to calculate AH,. Figure 9-2, taken from Beeck, shows such isosteric heats of adsorption as a function of 9 for hydrogen on several metal films. These results are typical of almost all heats-of adsorption in showing a decrease in AH, with increasing surface coverage. 

For the adsorption of a mole of any gaseous molecule onto an inert surface (one that is not changed by the adsorption itself), it can be shown from thermodynamic principles (see Chapter 1) that this vapor-adsorbate phase change is described by the Clausius-Clapeyron equation ... 

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... 

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. 

The exponential distribution law employed defines the nature of the change of the heat of chemisorption with surface coverage. The Clausius-Clapeyron equation is the thermodynamic relation defining the heat effect accompanying a change of... 

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