Characteristic temperature Clapeyron equation

As already mentioned, the system ofEqs. (8.1-8.5) is supplemented by the Clausius-Clapeyron equation, as well as by the correlation that determines the dependence of enthalpy on temperature and describes the thermohydrodynamical characteristics of flow in a heated capillary. It is advantageous to analyze parameters of such flow to transform the system of governing equations to the form that is convenient for significant simplification of the problem. 

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. ... 

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 Clausius-Clapeyron equation relates the equilibrium vapor pressure, the cell temperature, and the physical characteristics specific to the organic compound ... 

If the characteristic time of relaxation tends to zero, the transition occurs instantaneously in a sharply defined front. We denote the velocity of the front by U and the velocity of the mixture by u. The state of the mixture is characterized by pressure p, temperature T and liquid mass fraction fp. The gas and the vapour are considered perfect so they obey Dalton s law. For simplicity we shall from now on refer to the supersaturated vapour state and to the state of liquid-vapour equilibrium as state 1 and state 2, respectively. It is dear there exists no liquid phase in state 1, fp = 0, and the vapour mass fraction f is equal to the total mass fraction of the condensable component of the mixture. The vapour pressure p 2 state 2 has to satisfy the Clausius-Clapeyron equation (2). The laws of conservation of mass, momentum and energy are, applied to the condensation discontinuity ... 

The use of the low-pressure experimental conditions also impacts on the observed adsorption characteristics. For surface states with low heats of adsorption, it follows from the Clausius-Clapeyron equation that very low temperatures are required to achieve reasonable coverages at the pressures used in most single-crystal studies (<10" torr). In the case of hydrogen, for instance, where the heat of adsorption is very low (43-67 kJ mor ), " temperatures of around 160 to 280 K are required. The even less strongly bound molecular precursor states, which can have heats of adsorption below 20 kJ mol", require experimental temperatures as low as 20 Clearly, if adsorption studies are carried out at ambient temperature then these states will not be observed. 

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