Sublimation Clapeyron equation

Worked Example 5.3 The Clausius-Clapeyron equation need not apply merely to boiling (liquid-gas) equilibria, it also describes sublimation equilibria (gas-solid). 

The Clapeyron equation, Equation (5.1), yields a quantitative description of a phase boundary on a phase diagram. Equation (5.1) works quite well for the liquid-solid phase boundary, but if the equilibrium is boiling or sublimation - both of which involve a gaseous phase - then the Clapeyron equation is a poor predictor. 

A sublimation process is controlled primarily by the conditions under which phase equilibria occur in a single-component system, and the phase diagram of a simple one-component system is shown in Figure 15.30 where the sublimation curve is dependent on the vapour pressure of the solid, the vaporisation curve on the vapour pressure of the liquid, and the fusion curve on the effect of pressure on the melting point. The slopes of these three curves can be expressed quantitatively by the Clapeyron equation ... 

Although there are few data available on sublimation-desublimation, a considerable amount of information can be calculated using the Clausius-Clapeyron equation provided that information on vapour pressure is available at two or more temperatures. In this way ... 

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. 

This expression, therefore, gives the correct value of the heat of formation. The heats of sublimation of elements can usually be determined directly, but are mostly derived from the change of vapour pressure of the solid with the change of temperature, making use of the Clapeyron equation which relates these quantities. 

Some of these ambiguities can be partially solved using a simple approach recently proposed by Gamier et al. [62], The sublimation pressure of a solid can be estimated using experimental fusion properties and the vaporization enthalpy derived from the equation of state. Using the Clapeyron equation P b can be approximated by ... 

Equation (9) is valid for evaporation and sublimation processes, but not valid for transitions between solids or for the melting of solids. Clausius-Clapeyron equation is an approximate equation because the volume of the liquid has been neglected and ideal behaviour of the vapour is also taken into account. 

The equation of the sublimation curve is conveniently obtained from the integrated Clausius-Clapeyron equation,... 

The molar enthalpy of sublimation A// is rigorously related to the slope dpIdTof the sublimation pressure curve by the Clapeyron equation, which has been discussed in Exp. 13. The exact expression is... 

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. 

The Clausius-Clapeyron equation can also be applied to estimate the vapour pressure of a solid precursor. In this case, the enthalpy of sublimation (AHsub) should replace the enthalpy of vaporisation. 

This problem involves the application of the Clausius-Clapeyron equation. We will assume that the heats of fusion, sublimation and vaporization are all constant. Thus we will use... 

Changes of sublimation pressure with temperature can be predicted and quantitatively calculated by means of the theorem of Lc Chatclier and the Clausius-Clapeyron equation, in the same way as changes of vapour pressure of a liquid (pp. 18 and 19). 

Univariant Systems.—Equilibrium between liquid and vapour. Vaporisation curve. Upper limit of vaporisation curve. Theorems of van t Hoff and of Le Chatelier. The Clausius-Clapeyron equation. Presence of complex molecules. Equilibrium between solid and vapour. Sublimation curve. Equilibrium between solid and liquid. Curve of fusion. Equilibrium between solid, liquid, and vapour. The triple point. Complexity of the solid state. Theory of allotropy. Bivariant systems. Changes at the triple point. Polymorphism. Triple point Sj—Sg— V. Transition point. Transition curve. Enantiotropy and monotropy. Enantiotropy combined with monotropy. Suspended transformation. Metastable equilibria. Pressure-temperature relations between stable and metastable forms. Velocity of transformation of metastable systems. Metastability in metals produced by mechanical stress. Law of successive reactions. 

The heat of sublimation of naphthalene is not given. However, we can compute this quantity from the vapor pressure curve of the solid and the Clausius-Clapeyron equation (Eq. 7.7-5a) by taking P to be equal to the sublimation pressure P " , AW to equal the heat of sublimation, and setting A V = V - = AsubY = RT/P. Thus... 

Note that (17.94) is similar to the Clausius-Clapeyron equation for water vapor-water equilibrium (17.8), with the enthalpy of sublimation replacing the enthalpy of evaporation. 

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. 

The criteria for equilibria involving solid phases are exactly those given in 7.3.5 for any phase-equilibrium situation phases in equilibrium have the same temperatures, pressures, and fugacities. Moreover, pure-component solid-fluid equilibria obey the Clapeyron equation (8.2.27). This means the latent heat of melting is proportional to the slope of the melting curve on a PT diagram and the latent heat of sublimation is proportional to the slope of the sublimation curve. In the case of solid-gas equilibria, the Clausius-Clapeyron equation (8.2.30) often provides a reliable relation between temperature and sublimation pressures, analogous to that for vapor-liquid equilibria. 

For vapor pressure values given only in the form of the Clausius-Clapeyron equation a pseudo Third Law value is calculated by evaluating the enthalpy of sublimation at the temperature extremes and averaging. The value obtained gives a reasonable estimate compared to the value that would have been obtained if all the data had been available. For vapor pressure measurements, the Revised method is used to calculate the Second Law enthalpy of sublimation, but for mass spectrographic measurements fitted to the Clausius-Clapeyron equation, the Traditional method is used. 

The temperature dependence of the vapour pressure is related to the enthalpy of sublimation (A//s) of the crystals by the Clapeyron equation ... 

The gas-solid phase equilibrium at sublimation or desublimation is described by the generaUzed Clausius-Clapeyron equation, see Eq. (1.71) and Fig. 1-32... 

The melting pressure curve may be given based on the general form of the Clausius-Clapeyron equation in a similar manner to the vapor pressure and sublimation curves (Eq. (1-71))... 

The Clausius-Clapeyron equation can also be applied to melting or sublimation processes. 

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