Solids Clausius-Clapeyron equation

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 Clausius-Clapeyron equation The Clapeyron equation can be used to derive an approximate equation that relates the vapor pressure of a liquid or solid to temperature. For the vaporization process... 

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

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. 

Equation 6.56 is known as the equation of lowering of freezing point and is valid for solid mixtures crystallizing from multicomponent melts. Like the Clausius-Clapeyron equation, it tells us how the system behaves, with changing T, to maintain equilibrium on the univariant curve. However, whereas in the Clausius-Clapeyron equation equilibrium is maintained with concomitant changes in 7) here it is maintained by appropriately varying the activity of the component of interest in the melt and in the solid mixture. 

The vapor pressure, pv, is the pressure exerted by fluids and solids at equilibrium with their own vapor phase. The vapor pressure is a strong function of T, as expressed in the Clausius-Clapeyron equation [1] ... 

Clausius-Clapeyron equation. Because the magnitude of changes in X and kx in solid... 

If the gas phase activity of the host is controlled by the presence of a pure condensed phase, solid or liquid, the equilibrium between host and guest in a stoichiometric clathrate can be described in terms of the gas phase pressure of the guest. This is, in effect, a vapor pressure for the guest. At higher pressures the guest will condense to form clathrate, and at lower pressures the clathrate will decompose. Temperature variation of this pressure will follow the Clapeyron equation which, with the usual assumptions (ideal gas behavior of the vapor and negligible volume of the condensed phase), reduces to the Clausius-Clapeyron equation ... 

The above equations are variously labelled as Clausius-Clapeyron equations. Subject to the satisfactory nature of the assumptions made, a plot (Figure 26.1(a)) of the variation of the natural logarithm of the vapour pressure, In(P/P°), over a liquid measured at various temperatures against the reciprocal of temperature (1 /T) should be linear and have a gradient equal to — Avap H°/R so provides a means of measuring Avap H° for a variety of liquids (Figure 26.1(b)). Also from vapour pressure data for solids at two or more different temperatures one can measure AsubH°. 

Data were also obtained by this method for the solid states for the methyl ester of 2,4-D, the n-propyl ester of 2,4,5-T, and the butyl ester (liquid) of 2,4-D. The results are shown in Table III. These data were fitted by the least squares method to the Clausius-Clapeyron equations given in footnotes to Table III. These equations were used to estimate the vapor pressures at several temperatures, including the melting point. In Table IV, these are compared with estimates from other sources. Jensen s unpublished data with the Knudsen method compare favorably with those reported in this work, but the published values obtained by other methods are larger. 

In connection with the orifice test, the vapor pressure of the solid state of this pyrimidine was also determined over a range of temperature from 0° to 50°C. The Clausius-Clapeyron equation fitted by least squares to the 14 measurements made was ... 

Suppose we have a solid-liquid coexistence point T, p) for pore fluid on the bulk phase diagram. Though the bulk pressure is at p, the fluid in pore is supposed to have different pressure because of the pore-wall potential and the capillary effect. Not for the bulk pressure but for this pressure felt by fluid in pore, p, the Clausius-Clapeyron equation for the bulk is assumed to hold. 

Solids Below the triple point, the pressure at which the solid and vapor phases of a pure component are in equilibrium at any given temperature is the vapor pressure of the solid. It is a monotonic function of temperature with a maximum at the triple point. Solid vapor pressures can be correlated with the same equations used for liquids. Estimation of solid vapor pressure can be made from the integrated form of the Clausius-Clapeyron equation... 

T and pressure P. It should be noted that equation (33.26) is the exact form of the Clausius-Clapeyron equation (27.12). If the vapor is assumed to be leal, so that the fugacity may be replaced by the vapor pressure, and the total pressuic is taken as equal to the equilibrium pressure, the two equations become identical. In this simplification the assumption is made that the activity of the liquid or solid does not vary with pressure this is exactly equivalent to the approximation used in deriving the Clausius-Clapeyron equation, that the volume of the liquid or solid is negligible. 

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. 

There is a more or less generalized agreement that the isosteric adsorption heat is strongly affected by the microstructure of the adsorbent, particularly in the case of porous solids. This magnitude is better suited for structural analysis than other thermodynamic quantities. The use of the Clausius—Clapeyron equation to determine the isosteric adsorption heat has several limitations both theoretical and experimental, that are well known. 

Aeeording to the Clausius-Clapeyron equation the transition temperature at the solid —> liquid phase boundary of a binary mixture shifts by (5Z) upon a relative eoneentration ehange (dc) between the two eonstituents ... 

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. 

At the boiling point, the vapor pressure of a liquid equals the external pressure. The molar heat of vaporization of a liquid is the energy required to vaporize one mole of the liquid. It can be determined by measuring the vapor pressure of the liquid as a function of temperature and using the Clausius-Clapeyron equation [Equation (11.2)]. The molar heat of fusion of a solid is the energy required to melt one mole of the solid. 

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... 

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. 

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