Clapeyron equation solubility

The temperature dependence of the solubility, as follows from Eq. (18.3), obeys the Clausius-Clapeyron equation ... 

In order to extend the method of calculating ideal solubilities of gases to temperatures above the critical point, when the liquid cannot exist and direct determination of the vapor pressure is not possible, it is necessary to estimate a hypothetical vapor pressure by suitable extrapolation. This is best done with the aid of the integrated form of the Clausius-Clapeyron equation. If the vapor pressures at any two temperatures are known, the hypothetical value at a temperature above the critical point may be evaluate on the assumption that the heat of vaporization remains constant. 

Salmeterol xinafoate is known to exist in two polymorphic forms, forms I and II. Form I is stable, and form II is the metastable polymorph at ambient temperature. The enthalpies of solution (AHso ) of forms I and II determined from van t Hoff solubility-temperature plots are 32.1 and 27.6 kJ/mol, respectively, and the transition temperature obtained by linear extrapolation of the van t Hoff plots is 99 °C. The enthalpy of polymorphic conversion (AHii i) calculated from the plots of log solubility ratio of polymorphs versus the reciprocal of absolute temperature is negative (—4.55 kJ/mol) (5). However, the change in molar volume (AFu i) due to the conversion is positive. Therefore, according to the Clausius-Clapeyron equation,... 

By applying the Clausius-Clapeyron equation to the observed solubilities, S, of these di-univalent salts in the form ... 

Figure 2410. Vapor-phase concentrations of trichloroethylene (solid line) and tetrachloro-ethylene (dashed hne) as functions of temperature. (Note Calculated with the Clapeyron equation assuming ideal gas behavior). The arrows to the ordinate axis indicate the aqueous solubilities of the two compounds at 25 °C.

The difference between a freezing point determined in the presence of air at a pressure of one atmosphere and the triple-point temperature will depend on the effect of dissolved air and on the pressure difference. The presence of dissolved gas in the liquid phase always lowers the transition temperature. The amount by which it does so can be calculated from the freezing point depression equations if the solubility is known and ideal behaviour is assumed. The effect of pressure can be calculated from the Clapeyron equation... 

The solubility of common gases in hydrocarbon liquids is determined to meet requirements of aerospace industry. This test method is based on the Clausius-Clapeyron equation, Henry s law, and the perfect gas law. The results are important in the lubrication of gas compressors where dissolved gas may cause erosion due to cavitation. In fuels, dissolved gases may cause interruption of fuel supply and foaming in tank. The liquid density is determined experimentally. Using this density, the Ostwald coefficient is taken from a chart and used for e calculation of the Bunsen coefficient (solubility of gas). The solubility of the gas or mixture of gases and Henry s law constant are also calculated. 

In some cases, in order to apply the theory, it is necessary to extrapolate the vapour pressure of the liquefied gas beyond the critical point. For example, suppose that it is desired to estimate the ideal solubility of methane at a temperature of 25 C, which is far above critical. If the observed vapour pressures are extrapolated by means of the Clausius-Clapeyron equation, the estimated value of p at 25 is found to be 289 atm— but of course this does not correspond to a stable state of gas-liquid ecjuUibrium. The ideal solubility of methane at 25 is therefore 1/289=0.0035. Some of the observed solubilities, as quoted by Hildebrand and Scott, are given in the table. 

When spraying solutions of substances of low molecular weight, a first drying step occurs above the solubility limit of the solute, and the drop reduces in size until the solubility limit is reached. For this first step, which is usually called the first or constant rate period (CRP), the drying velocity, that is, the solvent mass evaporated per unit time, is given by the vapor pressure of the solution at the drop surface, p s, and the vapor pressure in the vicinity of the particle, pv.oo. The vapor pressure at the surface depends - assuming a well-mixed state within the droplet - on the drop temperature and on the water activity within the solution. The surface temperature remains low in the CRP as the solvent, due to its heat of evaporation Ah uses up the sensible heat (expressed by the specific heat capacity Cp of the air-vapor mixture) transferred to the particle by the gas in a hot atmosphere. The particle surface temperature Ts is more or less close to the wet bulb temperature of pure water, depending on the water activity in the case of dissolved matter (see Eq. 5.44 in Volume 1 of this series). The dependence of the vapor pressure of the solvent on the surface temperature Ts may be expressed by the Antoine or Clausius-Clapeyron equation, as... 

The solubilities of gases decrease with increasing temperature. Account is taken of this factor with the Clausius-Clapeyron equation. 

The variation of the solubilities of most substances with temperature is fairly regular, and usually increases with temperature. When water is the solvent, breaks may occur in solubility curves because of formation of hydrates. Figure 16.1(a) shows such breaks, and they can be also discerned in Figures 16.2(b) and (c). Unbroken lines usually are well enough represented by second degree polynomials in temperature, but the Clapeyron-type equation with only two constants, Inx = A + B/T, is of good accuracy, as appears for some cases on Figure 16.1(b). 

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