Molar volume Clausius-Clapeyron equation

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

If the vapor phase in VLE is ideal and the liquid molar volumes are negligible (assumptions inherent in Raoult s law), then the Clausius/Clapeyron equation applies (see Ex. 6.5) ... 

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,... 

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. 

Equilibrium concentrations and temperatures 7 are additionally influenced by the pressure. According to the Clausius-Clapeyron equation, a change in the molar volume Al " = V i — at the ceiling temperature T, generally gives... 

The molar volume of the gas phase is much larger them the molar olume of the liquid or solid phase, so Av = u as - condensed t gas = RTIP, if the ideal gas law applies. Substituting this expression into Equation (14.21) gives the Clausius-Clapeyron equation. 

As a first approximation we may use the molar volume of an ideal gas, l mg = RT/p. Substituting this expression in the place of and noting that dplp = d np), we arrive at the the Clausius-Clapeyron equation ... 

To move on to the Clausius-Clapeyron equation, we consider the case of vaporization. Because the molar volume of a gas is much larger than the molar volume of a liquid, we can replace AyapV = Vm(g) f in(l) by Vm(g) alone and write... 

Keep a note of any approximations made in a derivation, for they limit the range of applicability of an expression. We have made two approximations in the derivation of the Clausius-Clapeyron equation (1) the molar volume of a gas is much greater than that of a liquid and (2) the vapor behaves as a perfect gas. 

The Clausius-Clapeyron equation is obtained by integrating the Clapeyron equation in the case that one of the two phases is a vapor (gas) and the other is a condensed phase (liquid or solid). We make two approximations (1) that the vapor is an ideal gas, and (2) that the molar volume of the condensed phase is negligible compared with that of the vapor (gas) phase. These are both good approximations. 

Experimental values are determined either directly from calorimetric measurements. or indirectly through the Clausius-Clapeyron equation, using experimental data for the vapor pressure and the saturated molar volumes. Correlations of experimental data as a function of temperature for a large number of compounds is given by Daubert and Danner. 

The Clapeyron-Clausius equation as applied to liquid = vapour equilibrium, can be easily integrated. The molar volume of a substance in the vapour state is considerably greater than that in the liquid state. In the case of water, for example, the value of Vg at 100 C is 18 1670 = 30060 ml while that of V, is only a little more than 18 ml. Thus, V-Vt can be taken as Vwithout introducing any serious error. The Clapeyron-Clausius equation 1.51, therefore, may be written as... 

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