Condensed phases limiting molar

However, AH, the difference between the molar enthalpy of the gas and the condensed phase, depends in general on both the temperature and the pressure. The enthalpy for an ideal gas is independent of pressure and, fortunately, the enthalpy for the condensed phase is only a slowly varying function of the pressure. It is therefore possible to assume that AH is independent of the pressure and a function of the temperature alone, provided that the limits of integration do not cover too large an interval. With this final assumption, the integration can be carried out. When the molar heat capacities of the two phases are known as functions of the temperature, AR is obtained by integration. If ACP, the difference in the molar heat capacities of the two phases, is expressed as... 

For component i of a condensed-phase mixture, we take a constant pressure equal to the standard pressure p°, and a mixture composition in the limit given by Eqs. 9.5.20-9.5.24 in which the activity coefficient is unity. Hi is then the standard molar enthalpy H , and the activity is given by an expression in Table 9.5 with the pressure factor and activity... 

For reactions involving only condensed phases, including those occurring in liquid solutions, which are our chief concern, the situation is very different. Three choices of standard state are in common use. For the solvent (i.e., the substance present in largest amount), the standard state almost universally chosen is the pure liquid. This choice is also often made for other liquid substances that are totally or largely miscible with the solvent. The activity scale is then related to the mole fraction, through the rational activity coefficient f which is unity for each pure substance. For other solutes, especially those that are solid when pure, or for ionic species in solution in a nonionic liquid, activity scales are used that are related either to the molar concentration or the molality, depending on experimental convenience. On these scales, the activity coefficients become imity in the limit of low concentration. 

Here represents the radius of a sphere containing x atoms, each having the atomic volume calculated from the molar volume of the solid and W is the work function of the corresponding bulk material in the condensed (solid or liquid) phase. This theory works well for the rare gases and the data extrapolate to the limit Ip = W oi the bulk materials. 

Apart from the inherent interest in the gas phase ion clusters, the accumulation of solvent molecules around an ion should yield at the limit of very large values of n to a constant value of j H°(S,g). This would be the molar enthalpy of condensation of a solvent molecule into the bulk liquid solvent, because at this limit, the ion has no influence any more on the energetics of the process. Thus ... 

The Molar-Ratio Limiting-Value A molar-ratio >1 appears to be a limiting value for attaining a phase-separation-fiee sol-gel reaction, even at temperatures as high as 80°C. This observation, which is common to TEOS, TMOS and MTMS can be rationalized as follows we should keep in mind that all the hydrolysis and condensation processes (1) to (4) occur, as a matter of fact, via reversible reactions. Therefore, when we manage by elevated temperature to drive the hydrolysis to form partially-hydrolyzed siloxane species, there is always the possibility that given sufficient mobility and time these species will undergo disproportionation reactions ... 

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