Standard-state pressure

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

This method is invalid because the temperature in the denominator of the equation must be the temperature at which the liquid-vapor transition is at equilibrium. Liquid water and water vapor at 1 atm pressure (standard state, indicated by ) are in equilibrium only at 100° C = 373 K. 

You may wonder how a reaction, such as combustion of methane, can occur at 25°. The fact is that the reaction can be carried out at any desired temperature. The important thing is that the AH° value we are talking about here is the heat liberated or absorbed when you start with the reactants at 25° and finish with the products at 25°. As long as AH0 is defined this way, it does not matter at what temperature the reaction actually occurs. Standard states for gases are 1 atm partial pressure. Standard states for liquids or solids usually are the pure liquid or solid at 1 atm external pressure. 

Moles of solvent per mole of solute Number of moles, species i Absolute pressure Standard-state pressure Critical pressure Reduced pressure Reference pressure Partial pressure, species i Sahiration vapor pressure, species i Heat... 

Table 2.3 summarizes the essential relationships for pressure effects on chemical equilibrium for the variable-pressure standard-state convention. Note, that these relationships can apply to any consistent choice of standard part ial molar volumes, for example, one for which an ionic medium such as seawater is adopted as the solute reference state. For detailed discussion of applications to seawater see, for example, Millero (1969) and Whitfield (1975). A compie-... 

Integrating between unit pressure (standard state) and the internal pressure of the particle... 

VARIABLE PRESSURE STANDARD STATES 12.4.1. Standard States Based on Raoult s Law... 

Equations (12.10) and (12.14) are examples of expressions that use a variable pressure standard state. Although it is theoretically possible to keep P° fixed and to simply add BT In P) to the — term for each different P considered, in practice one usually considers that the pressure of the standard state (P°) and the pressure on the system or state of interest (P) are the same, so that p° is a function of the system pressure. 

In considering the effect of pressure on activity, we must recall that the standard state pressure (P°) is not always the same as the system pressure (P), so that the differentiation with respect to pressure is not always completely analogous to differentiation with respect to temperature. First of all, for variable pressure standard states, those that do have P° = P, we have... 

For gases and supercritical fluids, fugacities are normally used, and the standard state is normally chosen as the ideal gas at the system temperature (T) and one bar, i.e., a fixed pressure standard state P° = 1 bar), so that normally... 

Normally of course the expression for the variation of K with P is simpler than this, perhaps because all three states of matter may not be present, but also because it is quite unusual to use a variable pressure standard state for constituents whose fugacities are known or sought, (because this adds complexities rather than simplifying matters), and the In Qig) term is therefore essentially never required. To take a real example, let s consider the brucite-periclase reaction again. We have discussed the variation of the equilibrium constant for the brucite-periclase-water reaction with temperature at one bar, and showed that the equilibrium temperature for the reaction at one bar is about 265°C. Calculation of the equilibrium temperature of dehydration reactions such as this one at higher pressures was discussed briefly in 13.2.2. Here we will discuss the reaction in different terms to demonstrate the relationships between activities, standard states and equilibrium constants. 

Note that if you do set /° = 1, then and f° are independent of the system pressure. That is to say, they depend on P° but not on P. Once the standard state is chosen, it is a function only of the value of T, held constant during the integration. Another way of putting this is that we have a fixed pressure standard state. 

Again, because we often consider various system pressures, the standard state pressure will vary, so we have a variable pressure standard state. 

First of all, for variable pressure standard states, those that do have P° = P, we have... 

If the standard state is defined as having a fixed pressure of 1 bar, the reference state is also sometimes referred to as a state reached from the standard state by a change in pressure (Pitzer and Brewer, 1961, p. 249). Because in this text we use a variable pressure standard state, we have no need of the reference state in this sense either. 

Standard temperature and pressure, or STP, is the standard condition most typically associated with gas law calculations. STP conditions are taken as room temperature (298.15 K) and atmospheric pressure. (Standard-state pressure is actually defined as 1 bar = 100 kPa. Atmospheric pressure is taken as 1 atm = 101.325 kPa. These slight differences are usually ignored.)... 

There is an advantage in using the constant surface pressure standard state since it yields molar properties (enthalpies and entropies of adsorption) analogous to those associated with phase changes evaluated from the Clapeyron equation [80]. The use of the standard state with constant surface concentration provides differential quantities for the enthalpy and entropy changes which are not directly comparable with those calculated using the methods of statistical thermodynamics. The values of AS calculated by these two standard states differ only by the gas constant, B, and are readily interconverted. 

The a, method of Sing (49) plots a, versus x where a, x/x x is the amount of gas adsorbed, and x, is the amount adsorbed at a selected relative pressure (standard state). Usually by definition, a, is taken as 1.0 at p/pa 0.4. This pressure is selected because only monolayer coverage and micropore filling occur below this point and hysteresis loops owing to capillary filling effects occur at higher pressure. 

We also note that the standard pressure used is 1 bar rather than 1 atm, although such a change in the pressure standard state has little effect on the results with regard to the usual level of precision. 

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