Volume standard state

A point of occasional confusion arises with respect to units. In Eq. (15.22), all portions are unitless except for k T/h, which has units of sec , entirely consistent with the units expected for a unimolecular rate constant. In Eq. (15.23), the same is true with respect to the r.h.s., but a bimolecular rate constant has units of concentration" sec", which seems paradoxical. The point is diat, as with any thermodynamic quantity, one must pay close attention to standard-state conventions. Recall that die magnitude of die iranslational partition function depends on specification of a standard-state volume (or pressure, under ideal gas conditions). Thus, a more complete way to write Eq. (15.23) is... 

Figure 5.4 Corrected model allowing for distortion of hydrate due to guests (v v11). Process (1) in Figure 5.4 is done at constant volume and therefore, the van der Waals and Platteeuw statistical model can be used. Process (2) in Figure 5.4, the volume change of the hydrate from its standard state volume, is done at constant composition and is described by the activity coefficient in Equation 5.28.

Replacing the standard-state volume in Equation (4.246) with (RT ) we obtain... 

The pressure dependence of the activity is related to the partial molar volume and standard state volume by ... 

PURE calculates pure liquid standard-state fugacities at zero pressure, pure-component saturated liquid molar volume (cm /mole), and pure-component liquid standard-state fugacities at system pressure. Pure-component hypothetical liquid reference fugacities are calculated for noncondensable components. Liquid molar volumes for noncondensable components are taken as zero. 

Standard-state fugacities at zero pressure are evaluated using the Equation (A-2) for both condensable and noncondensable components. The Rackett Equation (B-2) is evaluated to determine the liquid molar volumes as a function of temperature. Standard-state fugacities at system temperature and pressure are given by the product of the standard-state fugacity at zero pressure and the Poynting correction shown in Equation (4-1). Double precision is advisable. 

The confinement term is unique because it alone causes a dependence of the binding free energy on the choice of unit concentration in the standard state the volume available per ligand molecule in the free state, and hence the compression factor, depend on the unit concentration. 

Clearly, a free energy of binding computed with (9), (10) and (13) refers to a highly restricted state of the dissociated ligand. In order to convert such a free energy to a free energy relative to a normal standard state with volume per molecule Vg and no restriction on the molecular orientation, the following term must be added... 

It is a common practice to evaluate the molal volume ( V) of an ideal gas at a set of reference conditions known as the standard state. If the standard state is chosen to be... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

Since the pure solvent has been chosen as the standard state, ai = Pi/Pij to the approximation that the vapor may be regarded as an ideal gas. For the osmotic pressure itVi=—(mi—Mi) where Vi is the molar volume of the solvent. Thus, according to Eq. (26)... 

The free energy of mixing polymer segments with solvent in the volume element 5F, obtained from 5(AaSm ) in conjunction with a term kTxi niV representing the standard state free energy of mixing (see Eq. 20), is... 

There is no continuity in sizing of chemical equipment. There are discrete changes in capacities of chemical equipment of all kinds, either due to state and international standards or due to internal standards of individual manufacturers of chemical equipment. For instance, according to DIN standards, the volumes of glass-lined reactors increase in... 

In the reaction shown above, the volume of the reaction products (2 mol CO) is seen to be much greater than that of the reactants (2 mol of solid carbon plus 1 mol of oxygen). The effect of pressure on the free energy of formation of an oxide associated with an increase in the number of gas molecules which is representative of the type of reaction in the present illustration is shown in Figure 4.2 (A). Applying the criterion of volume increase per mole accompanying reaction at standard state to the case of metal oxidation such as... 

The observation that the transition state volumes in many Diels-Alder reactions are product-like, has been regarded as an indication of a concerted mechanism. In order to test this hypothesis and to gain further insight into the often more complex mechanism of Diels-Alder reactions, the effect of pressure on competing [4 + 2] and [2 + 2] or [4 + 4] cycloadditions has been investigated. In competitive reactions the difference between the activation volumes, and hence the transition state volumes, is derived directly from the pressure dependence of the product ratio, [4 + 2]/[2 + 2]p = [4 + 2]/[2 + 2]p=i exp —< AF (p — 1)/RT. All [2 + 2] or [4 + 4] cycloadditions listed in Tables 3 and 4 doubtlessly occur in two steps via diradical intermediates and can therefore be used as internal standards of activation volumes expected for stepwise processes. Thus, a relatively simple measurement of the pressure dependence of the product ratio can give important information about the mechanism of Diels-Alder reactions. 

A more direct link with molecular volumes holds for alkali halides, because the lattice energy (IT) is inversely proportional to interatomic distance or the cube root of molecular volume (MV). The latter has been approximated by a logarithmic function which gives a superior data fit. Plots of AH against log(MV) are linear for alkali halides 37a). Presumably, U and AH can be equated because AH M, ) is a constant in a series, and AH (halide )) is approximately constant when the anion is referred to the dihalogen as the standard state. 

Equation 7.31 was derived from a least squares fit to the data given in W.N. Hubbard, D. W. Scott, G. Waddington. Standard States Corrections for Combustions inaBomb at Constant Volume. In Experimental Thermochemistry, vol. 1 F. D. Rossini, Ed. Interscience New York, 1956 p. 93. 

---