Open system calculation

For a closed system at constant volume, calculate AU = n AU (where n is the amount of the species being heated or cooled). For a closed system at constant pressure, calculate AH = n AH. For an open system, calculate AH = h AR. where h is the species flow rate. 

First-law—Open-System—Calculation Using the Thermodynamic Web... 

For an open-shell system, try converging the closed-shell ion of the same molecule and then use that as an initial guess for the open-shell calculation. Adding electrons may give more reasonable virtual orbitals, but as a general rule, cations are easier to converge than anions. 

Many transition metal systems are open-shell systems. Due to the presence of low-energy excited states, it is very common to experience problems with spin contamination of unrestricted wave functions. Quite often, spin projection and annihilation techniques are not sufficient to correct the large amount of spin contamination. Because of this, restricted open-shell calculations are more reliable than unrestricted calculations for metal system. Spin contamination is discussed in Chapter 27. 

Open shell systems—for example, those with unequal numbers of spin up and spin down electrons—are usually modeled by a spin unrestricted model (which is the default for these systems in Gaussian). Restricted, closed shell calculations force each electron pair into a single spatial orbital, while open shell calculations use separate spatial orbitals for the spin up and spin down electrons (a and P respectively) ... 

In Figure 2 the solubility and speciation of plutonium have been calculated, using stability data for the hydroxy and carbonate complexes in Table III and standard potentials from Table IV, for the waters indicted in Figure 2. Here, the various carbonate concentrations would correspond to an open system in equilibrium with air (b) and closed systems with a total carbonate concentration of 30 mg/liter (c,e) and 485 mg/liter (d,f), respectively. The two redox potentials would roughly correspond to water in equilibrium wit air (a-d cf 50) and systems buffered by an Fe(III)(s)/Fe(II)(s)-equilibrium (e,f), respectively. Thus, the natural span of carbonate concentrations and redox conditions is illustrated. 

Bones and teeth, however, are primary archaeological materials and are common to many archaeological sites. Bones bearing cut marks from stone tools are a clear proxy for human occupation of a site, and in the study of human evolution, hominid remains provide the primary archive material. Hence, many attempts have been made to directly date bones and teeth using the U-series method. Unlike calcite, however, bones and teeth are open systems. Living bone, for example, contains a few parts per billion (ppb) of Uranium, but archaeological bone may contain 1-100 parts per million (ppm) of Uranium, taken up from the burial environment. Implicit in the calculation of a date from °Th/U or Pa/ U is a model for this Uranium uptake, and the reliability of a U-series date is dependent on the validity of this uptake model. 

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

The first and most critical step in developing a geochemical model is conceptualizing the system or process of interest in a useful manner. By system, we simply mean the portion of the universe that we decide is relevant. The composition of a closed system is fixed, but mass can enter and leave an open system. A system has an extent, which the modeler defines when he sets the amounts of fluid and mineral considered in the calculation. A system s extent might be a droplet of rainfall, the groundwater and sediments contained in a unit volume of an aquifer, or the world s oceans. 

The complex scattering wave function can be specified by nodal points at which u = 0,v = 0. They have great physical significance since they are responsible for current vortices. We have calculated distribution functions for nearest distances between nodal points and found that there is a universal form for open chaotic billiards. The form coincides with the distribution for the Berry function and hence, it may be used as a signature of quantum chaos in open systems. All distributions agree well with numerically computed results for transmission through quantum chaotic billiards. 

Once we open the system to allow exchange of ligands between the sites and the reservoir, the number of occupancy states of our system is not Q) (or 6 in the case of Fig. 1.1), but 2 " (or 2 = 16 as in Fig. 1.2). This is so because any site can be either empty or occupied, i.e., 2 states for each site, hence 2 " states for the m sites. Clearly, in an open system these 2 " configurations are not equally probable. For calculating the probabilities of the various events statistical mechanics provides a general recipe which differs from the classical method used above. The latter is applicable only when there are Q equally probable events (say, six outcomes of casting a die with probability 1/6 for each outcome). 

Several models have been proposed (Cast, 1968 Shaw, 1970 Hertoghen and Gijbels, 1976), but the most satisfactory are those based on iterative calculations, because they allow the evaluation of discontinuous events with sudden modifications of melting proportions, which are common in geological open systems (note that such discontinuities, or even the evolution of melting proportions, cannot be accounted for, if the distribution equations are obtained by integration at the limit). The following distribution model (Ottonello and Ranieri, 1977) is a partial modification of Cast s (1968) approach ... 

A final caveat that must be applied to phase diagrams determined using DFT calculations (or any other method) is that not all physically interesting phenomena occur at equilibrium. In situations where chemical reactions occur in an open system, as is the case in practical applications of catalysis, it is possible to have systems that are at steady state but are not at thermodynamic equilibrium. To perform any detailed analysis of this kind of situation, information must be collected on the rates of the microscopic processes that control the system. The Further Reading section gives a recent example of combining DFT calculations and kinetic Monte Carlo calculations to tackle this issue. 

An open system calorimetric test will tend to measure QGmax, rather than the sum of QGmax and Qvmax, because the vapour produced will tend to condense in the relatively cold containment vessel. A closed system test will also underestimate Qvmax because the high pressure will suppress vaporisation. Qvmax could also be calculated from ... 

In an open system with gas production, the volume of gas can be obtained from Equation 10.3, and the volume calculated either for a toxicity limit, as for example the level called Immediately Dangerous to Life and Health (IDLH) ... 

For open systems the change in the mole number that corresponds to the addition or removal of material from the system would be done at constant temperature and pressure. The molar enthalpy, the molar entropy, and the chemical potential would then be constant and the change in the enthalpy, entropy, and Gibbs energy of the system would be the product of the molar quantities and the change in the number of moles. Again, such changes cannot be calculated, because the absolute values are not known. However, the concept of the operation is used in later chapters. 

Carbonate equilibria in an open system. What is the pH of water in equilibrium with atmospheric C02 gas To answer such a question involves a knowledge of acid-base chemistry, the use of Henry s Law constant for the solubility of carbon dioxide and the use of the ENE to calculate the proton concentration of the equilibrium solution. The details of the equilibrium constants used are detailed below. 

We emphasize that the density matrix calculated from Eq. (6) is equivalent to that from Eq. (4), but Eq. (6) is much easier to compute for open systems. To see why this is so, let us consider zero temperature and assume ftL — ftR = eV], > 0. Then, in the energy range -oo < E < pR the Fermi functions = fR = 1. Because the Fermi functions are equal, no information about the non-equilibrium statistics exists and the NEGF must reduce to the equilibrium Green s function GR. In the range pR < E < pR, fL 7 fR and NEGF must be used in Eq. (6). A more careful mathematical manipulation shows that this is indeed true [30], and Eq. (6) can be written as a sum of two terms ... 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

In an open system such as a CSTR chemical reactions can undergo self-sustained oscillations even though all external conditions such as feed rate and concentrations are held constant. The Belousov-Zhabotinskii reaction can undergo such oscillations under isothermal conditions. As has been demonstrated both by experiments [1] and by calculations 12,3] this reaction can produce a variety of oscillation types from simple relaxation oscillations to complicated multipeaked periodic oscillations. Evidence has also been given that chaotic behavior, as opposed to periodic or quasi-periodic behavior, can take place with this reaction [4-12]. In addition, it has been shown in recent theoretical studies that chaos can occur in open chemical reactors [11,13-17]. 

The interaction-energy curves for alkali metal-rare gas pairs are also of interest experimentally in scattering and radiation problems, and theoretically because of the expected reliability of the HF energy for this class of half-open-closed-shell systems. Calculations on LiHe and NaHe (X22+, A2U, if 22+) and their X1 + ions have been reported by Krauss et al.285 from R = 3 to 10 bohr. Both STO and GTO expansion bases were used, with comparable results except for the ASH state of NaHe. The variation of dipole and quadrupole moments with R was investigated. The X2S+ curve is... 

Wang, F. and Ziegler T., Excitation energies of some dl systems calculated using time-dependent density functional theory an implementation of open-shell TDDFT theory for doublet-doublet excitations. Mol.Phys (2004) 102 2585 -2595. 

C02(g) under batch reaction calculations - saturation indices" - P(C02) = 3.02 vol%), since only a limited amount of gas (lliter) is assumed for the reaction. Because in the open system the partial pressure P(C02) = 2 vol% is lower than in the closed system ( 3.02 vol%), the calcite dissolution is less. 

---