Grand equilibrium method

Grand equilibrium method application to the calculation of solubility of gases in polystyrene... 

In the grand equilibrium method, a simulation of the condensed phase is done to calculate the excess chemical potentials, /x,ex, and the partial molar volumes, V,-, of all components. One may use the test-particle insertion method [59] to calculate the excess chemical potentials and the partial molar volumes as... 

Another technique to determine the vapor-liquid equilibrium of pure substances or mixtures, which has some similarities with what is described in [190, 204-206], is the grand equilibrium method [192]. It is a two-step procedure, where the coexisting phases are simulated independently and subsequently. In the first step, one NpT simulation of the liquid phase is performed to determine the chemical potentials p] and the partial molar volumes v of all components i. These entropic properties can be determined by Widom s test molecule method [207] or more advanced techniques, such as gradual inserticMi [208-210] (see below). On the basis of the chemical potentials and partial molar volumes at a specified pressure po, first order Taylor expansions can be made for the pressure dependence ... 

Both the GEMC and the grand equilibrium method have been applied to evaluate vapor-liquid equilibrium data for ammonia. Kristdf et al. [246] calculated the vapor pressure and saturated densities using the force field by Impey and Klein [108] and found systematic deviations from experimental data cf. Fig. 3. Therefore, they proposed a new ammonia force field that was optimized to vapor-liquid equilibria [246], achieving a better accuracy. Simulated saturated densities and enthalpies based on this force field agree with the experimental data within 1 and 3%, respectively. However, it shows a mean deviation of 13% from experimental... 

Vrabec J, Hasse H (2002) Grand equilibrium vapour-liquid equilibria by a new molecular simulation method. Mol Phys 100 3375-3383... 

For the case of equilibrium methods, for mesoporous materials gas porosimetry is complemented by Small Angle Neutron Scattering to obtain information on pore size distribution. For microporous membranes the extraction of structural information from the equilibrium sorption measurements can be based on techniques like Grand Canonical Monte Carlo Simulation. 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

The grand canonical ensemble is appropriate for adsorption systems, in which the adsorbed phase is in equilibrium with the gas at some specified temperature. The use of a computer simulation allows us to calculate average macroscopic properties directly without having to explicitly calculate the partition function. The grand canonical Monte Carlo (GCMC) method as applied in this work has been described in detail earlier (55). The aspects involving binary fluid mixtures have been described previously in our Xe-Ar work (30). 

Example 4.32 Column grand composite curves in methanol plant Table 4.16 describes the existing base case operations for columns 1 and 2 of the methanol plant obtained from the converged simulations using the RKS equation of state to estimate the vapor properties. The activity coefficient model, NRTL, and Henry components method are used for predicting the equilibrium and liquid properties. 

Although the capillary tube method has recently fallen into disfavour, which is unfortunate, it can with careful work give very accurate results, and some of the best values of g have been found by this method. The tube (which may be freshly drawn) must be free from grease Volkmann washed the tube with potash solution and then distilled water (the chromic-sulphuric acid mixture should always be avoided). Schultzei said the walls of the tube should first be wetted, to fill the capillary pores of the glass surface. The rate of rise of a liquid in a capillary was studied by Le Grand and Reuse. Muller 12 determined surface tension and viscosity simultaneously from the rate of flow back to the equilibrium position in a capillary. 

The Molecular Layer Structure Theory (MLST) was first developed [4, 5] to study the vapour liquid equilibrium and surface tension of many substances over a wide range of temperature. In this method, the fluid is considered as parallel molecular layers, whose surface densities are different for inhomogeneous fluids. Similar to the DFT method, we define the following grand potential ... 

The transport of a sub-critical Lennard-Jones fluid in a cylindrical mesopore is investigated here, using a combination of equilibrium and non-equilibrium as well as dual control volume grand canonical molecular dynamics methods. It is shown that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. It is also demonstrated that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities. 

Cerius2 (MSI Inc.) was used throughout the simulations. Forcefield parameters obtained by Mellot et all. [3] are listed in Table 1. Ilie Grand Canonical Monte Carlo method (under constant chemical potential (p), volume (V), temperature (T)) was used to get the equilibrium amount adsorbed. 

The adsorption equilibria of methane, ethane and their mixture into single-walled carbon nanotuhes (SWNTs) were studied by using a Grand Canonical Monte Carlo (GCMC) method. The equilibrium isotherms of methane and ethane and the selectivity from their equimolar mixture were reported. 

Up to now, numerous studies have been conducted on their synthesis [9,10], treatment [5,13] and physical properties [4], However only limited number of studies has been carried out on die adsorption of gas in CNTs, including experimental works [8,11] and molecular simulations [3,7,14-lS]. Adsorption behavior depends strongly on the microporous structure of the particular adsorbent. In this work the effect of pore size on the adsorption behavior is of interest. The adsorption equilibria of methane, ethane and their mixture into SWNTs were studied by using a Grand Canonical Monte Carlo (GCMC) method. We reported equilibrium isotherms of methane and ethane, and the selectivity from their equimolar mixture. 

The Horvath and Kawazoe (HK) method [39] was developed to determine the PSD of active carbons from nitrogen adsorption isotherm. All pores are assumed to have slit shape. This method rests on the assumption that the adsorption state of a pore is either empty or completely fiUed. The demarcation pressure between these two states is called the pore-filling pressure, and it is a function of pore width. The equilibrium of a pore exposed to a bulk phase of constant chemical potential is obtained from the minimization of the following grand thermodynamic potential ... 

The system is said to be at equilibrium when the grand thermodynamic potential is a minimum. To perform this minimization, we need to determine the molecular Helmholtz free energy, and this is the crucial part of the DFT method as we shall show below. The molecular Helmholtz free energy/(r) may be expressed as a sum of four contributions ... 

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

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