Chemical potentials, components

In the mitochondrion ApH is about 1.4, thereby giving a value of 0.06 x 1.4 = 0.084V for the chemical potential component, and Ai ) is about 0.14V thus the PMF has a value at 30 °C of about 0.224V. Ibis corresponds to a AG value of about 21.6 kJ per mole of H transported, as can be calculated by insertion of the ApH and Atti values into equation (vii). Hius it can be deduced that at least two moles of H have to be transported from the mitochondrial matrix to the intermembrane space by exergonic flow of electrons down the electron transport chain to drive the ender-gonic generation of one mole of ATP (ADP + P -> ATP + HjO AG" = + 30.5 kj.mol ). 

It follows from Eq. (73) that in the open molecular syst s, e.g., an adsorbate, there is also the "reservoir" (catalyst) chemical potential component of respmses in nuclear positions, in addition to the usual (closed-system) force component of such an equilibrium relaxation of the syston geometry. 

The chemical potential pi, has been generalized to the electrochemical potential Hj since we will be dealing with phases whose charge may be varied. The problem that now arises is that one desires to deal with individual ionic species and that these are not independently variable. In the present treatment, the difficulty is handled by regarding the electrons of the metallic phase as the dependent component whose amount varies with the addition or removal of charged components in such a way that electroneutrality is preserved. One then writes, for the ith charged species. 

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

The grand canonical ensemble is a collection of open systems of given chemical potential p, volume V and temperature T, in which the number of particles or the density in each system can fluctuate. It leads to an important expression for the compressibility Kj, of a one-component fluid ... 

Then, since the chemical potential for a one-component system is just p. = (i= A+pV, a reduced chemical potential can be written in tenns of a reduced density p. = p/p ... 

The chemical potential is an example of a partial molar quantity /ij is the partial molar Gibbs free energy with respect to component i. Other partial molar quantities exist and share the following features ... 

The first point in developing the thermodynamic method is the observation that for equilbrium between two phases-say, a and 3-the chemical potential must be equal in both phases for all components ... 

The criterion for phase equilibrium is given by Eq. (8.14) to be the equality of chemical potential in the phases in question for each of the components in the mixture. In Sec. 8.8 we shall use this idea to discuss the osmotic pressure of a... 

Next suppose AS , and AH , are both positive. In this case these two partially offset one another, and a plot of AGj resembling that shown in Fig. 8.2b may result. We are particularly interested in the two minima in this curve and the hump between them. A common tangent can always be drawn to two such minima so the above discussion shows that the minima at points P and Q in Fig. 8.2b each have the same values of AjUi and A/i2. Since AjUj is simply the difference between juj and its value for the pure component, the chemical potential for each component is seen to have the same value for both solution P and solution Q in Fig. 8.2b. 

The stabiHty criteria for ternary and more complex systems may be obtained from a detailed analysis involving chemical potentials (23). The activity of each component is the same in the two Hquid phases at equiHbrium, but in general the equiHbrium mole fractions are greatiy different because of the different activity coefficients. The distribution coefficient m based on mole fractions, of a consolute component C between solvents B and A can thus be expressed... 

In general, equaHty of component fugacities, ie, chemical potentials, in the vapor and Hquid phases yields the foUowing relation for vapor—Hquid equiHbrium ... 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

An empirical formula, due to Pick, shows that, under simple cucumstances where the chemical potential of a component in a system is dehned by the equation... 

Another problem in the construction of tlrese devices, is that materials which do not play a direct part in the operation of the microchip must be introduced to ensure electrical contact between the elecuonic components, and to reduce the possibility of chemical interactions between the device components. The introduction of such materials usually requires an annealing phase in the construction of die device at a temperature as high as 600 K. As a result it is also most probable, especially in the case of the aluminium-silicon interface, that thin films of oxide exist between the various deposited films. Such a layer will act as a banier to inter-diffusion between the layers, and the transport of atoms from one layer to the next will be less than would be indicated by the chemical potential driving force. At pinholes in the AI2O3 layer, aluminium metal can reduce SiOa at isolated spots, and form the pits into the silicon which were observed in early devices. The introduction of a tlrin layer of platinum silicide between the silicon and aluminium layers reduces the pit formation. However, aluminium has a strong affinity for platinum, and so a layer of clrromium is placed between the silicide and aluminium to reduce the invasive interaction of aluminium. 

Mass transfer Irreversible and spontaneous transport of mass of a chemical component in a space with a non-homogeneous field of the chemical potential of the component. The driving force causing the transport can be the difference in concentration (in liquids) or partial pressures ( in gases) of the component. In biological systems. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

Let us consider an V-component fluid in a volume V, at temperature T, and at chemical potentials /r = mi, > Mv - The fluid is in contact with an impermeable solid surface. We assume that the fluid particles interact between themselves via the pair potential denoted by u pir), and interact with the confining surface via the potential (a,f3= 1,2,. ..,V). The potential v ir) contains a hard-wall term to ensure that the solid surface is impermeable. For the sake of convenience, the hard-wall term is assumed to extend into the bulk of the solid [46,47], such that the Boltzman factor (r), and the local density Pa r) are cutoff at a certain distance z = z, ... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

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. 

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