Closed systems equilibria

In a strict sense, the term Rayleigh fractionation should only be used for chemically open systems where the isotopic species removed at every instant are in thermodynamic and isotopic equilibrium with those remaining in the system at the moment of removal. Furthermore, such an ideal Rayleigh distillation is one where the reactant reservoir is finite and well mixed, and does not re-react with the product (Clark and Fritz, 1997). However, the term Rayleigh fractionation is commonly applied to equilibrium closed systems and kinetic fractionations as well because the situations may be computationally identical. Isotopic fractionations are strongly affected by whether a system is open or closed. 

Because a closed system must eventually reach equilibrium, closed systems can sustain oscillating chemical reactions for only a limited time. Sustained oscillating reactions require an open system with a constant influx of reactants, energy and removal of products. 

Characteristic pressure of a vapor above a confined liquid or solid when they are in dynamic equilibrium closed system)... 

Thermodynamics. Consider an equilibrium closed system of a liquid, L 2, in contact with a plane surface of an isotropic solid, S (which is not soluble in L 2), and saturated vapor, Vj, of liquid L2. The Young-Dupr equation is... 

From all possible statistical systems the equilibrium closed systems are allocated. For such systems the physical theory, referred to as equilibrium thermodynamics or simply thermodynamics, is well developed. Thermodynamics is the phenomenological doctrine of heat. Classical thermodynamics asserts that the isolated thermodynamic system cannot spontaneously change it s state . This statement is sometimes referred to as the zeroth law of thermodynamics, another assertion of which is that If two thermodynamic systems are in thermodynamic equilibrium with some third body they are in thermodynamic equilibrium with each other . 

An explicit example of an equilibrium ensemble is the microcanonical ensemble, which describes closed systems with adiabatic walls. Such systems have constraints of fixed N, V and E < W< E + E. E is very small compared to E, and corresponds to the assumed very weak interaction of the isolated system with the surroundings. E has to be chosen such that it is larger than (Si )... 

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

Fig. 4. Drydown path to equilibrium ia a closed system. A represents the equiUbrium curve and B, the material balance line.

For a closed system which interacts with its surroundings, a final static condition may likewise be reached such that the system is not only internally at equihbrium but also in external equilibrium with its surroundings. 

Although derived for a reversible process, this equation relates properties only and is valid for any change between equilibrium states in a closed system. It may equally well be written... 

For the case of equilibrium with respect to chemical reaciion within a single-phase closed system, combination of Eqs. (4-16) and (4-271) leads immediately to... 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

Process Applications The production of esters from alcohols and carboxylic acids illustrates many of the principles of reactive distillation as applied to equilibrium-limited systems. The equilibrium constants for esterification reactions are usually relatively close to unity. Large excesses of alcohols must be used to obtain acceptable yields with large recycles. In a reactive-distiUation scheme, the reac-... 

The vapor pressure (P ) of a pure liquid at a given temperature (T) is the pressure exerted by its vapor in equilibrium with the liquid phase in a closed system. All liquids and solids exhibit unique vapor pressure-temperature curves. For instance, in Figure 2-79, lines BA and AC represent the equilibrium vapor pressure curves of the solid and liquid phases, respectively. 

Now we can give a complete statement about recognizing equilibrium equilibrium is recognized by the constancy of macroscopic properties in a closed system at a uniform temperature. 

Consider two separate closed systems, each at equilibrium ... 

The near-equilibrium desorption in a closed system (dP/dt dn,/dt) was used in practice, for example, by Procop and Volter (45h) and by Dawson and Peng (98). 

Equations (5.63) and (5.64) are actually more general than is apparent from the derivation. Consider a closed system at a given temperature and pressure with /t , moles of the components 1.2,3,... distributed among the phases A,B, C,... For the flow of mass between the phases due to an infinitesimal reversible (equilibrium) displacement we can write... 

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. 

Generally, in a system that is energetically and materially isolated from the environment without a change in volume (a closed system), the entropy of the system tends to take on a maximum value, so that any macroscopic structures, except for the arrangement of atoms, cannot survive. On the other hand, in a system exchanging energy and mass with the environment (an open system), it is possible to decrease the entropy more than in a closed system. That is, a macroscopic structure can be maintained. Usually such a system is far from thermodynamic equilibrium, so that it also has nonlinearity. 

Chemical equilibria with reactants and products that are all in the same phase are called homogeneous equilibria. Equilibria C, D, and E are homogeneous. Equilibria in systems having more than one phase are called heterogeneous equilibria. Equilibrium F is heterogeneous so too is the equilibrium between water vapor and liquid water in a closed system ... 

Assuming that the carbon cycle of Fig. 4-12 will remain a closed system over several thousands of years, we can ask how the equilibrium distribution within the system would change after the introduction of a certain amount of fossil carbon. Table 4-2 contains the answer for two different assumptions about the total input. The first 1000 Pg corresponds to the total input from fossil fuel up to about the year 2000 the second (6000 Pg) is roughly equal to the now... 

Schematic view of evaporation and condensation in a closed system, (a) Initial conditions, (b) Intermediate conditions, (c) Equilibrium conditions partial pressure = vapor pressure.

The easiest of the colligative properties to visualize is the effect of solute molecules on the vapor pressure exerted by a liquid. In a closed system, the solvent and its vapor reach dynamic equilibrium at a partial pressure of solvent equal to the vapor pressure. At this pressure, the rate of condensation of solvent vapor equals the rate of evaporation from the liquid. 

In our calculation we assume that the gas mixture approaches equilibrium under conditions where the pressure is constant. This situation corresponds, for instance, to a volume of gas moving through a plug flow reactor with a negligible pressure drop. (Note that if the ammonia synthesis were carried out in a closed system, the pressure would decrease with increasing conversion.)... 

As we all know from thermodynamics, closed systems in equilibrium have minimum free energy and maximum entropy. If such a system were brought out of equilibrium, i.e. to a state with lower entropy and higher free energy, it would automatically decay to the state of equilibrium, and it would lose all information about its previous states. A system s tendency to return to equilibrium is given by its free energy. An example is a batch reaction that is run to completion. 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

They calculated the change in 8 0 values of hydrothermally altered volcanic rocks as a function of water to rock ratio by weight and temperature, assuming that oxygen isotopic equilibrium is attained in a closed system, and demonstrated that the increase in 8 0 values of altered andesitic rocks from the veins towards peripheral zones can be interpreted as a decrease in temperature from the vein system (Fig. 1.135). In their calculations, the effect of mixing of hydrothermal solution with groundwater was not considered. 

Secular equilibrium materials. For materials that have remained a closed system for sufficient time that secular equilibrium has been achieved, the half-lives of nuclides within the decay chain can be calculated from the relationship A,pP = A,dD. If the atom ratio P/D is measured, and one of the decay constants is well known, then the other can be readily calculated. Limitations on this approach are the ability to measure the atom ratios to sufficient precision, and finding samples that have remained closed systems for a sufficient length of time. This approach has been used to derive the present recommended half lives for °Th and (Cheng et al. 2000 Ludwig et al. 1992). 

Figure 4. Evolution of the (N2/N1) ratio in a reservoir in the two cases of closed system evolution (as a function of t/T2, where t is the time since fractionation), or in an open-system, steady-state reservoir (the steady-state (N2/N1) ratio is plotted as a function of x/ T2, where x is the residence time of the magma in the reservoir). Initial fractionation results in an arbitrarily chosen ratio of 2, which is kept constant for the iirfluent magma in the continuously replenished reservoir. The diagram shows that radioactive equilibrium is reached sooner in a closed system evolution. It also illustrates the fact that the radioactive parent-daughter pair should be chosen such as T2 is commensmate with the residence time of the magma in the reservoir (e.g., x/ T2 between 0.1 and 10). If T2 is much longer than the residence time x, then the (N2/N1) ratio will remain close to the initial value (here 2). If T2 is much shorter than x, equilibrium will be nearly established in the reservoir.

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