Approaching Equilibrium

We demand now that the two gases in Fig. 6.2 form an isolated system and we want a free exchange of entropy between the two chambers. The respective mol numbers we treat as constant, dn = 0 and d = 0. The isolated system means that the total energy is constant. We formulate three equations of constraint  

In the set of equations of (6.15), we have fixed the total energy Utot, the total entropy Stot, and the total volume Vtot. We have three equations and four variables S, V, S , V . For this reason, one variable may be fixed deliberately. On the other hand, the variables may move in some way. This means that the system may approach equilibrium. However, we show that the movement to equilibrium may occur only in special cases. We presume that the condition of equilibrium would be equal temperature and equal pressure  

These thermodynamic functions are not fixed, but must be simply equal. 

Example 6.1. We consider a special example. Allow the general energy functions of the gases to be U S, V) = exp(5)V , thus 

Here we have suppressed the mol number and the constant that establishes the physical dimension of an energy on the right-hand side. Further, we measure the entropy in units of the heat capacity. The temperature and the pressure would be in general 

---

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

In physical systems, in the absence of external forces, A and B approach equilibrium ... 

In many molecular dynamics simulations, equilibration is a separate step that precedes data collection. Equilibration is generally necessary lo avoid introducing artifacts during the healing step an d to en su re th at the trajectory is aciii ally sim u laiin g eq u i librium properties. The period required for equilibration depends on the property of Interest and the molecular system. It may take about 100 ps for the system to approach equilibrium, but some properties are fairly stable after 1 0-20 ps. Suggested tim es range from. 5 ps to nearly 100 ps for medium-si/ed proteins. 

We also give calculations of the performance of some of these various gas turbine plants. Comparison between such calculations is often difficult, even spot calculations at a single condition with state points specified in the cycle, because of the thermodynamic assumptions that have to be made (e.g. how closely conditions in a chemical reformer approach equilibrium). Performance calculations by different inventors/authors are also dependent upon assumed levels of component performance such as turbomachinery polytropic efficiency, required turbine cooling air flows and heat exchanger effectiveness if these are not identical in the cases compared then such comparisons of overall performance become invalid. However, we attempt to provide some performance calculations where appropriate in the rest of the chapter. 

An LTX unit is not a very efficient stabilizer because the absence of trays or packing keeps the two phases from approaching equilibrium at the various temperatures that exist in the vessel. In addition, it is difficult to control the process. Typically, for a 100-psi to 200-psi operating pressure, a 300°F to 400°F bottoms temperature is required to stabilize completely the condensate. The heating coil in an LTX separator is more Uke-... 

For a Michaelis-Menten reaction, ki = 7X10VAf- sec, /f i = 1 X lOVsec, and fe = 2 X lOVsec. What are the values of Ks and Does substrate binding approach equilibrium or does it behave more like a steady-state system ... 

There are many ways to approach equilibrium in the N204-N02 system. Table 12.2 gives data for three experiments in which the original conditions are quite different. Experiment 1 is that just described, starting with pure N204. Experiment 2 starts with pure N02 at a partial pressure of 1.00 atm. As equilibrium is approached, some of the N02 reacts to form N204 ... 

Example 12.4 illustrates a principle that you will find very useful in solving equilibrium problems throughout this (and later) chapters. As a system approaches equilibrium, changes in partial pressures of reactants and products—like changes in molar amounts—are related to one another through the coefficients of the balanced equation. 

In bulb B we approached equilibrium from a higher temperature (which favors NOj) then redaction (5) predominated at first. Using the same sort of argument we applied to bulb A, we see jthat as time progresses, reaction (4) becomes more and more rapid (as N204 is produced) and reaction (5) becomes slower (as N02 is used up). Finally, when the rates become equal, equilibrium is reached and the equilibrium expression (6) is applicable in bulb B. 

Consider two reactions for which ° shows that products are favored, one an exothermic reaction, and the other an endothermic reaction. For the exothermic reaction, when the reactants are mixed they are driven toward equilibrium in accord with the tendency toward minimum energy. Now contrast the endothermic reaction for which ° shows that equilibrium favors products. When these reactants are mixed, they approach equilibrium against the tendency toward minimum energy (since heat is absorbed). This reaction is driven by the tendency toward maximum randomness. 

The requisite condition for neglecting the back reaction is [P]2[Q]/[A][B]2 overall reaction can be said to be irreversible. Note that a product, P, retards the rate even though the overall reaction is irreversible. This point is important retardation by a product does not necessarily mean that the system is approaching equilibrium. Here, the product lowers the reaction rate by diverting some proportion of the intermediate back to reactant. 

The anomalous increase of the water uptake observed in Fig. 10 when approaching equilibrium at 60 °C has been associated to the damage. The abrupt upturn of the sorption curve may be explained considering a possible crazing of the low crosslinked internodular matrix induced by the differential swelling stresses that can arise, at high water contents, between areas of different crosslinking density. 

Gibbs free energy of reaction depends on the composition of the reaction mixture and how it changes as the reaction approaches equilibrium. 

BrCl(g), K = 0.2. Construct a plot of the Gibbs free energy of this system as a function of partial pressure of BrCl as the reaction approaches equilibrium. 

If we were to place a piece of zinc metal into an aqueous copper(II) sulfate solution, we would see a layer of metallic copper begin to deposit on the surface of the zinc (see Fig. K.5). If we could watch the reaction at the atomic level, we would see that, as the reaction takes place, electrons are transferred from the Zn atoms to adjacent Cu2 r ions in the solution. These electrons reduce the Cu2+ ions to Cu atoms, which stick to the surface of the zinc or form a finely divided solid deposit in the beaker. The piece of zinc slowly disappears as its atoms give up electrons and form colorless Zn2+ ions that drift off into the solution. The Gibbs free energy of the system decreases as electrons are transferred and the reaction approaches equilibrium. However, although energy is released as heat, no electrical work is done. 

Figure 6.3 shows a free energy profile for a reaction in which B is thermodynamically more stable than C (lower AG), but C is formed faster (lower AG ). If neither reaction is reversible, C will be formed in larger amount because it is formed faster. The product is said to be kinetically controlled. However, if the reactions are reversible, this will not necessarily be the case. If such a process is stopped well before the equilibrium has been established, the reaction will be kinetically controlled since more of the faster-formed product will be present. However, if the reaction is permitted to approach equilibrium, the predominant or even exclusive product will be B. Under these conditions, the C that is first formed reverts to A, while the more stable B does so much less. We say the product is thermodynamically controlled.Of course. Figure 6.3 does not describe all reactions in which a Compound A can give two different products. In many cases, the more stable product is also the one that is formed faster. In such cases the product of kinetic control is also the product of thermodynamic control. 

Determine t and Ugut for a CSTR that approaches equilibrium within 5% that is. 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

Solution A rigorous treatment of a reversible reaction with variable physical properties is fairly complicated. The present example involves just two ODEs one for composition and one for enthalpy. Pressure is a dependent variable. If the rate constants are accurate, the solution will give the actual reaction trajectory (temperature, pressure, and composition as a function of time). If ko and Tact are wrong, the long-time solution will still approach equilibrium. The solution is then an application of the method of false transients. 

A zeolite catalyst operated at 1 atm and 325-500 K is so active that the reaction approaches equilibrium. Suppose that stack gas having the equilibrium composition calculated in Example 7.17 is cooled to 500 K. Ignore any reactions involving CO and CO2. Assume the power plant burns methane to produce electric power with an overall efficiency of 70%. How much ammonia is required per kilowatt-hour (kWh) in order to reduce NO , emissions by a factor of 10, and how much will the purchased ammonia add to the cost of electricity. Obtain the cost of tank car quantities of anhydrous ammonia from the Chemical Market Reporter or from the web. 

Thermal Effects in Addition Polymerizations. Table 13.2 shows the heats of reaction (per mole of monomer reacted) and nominal values of the adiabatic temperature rise for complete polymerization. The point made by Table 13.2 is clear even though the calculated values for T dia should not be taken literally for the vinyl addition polymers. All of these pol5Tners have ceiling temperatures where polymerization stops. Some, like polyvinyl chloride, will dramatically decompose, but most will approach equilibrium between monomer and low-molecular-weight polymer. A controlled polymerization yielding high-molecular-weight pol)mier requires substantial removal of heat or operation at low conversions. Both approaches are used industrially. 

The major species in an aqueous solution determine which categories of equilibria are important for that solution. Each major species present in the solution must be examined in light of these general categories. Are any of the major species weak acids or weak bases Are there ions present that combine to form an insoluble salt Do any of the major species participate in more than one equilibrium Any chemical reaction can approach equilibrium from either direction. Consequently, there are six different t q)es of aqueous equilibria in which major species are reactants ... 

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.)... 

Even if a system is not in chemical equilibrium it is possible to calculate the rate at which it is approaching equilibrium if we have sufficiently detailed knowledge of the energies involved in the transition state (so that it is possible to calculate the partition functions - the crucial step). However, computational chemistry has advanced to a level that good estimates of reaction rates can almost be obtained routinely. We will discuss examples in Chapter 6. 

Figure 8.22. Schematic drawing of an adiabatic two-bed radial flow reactor. There are three inlets and one outlet. The major inlet comes in from the top (left) and follows the high-pressure shell (which it cools) to the bottom, where it is heated by the gas leaving the reactor bottom (left). Additional gas is added at this point (bottom right) and it then flows along the center, where even more gas is added. The gas is then let into the first bed (A) where it flows radially inward and reacts adiabatically whereby it is heated and approaches equilibrium (B). It is then cooled in the upper heat exchanger and move on to the second bed (C) where it again reacts adiabatically, leading to a temperature rise, and makes a new approach to equilibrium (D). (Courtesy of Haldor Topspe AS.)...

Dekant et al. 1986b Filser and Bolt 1979 Prout et al. 1985). Male mice can metabolize inhaled trichloroethylene to a greater extent than male rats (Stott et al. 1982). In this study, virtually 100% of the net trichloroethylene uptake by mice was metabolized at both 10- and 600-ppm exposure concentrations, and there was no evidence of metabolic saturation. In rats, however, 98% of the net trichloroethylene uptake from the 10-ppm exposure was metabolized, but only 79% was metabolized at the 600-ppm exposure level. This suggested an incremental approach to the saturation of metabolism in this exposure range in the rat. Rats exposed by inhalation to trichloroethylene concentrations of 50 or 500 ppm for 2 hours showed metabolic saturation at 500 ppm (Dallas et al. 1991). This was indicated by the fact that the trichloroethylene blood levels of the 500-ppm animals progressively increased over the 2-hour period, rather than approaching equilibrium after 25 minutes, as was the case at 50 ppm. 

Excluding gold complexes that caimot transverse the membrane such as AuSTm, intracellular gold concentrations must approach equilibrium with extracellular gold concentrations if active transport is not at work. The balance of gold between the environment of a cell and within the cell will depend on the ligands available. 

---