Equilibrium points

If the spring follows Hooke s law, the force it exerts on the mass is directly proportional and opposite to the excursion of the particle away from its equilibrium point Xe- The particle of mass m is accelerated by the force F = —kx of the spring. By Newton s second law, F = ma, where a is the acceleration of the mass... 

Although Eq. (10.50) is still plagued by remnants of the Taylor series expansion about the equilibrium point in the form of the factor (dn/dc2)o, we are now in a position to evaluate the latter quantity explicitly. Equation (8.87) gives an expression for the equilibrium osmotic pressure as a function of concentration n = RT(c2/M + Bc2 + ) Therefore... 

F is zero at the equilibrium point r = ro) however, if the atoms are pulled apart by distance (r - Tq) a resisting force appears. For small (r - Tq) the resisting force is proportional to (r - rg) for all materials, in both tension and compression. 

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

Thesolutionsaremorecomplexthanwehaveseenbefore,butthecheckwehaveputthem throughindicatestheirvalidity.Thecomplexityarisesfromthefactthatthisproblemisone that is fully transient until the equilibrium point is reached. It is important to realize that there is a marked difference between equilibrium and steady state, as we will see when we examine flow reactors. We canhave a steady state in a flow reactor, which is far from... 

Repeat your analysis for tautomeric equilibria between 4-hydroxypyridine and 4-pyridone, 2-hydroxypyrimidine and 2-pyrimidone and 4-hydroxypyrimidine and 4-pyrimidone. For each, identify the favored (lower-energy) tautomer, and then use equation (1) to calculate the ratio of tautomers present at equilibrium. Point out any major differences among the four systems and rationalize what you observe. (Hint Compare dipole moments and electrostatic potential maps of the two pyridones and the two pyrimidones. How are these related to molecular stability )... 

Oil-canning The property of a panel that flexes past a theoretical equilibrium point, and then returns to the original position. This motion is analogous to the bottom of a metal oilcan when pressed and released. Part flexing can cause stress, fracturing, or undesirable melting of thin-sectioned, flat parts. 

Fig. 9-3 Conceptual model to describe the interaction between chemical weathering of bedrock and down-slope transport of solid erosion products. It is assumed that chemical weathering is required to generate loose solid erosion products of the bedrock. Solid curve portrays a hypothetical relationship between soil thickness and rate of chemical weathering of bedrock. Dotted lines correspond to different potential transport capacities. Low potential transport capacity is expected on a flat terrain, whereas high transport is expected on steep terrain. For moderate capacity, C and F are equilibrium points. (Modified with permission from R. F. Stallard, River chemistry, geology, geomorphology, and soils in the Amazon and Orinoco basins. In J. I. Drever, ed. (1985), "The Chemistry of Weathering," D. Reidel Publishing Co., Dordrecht, The Netherlands.)...

Solution Assume the reverse reaction has the form =kr c Setting the overall reaction rate equal to zero at the equilibrium point gives a second... 

Helgeson (1967) constructed an activity diagram depicting chemical equilibrium points (albite-sericite-K-feldspar and albite-sericite-Na-montmorillonite) of NazO-K20-Si02-Al203-H20 system at elevated temperatures. At these points,... 

Fig. 2.37. Phase diagram for Ca0-Na20 Si02-(Al203)-H20 system in equilibrium with quartz at 400°C and 400 bars. Plagioclase solid solution can be represented by the albite and anorthite fields, whereas epidote is represented by clinozoisite. Note that the clinozoisite field is adjacent to the anorthite field, suggesting that fluids with high Ca/(H+) might equilibrate with excess anorthite by replacing it with epidote. The location of the albite-anorthite-epidote equilibrium point is a function of epidote and plagioclase composition and depends on the model used for calculation of the thermodynamic properties of aqueous cations (Berndt et al., 1989).

THE HOPF BIFURCATION OR THE CHANGING NATURE OF EQUILIBRIUM POINTS PROBLEM OP WALAS... 

Mathematically, these are trajectories connecting equilibrium points of a system of autonomous ordinary differential equations. 

If this reaction is implemented at temperatures where the iron yielded is a solid, a sectioned, fractionally reacted iron oxide might appear as shown in Figure 3.26. As shown, for the hydrogen to reach the iron oxide with which it reacts, it has to diffuse through a layer of iron (mostly porous). In addition, the water vapour produced as a consequence of the reaction must be transported away from the iron-iron oxide interface by diffusion. Failing this, there will be accumulation of water vapour at the interface which will permit equilibrium point to be attained and the reaction ceases from further occurring. 

In a dialysis experiment, a dialysis bag containing the dissolved humic materials is placed in a solution of a pollutant (preferably radiolabeled). The dialysis tubing is chosen so the pollutant is free to diffuse through the bag while the humic materials are retained inside the bag. The solution is shaken at constant temperature until it comes to an equilibrium point. At equilibrium, the pollutant inside the dialysis bag consists of two fractions that truly dissolved and the bound to the humic materials. The concentration of pollutant on the outside of the dialysis bag consists only of the free, truly dissolved fraction. Any increase of the pollutant concentration on the inside of the dialysis bag is due to binding by dissolved humic materials. A series of dialysis experiments, therefore, can measure the bound fraction concentration as a function of the free concentration. 

A catalyst cannot change the ultimate equilibrium point set by thermodynamics, but it can affect the rate at which this point is approached. However, it can facilitate approach to equilibrium with respect to a desired reaction while not influencing the rates of other less desirable reactions. In optimizing yields of desired products, chemical engineers are very concerned with the selectivity or specificity of a catalyst. For commercial applications, selectivity is often more important than activity per se. 

Step (18) in the above is the analog of step (8), which is required for H2—D2 equilibration it is a necessary step if we view the jr-allyl as an immobile species on the surface. The products of step (19) can be viewed as propylene in the form of a loosely held w-complex which on desorption yields isomerized propylene. Readsorption of the isomerized propylene or further reaction of the x-complex would yield surface OD groups. When equilibrium is achieved, the concentration of surface OD groups should equal 40% of the initial concentration of OH groups. Figure 21 shows a plot versus time of the intensity (multiplied by a scale factor to yield concentration) of the surface OH and OD. The expected equilibrium points are indicated by arrows. Corresponding data for CD3—CH=CH2 are also shown. Except for the OH species from CD3—CH=CH2, which is a relatively weak band on the side of a surface hydroxyl, the curves approach the expected value. 

The transfer of the electron takes place very rapidly compared to nuclear motion, and will only take place when the combination of internal and librational coordinates is such that the curves interact. Thus, the [Fe(H20)6] + species must first distort and/or experience a dipole moment field from the instantaneous positions of the water molecules such that it attains the cross-over point. At this point, the electron may tunnel from the [Fe(H20)6]2+ ion to the metal, leaving behind an [Fe(H20)6]3 + ion with a non-equilibrium geometry, This then relaxes by heat transfer to the solvent to the equilibrium point, q0. 

Aqueous geochemists work daily with equations that describe the equilibrium points of chemical reactions among dissolved species, minerals, and gases. To study an individual reaction, a geochemist writes the familiar expression, known as the mass action equation, relating species activities to the reaction s equilibrium constant. In this chapter we carry this type of analysis a step farther by developing expressions that describe the conditions under which not just one but all of the possible reactions in a geochemical system are at equilibrium. 

Fig. 3.1. Variation in free energy G with reaction progress for the reaction bB + cC dD + eE. The reaction s equilibrium point is the minimum along the free energy curve.

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

Knowing the chemical potential function for each species in a reaction defines the reaction s equilibrium point. Consider a hypothetical reaction,... 

We can find the reaction s equilibrium point from Equation 3.3 as soon as we know the form of the function representing chemical potential. The theory of ideal solutions (e.g., Pitzer and Brewer, 1961 Denbigh, 1971) holds that the chemical potential of a species can be calculated from the potential pg of the species in its pure form at the temperature and pressure of interest. According to this result, a species chemical potential is related to its standard potential by... 

Here, R is the gas constant, 7k is absolute temperature, and XB is the mole fraction of B in the solution phase. Using this equation, we can calculate the equilibrium point of reactions in ideal systems directly from tabulated values of standard potentials p°. 

Despite the authority apparent in its name, no single rate law describes how quickly a mineral precipitates or dissolves. The mass action equation, which describes the equilibrium point of a mineral s dissolution reaction, is independent of reaction mechanism. A rate law, on the other hand, reflects our idea of how a reaction proceeds on a molecular scale. Rate laws, in fact, quantify the slowest or rate-limiting step in a hypothesized reaction mechanism. 

In considering the energetics of a microbially catalyzed reaction, it is important to recall that progress of the redox reaction (e.g., Reaction 18.7) is coupled to synthesis of ATP within the cell, so the overall reaction is the redox reaction combined with ATP synthesis. The free energy liberated by the overall reaction is the energy liberated by the redox reaction, less that consumed to make ATP. The overall reaction s equilibrium point is where this difference vanishes at this point,... 

Several chemical geothermometers are in widespread use. The silica geothermometer (Fournier and Rowe, 1966) works because the solubilities of the various silica minerals (e.g., quartz and chalcedony, Si02) increase monotonically with temperature. The concentration of dissolved silica, therefore, defines a unique equilibrium temperature for each silica mineral. The Na-K (White, 1970) and Na-K-Ca (Fournier and Truesdell, 1973) geothermometers take advantage of the fact that the equilibrium points of cation exchange reactions among various minerals (principally, the feldspars) vary with temperature. 

Let us assume the existence of a Taylor series for the Gibbs energy at the equilibrium point. This implies that the Gibbs energy and all its derivatives vary continuously at this point. The Taylor series is given as... 

---