Periodic equilibrium curve

We start by noting that in the general case of a nonlinear equilibrium relation, the adsorption and desorption periods will always differ because of the different vines of Aq/AY and H (Eigure 7.22). Because the latter quantity is invariably the greater of the two, we conclude that for nonlinear isotherms the adsorption step will proceed at a faster pace than the corresponding desorption step. This difference becomes more pronounced, the steeper the equilibrium curve at the origin. 

In the distillation process, it is assumed that the vapor formed within a short period is in thermodynamic equilibrium with the liquid. Hence, the vapor composition y is related to the liquid composition x by an equilibrium relation of the functional form y = f x). The exact relationship for a particular mixture may be obtained from a thermodynamic analysis and is also dependent upon temperature and pressure. Figure 4.1 shows an example equilibrium curve for a system consisting of CS2 and CCI4 at 1 atmosphere pressure. 

Figure B2.5.4. Periodic displacement from equilibrium through a sound wave. The frill curve represents the temporal behaviour of pressure, temperature, and concentrations in die case of a very fast relaxation. The other lines illustrate various situations, with 03Xj according to table B2.5.1. 03 is the angular frequency of the sound wave and x is the chemical relaxation time. Adapted from [110].

FIG. 16-33 Elution curves under trace linear equilibrium conditions for different feed loading periods audN = 80. Solid lines, Eq. (16-172) dashed line, Eq. (16-174) for Tp= 0.05. 

The period after which this can be repeated will depend upon the heating curve and the thermal time constant of the motor, i.e. the time the motor will take to reach thermal equilibrium after repeated starts (See Chapter 3). 

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

During the lifetime of a root, considerable depletion of the available mineral nutrients (MN) in the rhizosphere is to be expected. This, in turn, will affect the equilibrium between available and unavailable forms of MN. For example, dissolution of insoluble calcium or iron phosphates may occur, clay-fixed ammonium or potassium may be released, and nonlabile forms of P associated with clay and sesquioxide surfaces may enter soil solution (10). Any or all of these conversions to available forms will act to buffer the soil solution concentrations and reduce the intensity of the depletion curves around the root. However, because they occur relatively slowly (e.g., over hours, days, or weeks), they cannot be accounted for in the buffer capacity term and have to be included as separate source (dCldl) terms in Eq. (8). Such source terms are likely to be highly soil specific and difficult to measure (11). Many rhizosphere modelers have chosen to ignore them altogether, either by dealing with soils in which they are of limited importance or by growing plants for relatively short periods of time, where their contribution is small. Where such terms have been included, it is common to find first-order kinetic equations being used to describe the rate of interconversion (12). 

Figure 12. The diffusive modes of the periodic Yukawa-potential Lorentz gas represented by their cumulative function depicted in the complex plane ReFk,hnFk) for two different nonvanishing wavenumbers k. The horizontal straight line is the curve corresponding to the vanishing wavenumber k = 0 at which the mode reduces to the invariant microcanonical equilibrium state.

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