Flux ratio equation

This simple experiment was important in that it clearly established the key notion that cellular extrusion of sodium ions by the sodium pump was coupled to metabolism. Because in this and subsequent experiments of the same sort the electrochemical gradient for sodium was known precisely, and since the fluxes of sodium (and later potassium) both into and out of the cell could be measured independently, this study also laid the groundwork for a theoretical definition of active transport, a theory worked out independently by Ussing in the flux ratio equation for transepithelial active transport of ions (see below). 

We begin by showing how active transport can directly affect membrane potentials. We then compare the temperature dependencies of metabolic reactions with those for diffusion processes across a barrier to show that a marked enhancement of solute influx caused by increasing the temperature does not necessarily indicate that active transport is taking place. Next we will consider a more reliable criterion for deciding whether fluxes are passive or not — namely, the Ussing-Teorell, or flux ratio, equation. We will then examine a specific case in which active transport is involved, calculate the energy required, and finally speculate on why K+ and Cl- are actively transported into plant cells and Na+ is actively transported out. 

Local concentration ratio at the permeate-membrane interface is given by the flux ratio (equation (7.2.2a)). 

Equation A may now be used to determine the diffusivity of sulfur dioxide in the gas mixture. The flux ratios may be determined from the reaction stoichiometry. 

Following the mass-flux equations developed for open-system magma ehambers by DePaolo (1981), we will define a flux ratio of Fe(II) into tbe pool relative to Fe(III) out of tbe pool as ... 

As long as this has not been done, the author (96) prefers, as regards the co-ions, the theoretical flux ratios based on the Nemst-Planck equations. 

Transport through pervaporation membranes is produced by maintaining a vapor pressure gradient across the membrane. As in gas separation, the flux through the membrane is proportional to the vapor pressure difference [Equation (9.1)], but the separation obtained is determined by the membrane selectivity and the pressure ratio [Equation (9.11)]. Figure 9.7 illustrates a number of ways to achieve the required vapor pressure gradient. 

The key calculation results in the heat flux ratio, ratioflux (t) = q"a o/Qna (0> which could be expressed as a function of time. But to examine the validity of Equation 19.4 for nanocomposites, the heat flux ratio is presented as a function of the pyrolyzed depth, i.e., the thickness of the pyrolyzed depth. 

In this section, the correlation in the heat flux ratio versus the pyrolyzed depth given by Equation 19.14 is incorporated into the numerical model to predict the pyrolysis process of the PA6 nanocomposite at different heat fluxes and thicknesses. The boundary conditions remain as those given by Equations 19.8 and 19.9. However, the MLR is now calculated from the heat flux ratio correlation in Equation 19.14 as... 

Equation (3.31) is the standard form for the steady state flux though a simple reversible Michaelis-Menten enzyme. This expression obeys the equilibrium ratio arrived at above (b/a)eq = Keq = k+ k+2/(k- k-2), when Jmm(g, b) = 0. In addition, from the positive and negative one-way fluxes in Equation (3.30), we note that the relationship J+/J = Keq(a/b) = e AG/RT is maintained whether or not the system is in equilibrium. Thus, as expected, the general law of Equation (3.12) is obeyed by this reaction mechanism. 

Net flux for highly reversible reactions is proportional to reverse flux Near equilibrium (for AG <thermodynamic driving force J = —AAG, where X is called the Onsager coefficient [154, 155], When the near-equilibrium approximation AG RT holds, the flux ratio J+/J is approximately equal to 1. In this case Equation (3.12) is approximated ... 

The ratio of the influx of species) to its efflux, as given by Equation 3.25, can be related to the difference in its chemical potential across a membrane. This difference causes the passive flux ratio to differ from 1. Moreover, we will use the chemical potential difference to estimate the minimum amount... 

Similarly, the expected flux ratio is 0.20 for K+ and 0.000085 for Cl- (values given in Table 3-1, column 5). However, the observed influxes in the light equal the effluxes for each of these three ions (Table 3-1, columns 6 and 7). Equal influxes and effluxes are quite reasonable for mature cells of N. translucens, which are in a steady-state condition. On the other hand, if 7 n equals 7°ut, the flux ratios given by Equation 3.25 are not satisfied for K+, Na+, or Cl-. In fact, active transport of K+ and Cl- in and Na+ out accounts for the marked deviations from the Ussing-Teorell equation for N. translucens, as is summarized in Figure 3-13. 

Foley 1999). Several of these results represent the characteristics of the hydrological cycle near Manaus. Nishizawa and Koike (1992) slightly overestimated the annual evapotranspiration ratio, since both evapotranspiration and drainage are overestimated. Differences in results are due to different areas considered for the basin, different precipitation networks and methods of assessment (isolated stations, Manaus- only rain, the EOS-DNAEE network) and the methods used to determine the annual water balance, either using the ET = P - R approximation, or using the water vapor flux convergence equation, for the entire Amazon basin, or near Manaus only. For more discussion of these vales, see Matsuyama (1992). 

Equation 6.1.11 is useful when the flux ratios are known and constant along the... 

For situations where the stoichiometry of a chemical reaction controls the flux ratios it is preferable to proceed somewhat differently. Equations 7.2.9 may be combined in the form of Eq. 7.2.15 with given by... 

In problems where the flux ratios are known (e.g., condensation and heterogeneous reacting systems where the reaction rate is controlled by diffusion) the mole fractions at the interface are not known in advance and it is necessary to solve the mass transfer rate equations simultaneously with additional equations (these may be phase equilibrium and/or reaction rate equations). For these cases it is possible to embed Algorithms 8.1 or 8.2 within another iterative procedure that solves the additional equations (as was done in Example 8.3.2). However, we suggest that a better procedure is to solve the mass transfer rate equations simultaneously with the additional equations using Newton s method. This approach will be developed below for cases where the mole fractions at both ends of the film are known. Later we will extend the method to allow straightforward solution of more complicated problems (see Examples 9.4.1, 11.5.2, 11.5.3, and others). 

Equation (2.3-55) is in the form of a rate being governed by two resistances in series—diffusion and chemical reaction. If I k SIOAB (fast surface reaction), die rale is governed by diffusion, while if Ilk 6/Dar (slow reaction rate), the rate is governed by cheraical kinetics. This additivity of resistances is only obtained when linear expressions relate rates and driving forces and wonld not be obtained, for example, if Ihe surface reaction kinetics were second order. More complex kinatic situations can be analyzed in a similar fashion where reaction stoichiometry at the surface provides information on (be flux ratio of various species. 

The flux of reduced carbon should be consistent with the observed ratio for carbonate to organic carbon in sedimentary rocks. Recall that carbonate carbon is concentrated in limestones, organic carbon in shales, and assume for simplicity that both types of rocks are weathered at the same rate. For steady-state conditions, the flux balance equations then read... 

Furthermore, according to water mass conservation equation, porosity does not affect liquid fluxes ratio. Therefore, darcean liquid flow still governs saturation kinetics. As can be seen on table 2, Qm" and Cim" calculations give the same saturation time, whereas in Cim" the saturation phenomenon is accelerated. As for thermal-hydraulics calculations, full saturation of the EB is reached earlier when both the heating source is being activated and liquid dynamics viscosity depends on temperature. Once again, this acceleration is only due to water dynamic viscosity decrease while heating. 

The difiusivity is thus dependent upon concentration and flux ratio, and strictly speaking we should not see a theory of catalyst effectiveness based on constant diflfusivity. Efowever, this is more detail than it is worth a good approximation is obtained from the limit of equation (7-72) for constant pressure conditions and Ab = —Na- Then... 

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