Equilibrium versus Steady State

The spatial variation of concentration of the suspended particles is time-independent when the right-hand side of Equation 6.66 is zero. This can arise when either the flux J is zero or a constant. In equilibrium, J is zero, and there 

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Equilibrium Versus Steady State in Flow Systems 79... 

EQUILIBRIUM VERSUS STEADY STATE IN FLOW SYSTEMS... 

Eq.(50) shows the variation of the equilibrium dimensionless temperature as a function of the maximum value of the dimensionless coolant flow rate X6max- Plotting XQmax versus X3e a bifurcation curve can be obtained, from which it is possible to determine the value of xsmax which gives a different behavior of the reactor in steady state. It is interesting to note that Eq.(50) is equal to Eq.(47) when we make the substitutions of Eq.(49) into Eq.(47). 

Finally, yet another issue enters into the interpretation of nonlinear Arrhenius plots of enzyme-catalyzed reactions. As is seen in the examples above, one typically plots In y ax (or. In kcat) versus the reciprocal absolute temperature. This protocol is certainly valid for rapid equilibrium enzymes whose rate-determining step does not change throughout the temperature range studied (and, in addition, remains rapid equilibrium throughout this range). However, for steady-state enzymes, other factors can influence the interpretation of the nonlinear data. For example, for an ordered two-substrate, two-product reaction, kcat is equal to kskjl ks + k ) in which ks and k are the off-rate constants for the two products. If these two rate constants have a different temperature dependency (e.g., ks > ky at one temperature but not at another temperature), then a nonlinear Arrhenius plot may result. See Arrhenius Equation Owl Transition-State Theory van t Hoff Relationship... 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

We obtain the same final velocity equation for steady-stale conditions, except replaces Kg. This is not surprising since the steady-state assump. tion does not change the form of the velocity equation for the uninhibited reaction while the reaction between E and 1 to yield El must be at equilibrium, (There is nowhere for El to go but back to E-H.) The velocity equation differs from the usual Michaelis-Menten equation in that the K term is multiplied by the factor [1 4- ([I]/J i)]- The above derivation confirms our original prediction that is unaffected by a competitive inhibitor, but that the apparem K value is increased. The increase in the value does. loi mean that the El complex has a lower affinity for the substrate. El has no affinity at all for the substrate, while the affinity of E (the only form that can bind substrate) is unchanged. The apparent increase in results from a distribution of available enzyme between the full affinity and "no affinity forms. The factor [14-([I]/ffi)] may be considered as an [I]-dependent statistical factor describing the distribution of enzyme between the E and El forms. Figure 4-21 shows the effect of a competitive inhibitor on the v versus [S] plot. 

Unlike the steady-state system, the slope of the 1/u versus 1/[A] plot for the rapid equilibrium system goes to zero as [B] approaches infinity. (As [B] increases, the Ife/fB] term of the slope factor becomes very small.) Also, unlike the steady-state system, the plots of 1/u versus 1/[B] intersect on the vertical axis at 1/Vmax. (There is no intercept factor— the denominator [B] term is not multiplied by an [A]-containing term.)... 

Figure 11. Plots of the ratios °Be/ Al and Ne/ Al, respectively, versus the A1 concentration for rocks with a simple exposure history. The lower solid lines represent the temporal evolution of these quantities in the absence of erosion, the upper solid lines in erosion equilibrium. Dotted lines indicate the temporal evolution for erosion rates of 10, 1, 0.1, and 0.01 m/Ma (lm/Ma=10 cm/a). The areas between the curves ( steady-state erosion island Lai 1991) comprise all possible combinations of exposure ages and erosion rates for simple exposure histories, data outside of these areas indicate complex exposure histories or experimental error. Curves are plotted for °Be, Ne, and A1 production rates of 5.42, 19.0, and 35.2 atoms a respectively (modified from Kubik et al. 1998 Niedermarm 2000 see Table...

TABLE 11.5 Cleland nomenclature for bisubstrate reactions exemplified. Three common kinetic mechanisms for bisubstrate enzymatic reactions are exemplified. The forward rate equations for the order bi bi and ping pong bi hi are derived according to the steady-state assumption, whereas that of the random bi bi is based on the quasi-equilibrium assumption. These rate equations are first order in both A and B, and their double reciprocal plots (1A versus 1/A or 1/B) are linear. They are convergent for the order bi bi and random bi bi but parallel for the ping pong bi bi due to the absence of the constant term (KiaKb) in the denominator. These three kinetic mechanisms can be further differentiated by their product inhibition patterns (Cleland, 1963b)... 

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