Equilibrium transient

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

Transient Equilibrium—If the half-life of the parent is short enough so the quantity present decreases appreciably during the period under consideration, but is still longer than that of successive members of the series, a stage of equilibrium will be reached after which all members of the series decrease in activity exponentially with the period of the parent. At this time, the ratio of the parent activity to the daughter activity is constant. 

We can now distinguish three general cases, depending on whether the first decaying species has a longer, a much longer, or a shorter half-hfe than that of the daughter nuclide. These three cases are transient equilibrium, secular equilibrium, and nonequilibrium. 

Figure 11,8 Composite decay curves for (A) mixtures of independently decaying species, (B) transient equilibrium, (C) secular equilibrium, and (D) nonequilibrium, a composite decay curve b decay curve of longer-lived component (A) and parent radio nuclide (B, C, D) c decay curve of short-lived radionuclide (A) and daughter radionuclide (B, C, D) d daughter radioativity in a pure parent fraction (B, C, D) e total daughter radioactivity in a parent-plus-daughter fraction (B). In all cases, the detection coefficients of the various species are assumed to be identical. From Nuclear and Radiochemistry, G. Friedlander and J. W. Kennedy, Copyright 1956 by John Wiley and Sons. Reprinted by permission of John Wiley and Sons Ltd.

Transient equilibrium It is similar to secular equilibrium but differs in that the half-lives differ only by a small factor. 

N a (t f -22.7 hrs) biochemically more important because of its shorter half-life period. Shorter half-life ensures attainment of transient equilibrium faster. 

A second special case of Equations (3.21) and (3.22) is called transient equilibrium (Figs. 3.8c and 3.9). In this case, the parent is significantly ( 10x) longer lived than the daughter and thus controls the decay chain. Thus... 

FIGURE 6JS Classical nucleation theory dependence of nuclei size on Gibbs free energy at a function of saturation ratio, S. AS > 0 4> nonspontaneous formation, AG = 0 transient equilibrium, AG < 0 spontaneous formation of a solids phase. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], oop3nn t 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK. 

In other words, Berthollet regarded chemical combination not as a permanent species, but as the result of a transient equilibrium of several forces acting on the system. These included physical as well as chemical forces. Consequently, the chemical and the physical properties of a body were often related, which implied an intimate relationship among the various sciences that dealt with these forces. The analysis of chemical action required a coordination of various sciences, the totality of which constituted la physique. As yet, a general theory that comprised all the processes, results, and causes of chemical action was not available. The Essai was devoted precisely to this end of accounting for all the causes of chemical action and their interdependence. 

Figure 4.5. Transient equilibrium activities of mother and daughter nuclide as a function of t/t /2 2) (h/2(l)/h/2(2) = 5).

Figure 1.5. The transient equilibrium is illustrated in the plot of activity vs. time on a semilogarithmic graph. The daughter activity increases initially with time, reaches a maximum, then transient equilibrium, and finally appears to follow the half-life of the parent. Note that the daughter activity is higher than the parent activity in equilibrium.

The square planar complex 73 undergoes P-elimination via liberation of L to form a transient equilibrium of the iminium-rhodium hydride o-complex 74a and the Jt-complex 74b, Eq. 16. These complexes 74a and 74b represent a unique nitrogen-triggered mechanism that is different from either the hydride addition-elimination pathway or the 7i-allyl mechanism resulting in the intramolecular 1,3-hydrogen shift. 

Parent and the short-lived daughter reach transient equilibrium, character-... 

Transient equilibrium is reached by Bi from the batch decay of Pb, as iUustrated in Fig. 2.7. The time to reach this transient equilibrium is a few times the half-life of Bi. The activities of Bi and Pb would approach secular equUibrium, i.e., equal activities, if the ratio of the half-life of Pb to that of Bi were even greater. The second daughter, T1, can also be said to be in transient equilibrium with Pb, at times much greater than 1/(X2 +X3), because both its half-life and that of its immediate precursor are both short compared with the Pb half-life. 

Figure 2.9 illustrates that when the parent nuclide Sr has not reached equilibrium and when its radioactive daughter has not reached transient equilibrium, the amount of the daughter nuclide continues to increase for a time period after the production of the initial member of the chain is discontinued. 

HVL (half-value layers) attenuates approximately by 10 . There is 0.64 mm of radon gas at STP in transient equilibrium with 1 Ci of radium. 

The case of transient equilibrium will be illustrated by an example, such as the decay chain... 

If the parent is shorter-lived than the daughter, the daughter activity grows to some maximum value and then decays with its own characteristic half-life. This contrasts to the case of transient equilibrium where the daughter has an apparent decay given by the half-life of the parent. An example of this is shown in Figure 4.13 for the decay chain... 

Distinguishing between interface-dominated currents and bulk-dominated (ohmic) currents often requires a self-consistent interpretation of a variety of electrical conductivity measurements. These may be steady-state and transient, equilibrium and nonequilibrium, and can use a variety of electrode materials and sample geometries. 

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