Equivalence postulate

K.T. BAINBRIDGE proves the mass-energy equivalence postulated by Einstein... 

It is possible to assume that all mer units in a crystal occupy geometrically equivalent positions with respect to the chain axis. This is known as the Equivalence Postulate. 

The equivalence postulate is seen in the structures given in Figure 6.6. Here, the chain mers repeat their structure in the next unit cell. 

All of the above follows from the two equivalent postulates by Kelvin and Clausius. They constitute the second law of thermodynamics. However, mathematical... 

Finally, the molecules in all layers above the first are postulated to have the same partition function 9, as in the bulk liquid, so that 9, = 9, j,j for i > 1. This is of course equivalent to the BET assumption of liquid-like properties for these higher layers. 

Figure S-14. (A) A parabolic potential barrier and a linear perturbation. (B) Sum of the parabolic and linear functions, showing shift in maximum in accord with the Hammond postulate. (C) Two parabolic potential wells aa and bb are equivalent to the parabolic barrier cc .

This indicates a change in the formal oxidation state of the iron from -F2.25 to -F2.5, and mixed Fe /Fe species have been postulated. Flowever, it appears likely that these clusters are best regarded as electronically delocalized systems in which all the Fe atoms are equivalent. 

EDTA, leading to a postulate that more than one equivalent of Ca2+ can be captured by X (e.g. one Ca2+ sequestered by the three amines and the three carboxylates and another Ca2 + by the remaining half the donor groups), as the Dreiding model suggests. The fact that there was no interaction at neutral pH of X with phosphate or oxalate anions was separately confirmed. Thus, the dissolution of Ca3(P04)2 and Ca(C204) is entirely due to the cation complexation mechanism. 

B. —The state vector of any system remains some well-defined function of time except in so far as interaction occurs with some other system, when it may change to some other state vector in 3 . Laboratory measurement of any property of a system necessarily involves interaction between the system and the measuring equipment, and in general changes the state vector of the system. Now any change of vector in is equivalent to a transformation in 3, or in other words to an operator in 3. This leads to the following postulate ... 

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

Note that all the zero-order rate constants are essentially equivalent except those for the poly-hydric alcohols which are exactly half the value of the others. Ingold et al (Ref 49a) interpret this to mean that the rate of attack of nitronium is the same for both OH groups of the glycol molecule. Since there are two such groups the overall rate constant k0 is Vi that for monohydric alcohols. The explanation for the observed k0 for glycerol is more complex. In essence it consists of postulating that the two outside OH s are readily nitrated, ie, the 1-OH is nitrated at the same rate as the 3-OH, but the middle OH is nitrated much more slowly... 

A second approach considers that the regions of equivalent parameter values must enclose parameters for which the loss function is nearly the same or at any rate less different than some threshold. In other words, the equivalence regions should take the form 015(0) < c 5(6) for some appropriate constant of. Note that in this case the shape of the regions would not necessarily be ellipsoidal, or even convex In fact, we might postulate in general the existence of multiple minima surrounded by disjoint equivalence neigh-... 

The authors favour a two-equivalent oxidation in order to avoid the necessity to postulate free hydrogen atoms as intermediates in the reaction, viz. 

The reactive species might be V(OH)2l, produced in a similar manner to the analogous species in the oxidation of bromide ion, which could undergo one-equivalent breakdown to V(IV) and atomic iodine. Ramsey et al postulate transfer of OH to iodide ion, but the intermediacy of-12 is referred to in a later study of the oxygen effect to account for the relation ... 

---