Bracket method

In an earlier section, measurements were described in which the equilibrium constant, K, for bimolecular reactions involving gas-phase ions and neutral molecules were detennined. Another method for detemiining the proton or other affinity of a molecule is the bracketing method [ ]. The principle of this approach is quite straightforward. Let us again take the case of a proton affinity detemiination as an example. In a reaction... 

Modifications have been made to the single-direction bracketing method, wherein the rates of reaction are measured in order to further clarify and refine the thermochemical assignment. ... 

Samples were prepared for Cu isotope analysis on the Multicollector Inductively-Coupled Plasma Mass Spectrometer (MC-ICPMS) at University of Arizona. The Cu-rich samples were loaded and dissolved in pure HNO3 and the Cu-poor samples were loaded and dissolved in a mixture of HCI and HNO3, Chromatographic separation of the Fe and Cu ions was deemed necessary for the Cu-rich samples. The diluted solutions were injected into the MC-ICPMS using a microconcentric nebulizer. Samples were run numerous times to increase precision. The Cu isotope ratios are reported in conventional per mil notation, relative to the NIST 976 standard. Mass bias was also accounted for by bracketing methods with the NIST 976 standard. 

The bracket method is a straightforward application of the Arrhenius equation that can be used if the value of the activation energy is known. Assuming that stability of a product at 50°C is 32 days, and it will be stored at 25°C, then, tf. = 32 days, Te = 273 + 50°C = 323 K, and Tg = 273 + 25°C = 298 K. We know that activation energy is = 10 kcal/mol. Stability at recommended storage temperature is calculated as ... 

The bracket method should not be confused with bracketing, which is an experimental design that allows one to test a minimum number of samples at extremes of certain factors, such as strength, container size, and container fiU. Bracketing assumes that the stability of any intermediate level is represented by the stability of the extremes and testing at those extremes is performed at all time points. 

To illustrate the application of the Q-Rule, let us assume that the stability of a product at 50°C is 32 days. The recommended storage temperature is 25°C and M = (50 - 25)/10 = 2.5. Let us set an intermediate value of Q = 3. Thus, Qn = (3)2.5 = 15.6. The predicted shelf life is 32 days x 15.6 = 500 days. This approach is more conservative when lower values of Q are used. Both Q-Rule and the bracket methods are rough approximations of stability. They can be effectively used to plan elevated temperature levels and the duration of testing in the accelerated stability testing protocol. 

All values are taken from ref 22 unless noted otherwise. Determined by the bracketting method. 

The transient character of unstable species is intrinsically because of at least one fast reaction which they undergo as soon as they are formed (for example coalescence reaction in the case of atoms and clusters). This reaction therefore induces competition with any redox reaction which could be regarded as determining the redox potential of a transient entity. In particular, the competition does not enable the establishment of a reversible equilibrium of electron transfer with another suitable system. Thus, the redox potential of short-hved species must be evaluated from kinetic methods - the pulse technique enables us to observe whether or not electron transfer involving the transient species and a series of donor/acceptor couples, used as monitors, is elfective, and thus to establish by a bracketing method the value of the imknown redox potential. Only elementary monoelectronic transfers are considered. Thus, note that one of the forms of the reference couple, reduced or oxidized, can also be a transient radical. 

Bracketing methods involving exothermic proton transfer have yielded both cationic and neutral metal-ligand bond energies. For example, M0H (M - Fe and Co) were reacted with a series of reference bases, reaction 5 (42). For CoOH , proton transfer was observed with... 

Finally, the proton affinities of several atomic metal anions, M" (M -V, Cr, Fe, Co, Mo, and W), have been determined by bracketing methods (47). Combining these data with measured electron affinities of the metals yielded homolytic bond energies for the neutral hydrides, D (M-H). The monohydride bond energies compare favorably with other experimental and theoretical data in the literature and were used to derive additional thermodynamic properties for metal hydride ions and neutrals. 

In the absence of a calculator, we may use what we call the bracketing method to arrive at a rough estimate of the pH in just a few seconds. Here s how that works. The hydronium ion concentration, in proper scientific notation, will always be written as a number between 1 and 10 multiplied by some power of 10 (e.g., 6 X 10 °). It will always be true that this number may be bracketed as shown here between 1 X 10 ° and 10 X10 °. Thus, if we know that the log of 1 is 0 and that the log of 10 is 1 (since 10 = 10) we can quickly follow the logic below. 

Use the bracketing method to obtain a pH range for a sample of rainwater that has a hydronium ion concentration of 3 X 10" M. Once you have your estimate, determine the exact pH of this solution by using your calculator to find the exact value of the logarithm of 3. 

Use your knowledge of the relationship between hydronium ion concentration and hydroxide ion concentration to determine the pH of a solution of seawater that has a hydroxide ion concentration of 2 X10" M. Use your knowledge of the relationship between hydronium ion and hydroxide ion concentrations to determine the hydronium ion concentration. Then use the bracketing method to obtain a pH range, and finally, use the calculator to determine the exact pH. 

Alternatively, if we don t need an exact answer, the bracketing method can be used to conclude that the hydronium ion concentration would be somewhere between 10 M and 10 M, because the pH is between 7 and 8. 

Table 5.3 Characteristics of metal powders (in brackets - methods of analysis for parameters).

Proton-transfer equilibrium measurements and proton-transfer bracketing methods are sources for proton affinity values of organometallic complexes. The determination of the site of proto-nation, i.e., metal atom vs. a ligand, is a fundamental dilemma of any study on the protonation of metal complexes. It was demonstrated, for example, that Fe(CO)5 was protonated exclusively at the metal atom, whereas the results for the proton transfer to ferrocene can be explained by the formation of a metal-protonated form and a ring-protonated form, involving the agostic interaction of the proton with the metal atom. 

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