A Very Useful Formula

we present a very simple but powerful formula. This formula is so useful that you can sit in a research seminar or lecture and do mental arithmetic on the spot and then make a very intelligent comment such as Yes, Professor, but the wavelength for that transition should be... Recall that Planck realized that the energy of a light wave is proportional to the frequency of the wave and evaluated the proportionality constant to be h. As mentioned above, the same number occurs in the slope of the data for the photoelectric effect as analyzed by A. Einstein in 1905. The modem value is 6.6260693 x 10 erg s. Assume there are two energy levels in a molecular system such that the 

This was first brought to our attention many years ago [4] and has proved to be very useful for quick estimates of wavelength or energies in electron volts doing mental arithmetic while sipping coffee in a seminar. 

---

For short lines, this is a very useful formula. 

Equation (11.16) is already a very useful formula however, to determine a simple approximate energy dependence of the total cross section, we approximate further the repulsive potential by K (/ ) = Cexp( — R/D) and... 

The stoichiometry curve can be subdivided into three regions. In the temperature region between 473 K and 673 K, bimolecular and/or secondary reactions are sterically possible. The stoichiometry factor of 1.6, found by Hair49,52 for the reaction of silica, pretreated at 573 K, with methyltrichlorosilane, is reflected in these experiments. Equation (13) proves to be a very useful formula for a relatively fast evaluation of the stoichiometry factor, provided that the initial condition (PS+SS = 1) is fulfilled. The question whether these secondary species originate from bimolecular or secondary reactions, cannot be solved by this curve. 

Venous Nomogra.phs, The alignment chart is restricted neither to addition operations, nor to three-variable problems. Alignment charts can be used to solve most mathematical problems, from linear ones having any number of variables, to ratiometric, exponential, or any combination of problems. A very useful property of these alignment diagrams is the fact that they can be combined to evaluate a more complex formula. Nomographs for complex arithmetical expressions have been developed (108). 

Thus, if 5 = 1mm, / = 15.8 Hz. This very simple result is quite useful for approximately evaluating the gravity driven deflections of a stmeture given its natural frequency, or visa versa. Of course this was derived for a specific and very simple system, so it does not perfectly apply to more complex systems. Still it is a very useful rule of thumb. For a mass on the end of a cantilever beam, the above formula is correct. The lowest natural frequency of a massive cantilever beam is about 1.2 x the prediction of the above formula. 

Forty-four five-membered heterocycles of type A (13, 19) have been described (Table I). If the atoms or groups a, b, c, d, e, and f are selected from suitably substituted carbon, nitrogen, oxygen, and sulfur atoms, then with these conditions it can be shown that 144 structural possibilities are provided by the general formula 19. The number of structural possibilities can be deduced in various ways, but a very useful approach is to regard type A meso-ionic molecules (19) as being derived by the union (-<—u— ) of 1,3-dipoles (34) and heterocumulenes (35). 

Fuchsitt test Dilute solutions of triphenylmethane dyestuffs, such as fuchsin (for formula, see Section IV. 15, reaction 9) and malachite green, are immediately decolourized by neutral sulphites. Sulphur dioxide also decolourizes fuchsin solution, but the reaction is not quite complete nevertheless it is a very useful test for sulphur and acid sulphites carbon dioxide does not interfere, but nitrogen dioxide does. If the test solution is acid, it should preferably be just neutralized with sodium hydrogen carbonate. Thiosulphates do not interfere but sulphides, polysulphides, and free alkali do. Zinc, lead, and cadmium salts reduce the sensitivity of the test, hence the interference of sulphides cannot be obviated by the addition of these salts. 

All programs provide a means of copying (replicating) formulae or cell contents when required and this is a very useful feature. When copying, references to cells may be either relative, changing with the row/column as they are copied, or absolute, remaining a fixed cell reference and not changing as the formulae are copied. This distinction between cell references... 

Interpretation of nonlinear molecular measurements on molecules, and indeed our intuitive understandings of any polarization, is almost always based on a state model of the molecule the applied fields mix the levels of the molecular Hamiltonian so that spectral analysis (in the sense of sums over states, or SoS) becomes a very useful description. While more recent and more sophisticated electronic structure calculations have important direct-response methods, the SoS techniques, like the very simple two-level formula of Oudar and Chemla, have tremendous advantages in terms of generality and understanding. 

Very few potential barrier models, including the rectangular barrier model discussed above, yield exact results for the tunneling problem. In general one needs to resort to numerical calculations or approximations. A very useful approximation is the WKB formula, which generalizes the solution exp( zhr) of the free particle Schrodinger equation to the form... 

The example in section 6 is a very simple, almost trivial example of the application of AI techniques to solve a problem using formulas. However, the same process (without change) could be used on problems of much greater complexity. 

The Hougen-Watson formula presupposes the surface reaction to occur in a single step. This is rarely the case, and may often not even be an acceptable approximation. To reduce the mathematical complexity introduced by a multistep surface reaction, the practical engineer will then look for other simplifications that do not entail undue loss of accuracy. Here, the concept of relative abundance of adsorbed species on the catalyst surface is a very useful tool. 

Comment This is a very strong formula which should only be used when absolutely necessary. In cases suffering from concomitant vacuity weakness, this formula can be alternated with a supplementing formula to support the righteous. 

Microsoft Excel has a very useful program called Solver. To use this program, we must first arrange an equation to give a formula equal to a known value. Solver effectively calculates the formula in reverse, by starting with the answer of the formula (the known value) and, by iteration, solving for the value or values of one or more variables in the formula that results in the correct answer. (Other spreadsheet programs have similar problem solvers.)... 

Herbert B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed., Macmillan Co., New York, 1962. This book is out of print, but if you can find a used copy, you will find that it is a very useful compilation of formulas, including trigonometric identities, derivatives, infinite series, and definite and indefinite integrals. 

One typically starts with an internal energy of a macroscopic system, expressed as the internal energy of the periodically repeated unit cell. This state function is part of another state function H, the enthalpy, which is a very useful energetic measure for conditions of constant pressure p. For a complete picture, one also needs to know the value of the state function T, the temperature, and that of the state function S, the entropy, a measure of chaos and also probability. These functions may be combined to yield the Gibbs energy (or free enthalpy) G, the true and final measure of stability. In its difference form, the so-called Gibbs-Helmholtz formula reads... 

Barium nitrate is often added to glitter formulas. For example, a very cheap formula that is used in Class C comets, given to me many years ago by Bill Withrow is called "Pearl Mix". 

Ionisation of atomised metal may seem an extreme case, so perhaps it is not important if students think of the Na+ ion as more stable than the atom. However students have a strong tendency to see any species with an octet of electrons as stable, and research shows that by the end of secondary education, students will commonly rate a whole range of dubious ions as more stable than atoms because they have foil shells or octets of electrons. So not only do students tend to think Na is a stable ion, they make the same judgement about the chemically quite ridiculous species Na shown in Figure 3.7. It is important, therefore, that teachers make sure that students do not over-generalise the octet rule from a very useful rule of thumb for identifying the most likely formulae for molecules and ions, and adopt it as an absolute principle to judge stability and explain why reactions occur. 

If we try to use formula (20.12) to compute the Euler characteristic of Horn (Cm, ii n)) we will need nontrivial reductions to get a nice answer. Using Proposition 20.21 instead yields a very simple formula directly. This reflects very well the reduction that often occurs in passing between tableaux in the spectral sequence computation, since using the filtration on the simphcial complex Hom i (T, G) as we did is the same as using the first tableau of our spectral sequence. 

The handbook has some useful appendices that include the CAS registry numbers, a biological activity index, a formula index, and a name index that also includes synonyms. When looking up a compound in one of the indexes, you need to remember that the numbers provided are compound numbers, rather than page numbers. There is also a very useful section on organic name reactions that includes references to the primary literature. 

---