Purely arithmetical

Again and again the question arises whether the results of pilot plant experiments could not be replaced by purely arithmetical scaling up. Since, however, there is si ill no method of precalculating thermal separations available, such experiments will... 

According to Skolnik [57] a purely arithmetical prediction is also possible. For the members of a homologous series (e.g., hydrocaiLons) and a second component (e.g., benzene), these linear relations hold ... 

If the purely arithmetical method is applied, both stability limits have to be evaluated in any case, as there are processes with very high values for... 

These are purely arithmetical descriptors. Arithmetical descriptors that are not purely arithmetical since they depend on come from the mappings... 

Example (Arithmetical descriptors) We recall a few basic arithmetical descriptors - Let M = (e, y) M be a molecular graph. The purely arithmetic indices A and A are the counts of all or of the non-H atoms, obtained from the function A with its values... 

These descriptors are purely arithmetical, which is not the case for the descriptors arising from the following mappings ... 

These descriptors are not purely arithmetical. They may have different values for compounds of the same molecular formula (and of the same state distribution). They are nevertheless considered arithmetic descriptors since they are based merely on numbers of bonds. 

With what ease this purely arithmetical seriation may be made to accord with a horizontal arrangement of the elements according to their usually received groupings, is shown in the following table, in the first three columns of which the numerical sequence is perfect, while in the other two the irregularities are but few and trivial. ... 

Finally the overall heat transfer coefficient is obtained from equation 8. The global heat transferred for each tube is computed with equation 9. We call A7 / semi logarithmic temperature difference . It is the best compromise between pure logarithmic temperature difference that has no sense here (only one tube) and pure arithmetic temperature difference that does not allow to follow the evolution of water properties along the tube. The heat exchange diagram of the 0TB is presented in figure 3. 

Crude petroleum is fractionated into around fifty cuts having a very narrow distillation intervals which allows them to be considered as ficticious pure hydrocarbons whose boiling points are equal to the arithmetic average of the initial and final boiling points, = (T, + Ty)/2, the other physical characteristics being average properties measured for each cut. 

Therefore Eq. (5-47) is applicable to first-order and to second-order rate constants, it being understood that the arithmetic operations are carried out on pure numbers generated as shown. We have not evaded the requirement of dimensional consistency, which is provided by Eq. (5-43). 

Looked upon purely as an arithmetic multiplication table, the products are all correct. This condition remains valid whatever row is selected initially4. Because of the close association between tables 1 and 2 each row in Table 1 may indeed be regarded as a representation of the symmetry elements. 

The concept is based on the assumption that the properties of mixtures can be described by the properties of pure components. As a result, the arithmetic expressions involved (regular mixing rule) are relatively simple. 

The aim of the book is to describe in all simplicity the combinatorial and quantitative science of nucleosynthesis, a pure stellar arithmetic, chaste and contemplative, cloaked in the ironic reticence of nnmbers. It is so penetrating and revealing that it may be considered as our most powerful tool for divulging the material history of the Cosmos. 

You re not quite done, because the problem asks for the boiling point of the solution, not the change in the boiling point. Luckily, the last step is just simple arithmetic. You must add your at to the boiling point of pure acetic acid, which, accordingto Table 13-2, is 118.1°C. This gives you a final boiling point of 118.1°C-i-2.5°C = 120.6°C for the solution. 

The original result in [21] differed from the one in (9.22) by two percent. A later purely numerical calculation of the light by light contribution in [22, 23] produced a less precise result which, however, differed from the original result in [21] by two percent. After a thorough check of the calculations in [21] a minor arithmetic mistake in one of the intermediate expressions in the original version of [21] was discovered. After correction of this mistake, the semianalytic calculations in [21] lead to the result in (9.22) in excellent agreement with the somewhat less precise purely numerical result in [22, 23]. 

A closed subscheme X c Pr of pure dimension n is arithmetically Cohen-... 

Figure 17. Density dependence of the band maximum position of the 3vs mode (vm) in pure C02 at temperatures between 298 and 500 K (with vg = 6972.6 cm" ). Below the critical density (0.45 g cm-3), P and R branches were observed, and vm is taken as the arithmetic mean of the P and R branch maxima. Points on the lower left represent hot bands. Reproduced from Ref. 77a with permission from Verlag der Zeitschrift fur Naturforschung.

The internal forces of a pair of liquids are seldom so nearly alike as to permit their mixture to obey Raoult s law very closely throughout the whole range of composition. In the absence of chemical interaction, the attraction between two different molecular species, provided their dipole moments are zero or small, is approximately the geometric mean of the attractions between the like molecules, Since a geometric mean is less than an arithmetic mean, the mixing is accompanied by expansion and absorption of heat. The partial molal heat of transfer per mole from pure liquid to solution is given with fair accuracy for many systems by the equation,... 

When the professor saw the final result for the specific rotation, +164.62 0.22, she was bewildered. The reported value exceededihe literature value of the pure dextrarotatory compound, +152.70, by 11.92, about 50 times the claimed limit of error This ridiculous result could mean only that a serious error or mistake had been made in this determination, unless the literature value itself were seriously in error, which seemed unlikely. She looked for, but could not find, a mistake in arithmetic. She examined the data and asked the student why the reading of 20.09° had been rejected. The student replied that the reading seemed too far out of line. In order to check on this, the professor suggested that they carry out a Q test. This test showed that the reading had been wrongfully rejected (calculated Q = 0.29 for = 10, = 0.41 see Table 1 and accompanying discussion). With a... 

The pronounced and R branch contours in the first and second overtone spectra of CO (Figures 6.2-1 and 6.2-2) clearly indicate appreciable rotational freedom in dense carbon monoxide. The wavenumbers of maximum absorption in the P and in the / branch, i>/>(max) and i jffmax), of the first overtone (Fig. 6.2-4) and of the second overtone (Fig. 6.2-5) at various temperatures are plotted as a function of the pure carbon monoxide density. As is to be expected, the separation between these two maxima increases with the temperature. In all experimental spectra, the solid line in both figures indicates the arithmetic mean, i> , of the maximum positions of the P and the R branch. slightly decreases with the density but is independent of the temperature. The straight line formed by Urn in Fig. 6.2-4 can be extrapolated to = 4261.9 cm, which is very close to the literature value = 4260.1 cm for the pure vibrational transition in the gas phase (Bouanich et al., 1981). Similarly, extrapolation of the second overtone data in Fig. 6.2-5... 

Figure 6.2-4 Density dependence of the wavenumber at the P and R branch first overtone absorption maxima, vpimdx) and i>/f(max), of pure CO at temperatures between 20 and 227 °C Vm is the arithmetic mean of vpimsx) and t>/ (max) -f is quoted from Vu et al. (1963) and -o- from Bouanich et al. (1981).

Equation (2-62) is the key to the application of colligative properties to polymer molecular weights. We started with Eq. (2-53), which defined an ideal solution in terms of the mole fractions of the components. Equation (2-62), which followed by simple arithmetic, expresses the difference in chemical potential of the solvent in the solution and in the pure state in terms of the mass concentrations of the solute. This difference in chemical potential is seen to be a power series in the solute concentration. Such equations are called virial equations and more is said about them on page 65. 

---