Ratio integer

The small uncertainties in the calculated exponents seem to preclude the possibility that the d = 3 exponents are rational numbers (i.e. the ratio of integers). (At an earlier stage this possibility had been suggested, since not only die classical exponents, bnt also tlie rational numbers pre-RG calculations had suggested p = 5/16 and y = 5/4.)... 

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

The ratio between the pitch diameter and the number of teeth is called the diametral pitch, and for two gears to mesh, they must have the same diametral pitch. The diametral pitch is standardized and is always an integer. For gears used in process machineiy, the diametral pitch varies between 6 (coarse) and 20 (fine). 

Round upwards to the nearest integer all fractional turns in the results. Next multiply this result by the desired turns ratio to determine the number of... 

Since the first bracket on the right-hand side is a constant and the second is an integer, it is evident that, for any particular /, some leeway must exist in the value of the ratio rj/G for the equality to be satisfied. Here too, the presence of screw helicity must affect either / , or G, or both. In view of the fairly small variations of G allowed if the hybridization of the C atoms is to remain sp, and since the deformation of the C orbitals decreases as the radius of the cylindrical sheets increases, the distance between successive cylinders must decrease and probably tend towards a value characteristic of turbostratic graphite. 

Reality Check A mole ratio of 1.00 A 1.00 B 333 C would imply a formula A3B3C10 if the mole ratio were 1.00 A 2.50 B 5.50 C, the formula would be A2B5Cn. In general, multiply through by the smallest whole number that will give integers for all the subscripts. 

Chemical analysis always leads to the simplest formula of a compound because it gives only the simplest atom ratio of the elements. As pointed out earlier, the molecular formula is a whole-number multiple of the simplest formula. That multiple may be 1 as in H20, 2 as in H202, 3 as in CjHg, or some other integer. To find the multiple, one more piece of data is... 

These simple, integer volume ratios confirm the usefulness of the interpretation that equal volumes contain equal numbers of molecules. This proposal was first made in 1811 by an Italian scientist, Amadeo Avogadro hence it is called Avogadro s Hypothesis. It has been used successfully in explaining the properties of gases for a century and a half. 

In general, different compounds of the same two elements have different atomic ratios. Since these atomic ratios are always ratios of integers, 1/1, 1/2, 2/1, 2/3, etc., the weight ratios will be simple multiples of each other. Thus the atomic theory explains the observation that different compounds of the same two elements have relative compositions by weight that are simple multiples of each other. 

Once again it is no surprise that the simple integer volume ratios are readily explained with the atomic theory. The atomic theory was devised for this purpose, as is indicated in Chapter 2. 

We have already learned that metals may be deformed easily and we have explained this in terms of the absence of directional character in metallic bonding. In view of this principle, it is not surprising that two-element or three-element metallic crystals exist. In some of these, regular arrangements of two or more types of atoms are found. The composition then is expressed in simple integer ratios, so these are called metallic compounds. In other cases, a fraction of the atoms of the major constituent have been replaced by atoms of one or more other elements. Such a substance is called a solid solution. These metals containing two or more types of atoms are called alloys. 

The numbers in parentheses correspond to 1 estimated standard deviation (esd), except those for the Cpf Cn ratio, which correspond to 3 esd. The values of n and m shown are the smallest integers for which the ratio (n + m)/n lies within the experimental error limits of the measured axial ratio < Fe-<=R — t + e. The c parameters in the bottom line refer to the supercells corresponding to the underlined (n + ro)/n values ... 

Low-resolution devices are those that can separate and measure m/z ratios to the nearest integer value and have a numerical resolution of up to around 1000. As such, they can separate (resolve), for example, ions at m/z 28 and 29, i.e. they allow the analyst to differentiate between CO+ and CHO, or C2H4+ and C2H5+. Using these types of instrnment, we need only consider the masses of the isotopes as integers, e.g. = 12 Da, = 1 Da, = 14 Da and = 16 Da. 

The quadrupole is classified as a low-resolution device, i.e. it is capable of measuring the m jz ratio of an ion to the nearest integer value, and thus is unable to provide the elemental composition of an ion. 

The fact that the ratios of the three elements all came out very close to integers gives us confidence that our analysis is correct. As we describe in Chapter 4, organic acids contain the -CO2 H group. Thus, a chemical formula containing two O atoms is consistent with an organic acid. 

Nonstoichiometric Compounds Intrinsic defects are stoichiometric defects (i.e., they do not involve any change in overall composition). Defects can also be nonstoichiometric. In the case of extrinsic defects where the host crystal is doped with aliovalent impurities, the solid so formed is a nonstoichiometric compound because the ratio of the atomic components is no longer the simple integer. There is also... 

In Europe the notion of the zero evolved slowly in various forms. Eventually, probably to express debts, it was found necessary to invent negative integers. The requirements of trade and commerce lead to the use of fractions, as ratios of whole numbers. However, it is obviously more convenient to express fractions in the form of decimals. The ensemble of whole numbers and fractions (as ratios of whole numbers) is referred to as rational numbers. The mathematical relation between decimal and rational fractions is of importance, particularly in modem computer applications. 

Is the ratio of the masses of oxygen a ratio of small integers ... 

The ratio of 1.50 1 is equal to the ratio 3 2, and the law of multiple proportions is satisfied. Note that the ratio of masses of iron to oxygen is not necessarily a ratio of small integers the ratio of mass of oxygen in one compound to mass of oxygen in the other compound is what must be in the small integer ratio. 

The ratio of masses of oxygen in the two compounds (for a given mass of Cu) is the ratio of small integers, as required by the law of multiple proportions. The ratio of mass of copper to mass of oxygen is not integral. 

If more than two elements are present, divide all the numbers of moles by the smallest to attempt to get an integral ratio. Even after this step, it might be necessary to multiply every result by a small integer to get integral ratios, corresponding to the empirical formula. 

Dividing each by 1.425 yields 1.000mol Fe for every 1.333mol O. This ratio is still not integral. Multiplying these values by 3 yields integers. (We can round off when a value is within 1 or 2% of an integer, but not more.)... 

The empirical formula (or any formula) must be in the ratio of small integers. Thus, we attempt to get the ratio of moles of carbon to moles of hydrogen into an integer ratio we divide all the numbers of moles by the smallest number of moles ... 

This is still not a whole number ratio, since 1.333 is much too far from an integer to round off. Since 1.333 is about l, multiply both numbers of moles by 3 ... 

This is close enough to an integer ratio, so the empirical formula is Fe,04. 

If each of the following mole ratios is obtained in an empirical formula problem, what should it be multiplied by to get an integer ratio ... 

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