Small quantities. Calculations with

Atoms, most molecules, and formula units of ionic compounds are extremely tiny. Their formula masses are measured in atomic mass units, which are useful for comparison purposes only, in order to get weighable quantities of matter, a huge collection of formula units is required. The mole is defined as the number of atoms in exactly 12 grams of A millimole is 0.001 mol, and is useful for calculations with small quantities of substances. The mole is abbreviated mol, not m or M, which are used for related quantities, and millimole is abbreviated mmol. 

Several aspects of measurement will be considered in this chapter. First, Section 2.1 presents the factor label method, which makes calculations with measured quantities easier. This method will be used in the sections that follow and throughout the book. Next, in Section 2.2, we consider how to calculate with extremely large and extremely small numbers, using exponential notation. Section 2.3 introduces the metric system, a system of weights and measures... 

Volumetric. VINEGAR is an aqueous solution of acetic acid, HCsHjOj, with small quantities of nonacidic components. 20.00 ml NaOH is titrated with 32.80 ml 0.0670 M HCl. A 5.00-ml portion of vinegar is diluted with water and titrated with 30.50 ml of the NaOH solution. Calculate (a) the mass of acetic acid (i) per 5 ml, ( ) per 100 ml vinegar (b) the mass percent of acetic acid, if the density of the vinegar is 1.001 g/ml. 

Determine the moisture content of crude cocaine by drying about 1 g in a vacuum desiccator. To determine the ecgonine content, dissolve about 0 5 g in 15 ml of 2N hydrochloric acid and boil the solution under reflux for five hours. After cooling slowly over a period of at least two hours, filter through a plug of cotton wool and wash the wool with small quantities of 2N acid until the filtrate measures 25 ml at 20". Mix well and measure the optical rotation in a 2-dm tube. Calculate the percentage of ecgonine in the raw cocaine by the formula... 

Ipecacuanha root of the B.P. consists of the dried root or the rhizome and root of either Cephaelis ipecacuanha (Rio root) or Cephaelis acuminata (Cartagena, Nicaragua, or Panama root). Good samples of the former variety contain between 2 and 2 5 per cent of total alkaloids, of which 60 to 70 per cent is non-phenolic (calculated as emetine) and about 25 per cent cephaeline with small quantities of other alkaloids such as psychotrine. C. acuminata may yield more total alkaloids and some samples have been examined which contain well over 3 per cent. The proportion of non-phenolic alkaloids is lower in this variety, however, being of the order of 20 to 40 per cent. At one time C. ipecacuanha was the only official variety of ipecacuanha and a requirement was, therefore, included for non-phenolic alkaloids. This standard no longer applies although the amount of non-phenolic alkaloids present in a sample is still required for some commercial transactions. 

The method is generally applicable when other modes of esterification are either slow, inefficient, or likely to cause isomerisation it is, however, time-consuming and expensive. Small quantities of acid impurities are sometimes produced, hence it is advisable to wash the ester with saturated sodium bicarbonate solution. The silver salt can usually be prepared by dissolving the acid in the calculated quantity of standard ammonium hydroxide solution and... 

Differential calculus is the part of mathematics that deals with the slopes of curves and with infinitesimal quantities. Suppose we are studying a function y(x). As explained in Appendix IE, the slope of its graph at a point can be calculated by considering the straight line joining two points x and x + 8x, where 8x is small. The slope of this line is... 

In connection with a discussion of alloys of aluminum and zinc (Pauling, 1949) it was pointed out that an element present in very small quantity in solid solution in another element would have a tendency to assume the valence of the second element. The upper straight line in Fig. 2 is drawn between the value of the lattice constant for pure lead and that calculated for thallium with valence 2-14, equal to that of lead in the state of the pure element. It is seen that it passes through the experimental values of aQ for the alloys with 4-9 and 11-2 atomic percent thallium, thus supporting the suggestion that in these dilute alloys thallium has assumed the same valence as its solvent, lead. 

The difficulty in calculating a small number as a difference of two large ones, both known with limited accuracy, is not new and arises in many other fields. It often justifies using perturbation theory, which is aimed at estimating the quantity of interest directly. We will attempt to follow the same approach here. To do so, we combine (2.60) with (2.11) from Sect.2.2, in which we substitute U for This yields... 

Solving the quadratic equation in gives the extent of the reaction = 0.00155, from which the mole fractions of the components can be calculated. This highlights a real problem with prebiotic reactions even if the equilibrium constants are known and the kinetics are favourable, the reaction may only produce small quantities of the required material simply because the concentrations of the reactants are too low. 

---