[320] In the most exact quantitative work, as demanded in the determinations of atomic weights, every known loss must, as far as possible, be measured and taken into account. Beautiful instances of such work are found in T. W. Richards' classic determinations of atomic weights. See, for instance, Richards, ‹Carnegie Institution Publications›, No. 125 (1910), Determinations of the Atomic Weights of Silver, Lithium and Chlorine («Stud.»). [321] ‹Z. Elektrochem.›, «11», 797 (1905). [322] ‹Ibid.›, «11», 936 (1905), and «12», 725 (1906). [323] ‹Z. phys. Chem.›, «55», 707 (1906), and «61», 638 (1907). [324] If ‹n› is the number of moles of solute dissolved in ‹N› moles of solute, the concentration of the solute may be expressed as ‹n› / (‹n› + ‹N›), which is called its "mole fraction." This form of expressing concentrations is in many particulars preferable to the mole / liter form. For very dilute solutions (‹n› is very small compared with ‹N›) the two forms become practically identical, but they are not so for more concentrated solutions, and the ‹mole-fraction› expression is then easier to treat rigorously. [325] See Walden, ‹loc cit.›, for more extended data. [p157] PART II «SYSTEMATIC ANALYSIS AND THE APPLICATION OF FUNDAMENTAL PRINCIPLES» CHAPTER IX «SYSTEMATIC ANALYSIS FOR THE COMMON METAL IONS. THE IONS OF THE ALKALIES AND OF THE ALKALINE EARTHS. ORDER OF PRECIPITATION OF DIFFICULTLY SOLUBLE SALTS WITH A COMMON ION» In systematic analysis it is most convenient to make separate examinations for the metal and for the acid ions. The examination for metal ions usually precedes that for the acid ions, and the scheme of analysis for the former will be considered first. The analytical grouping of the metallic elements is not a natural one, as far as their chemical behavior is concerned. Such a grouping is found in the Periodic System of Mendeléeff and is used in systematic inorganic chemistry.[326] The groups in analysis are based chiefly, but not exclusively, on the physical property of greater or smaller solubility of certain salts of the metals. According to the salts chosen, different systems vary somewhat in detail. Frequently elements of the same natural family are also found in the same analytical group, relationship in chemical properties being often coincident with relationship in the physical behavior of the salts of the metals. In the following list, the common metal ions are arranged in groups, which are given in the order in which they are precipitated in the method of systematic analysis adopted. In each case, a group name and the characteristic reagents used in separating a group from those following it, are given. «The Silver Group.»—Ions whose chlorides are insoluble in dilute acid solutions. The precipitating agent is HCl. [p158] «Pb»^{2+} (the chloride is somewhat soluble), «Ag»^{+}, «Hg»^{+}.