The above is one of the most common types of indirect analyses. Others are more complex but they can be reduced to algebraic expressions and solved by their aid. It should, however, be noted that the results obtained by these indirect methods cannot be depended upon for high accuracy, since slight errors in the determination of the common constituent, as chlorine in the above mixture, will cause considerable variations in the values found for the components. They should not be employed when direct methods are applicable, if accuracy is essential. PROBLEMS (The reactions necessary for the solution of these problems are either stated with the problem or may be found in the earlier text. In the calculations from which the answers are derived, the atomic weights given on page 195 have been employed, using, however, only the first decimal but increasing this by 1 when the second decimal is 5 or above. Thus, 39.1 has been taken as the atomic weight of potassium, 32.1 for sulphur, etc. This has been done merely to secure uniformity of treatment, and the student should remember that it is always well to take into account the degree of accuracy desired in a particular instance in determining the number of decimal places to retain. Four-place logarithms were employed in the calculations. Where four figures are given in the answer, the last figure may vary by one or (rarely) by two units, according to the method by which the problem is solved.) VOLUMETRIC ANALYSIS 1. How many grams of pure potassium hydroxide are required for exactly 1 liter of normal alkali solution? !Answer!: 56.1 grams. 2. Calculate the equivalent in grams (a) of sulphuric acid as an acid; (b) of hydrochloric acid as an acid; (c) of oxalic acid as an acid; (d) of nitric acid as an acid. !Answers!: (a) 49.05; (b) 36.5; (c) 63; (d) 63. 3. Calculate the equivalent in grams of (a) potassium hydroxide; (b) of sodium carbonate; (c) of barium hydroxide; (d) of sodium bicarbonate when titrated with an acid. !Answers!: (a) 56.1; (b) 53.8; (c) 85.7; (d) 84. 4. What is the equivalent in grams of Na_{2}HPO_{4} (a) as a phosphate; (b) as a sodium salt? !Answers!: (a) 47.33; (b) 71.0. 5. A sample of aqueous hydrochloric acid has a specific gravity of 1.12 and contains 23.81 per cent hydrochloric acid by weight. Calculate the grams and the milliequivalents of hydrochloric acid (HCl) in each cubic centimeter of the aqueous acid. !Answers!: 0.2667 gram; 7.307 milliequivalents. 6. How many cubic centimeters of hydrochloric acid (sp. gr. 1.20 containing 39.80 per cent HCl by weight) are required to furnish 36.45 grams of the gaseous compound? !Answer!: 76.33 cc. 7. A given solution contains 0.1063 equivalents of hydrochloric acid in 976 cc. What is its normal value? !Answer!: 0.1089 N.