1050 EXPRESSION OF RESULTS* 1050 A. Units Standard Methods uses the International System of Units (SI).1,2 Concentration units for chemical and physical results are generally expressed in mass units/L. The numerical values range from 0.1 to 999.9 mg/L. If the values are less, express them as micrograms per liter (g/L) and if greater as grams per liter (g/L), with the numerical values adjusted accordingly. Record and report all analytical results to the proper number of significant figures (see 1050B). TABLE 1050:I. COMMONLY USED EXPRESSIONS of MASS CONCENTRATION Unit of Expression Proportion Parts Parts Parts Parts per per per per hundred (%) thousand (0/00) million (ppm) billion (ppb) Value ⫺2 10 10⫺3 10⫺6 10⫺9 w/v w/w v/v g/dL g/L mg/L g/L g/100 g g/kg mg/kg g/kg mL/dL mL/L L/L nL/L Adapted from: MILLS, I., T. CVITAS, K. HOMANN, N. KALLAY & K. KUCHITSU. 1993. Quantities, Units and Symbols in Physical Chemistry, 2nd ed. Blackwell Scientific Publications, Oxford, U.K. 1. Radioactivity For information on reporting radiological results, see Section 7020D. b. Other concentration units: To obtain mass concentrations, the analyst may have to compute intermediate results in terms of mass, moles, volume, or equivalent. The most commonly used intermediate forms are defined as follows: 1) Mass fraction—analyte mass divided by the total mass of the solution or mixture, expressed as kg/kg; 2) Volume fraction—analyte volume divided by total volume of sample where measurements are at equal pressure and temperature, expressed as L/L; 3) Mole fraction—number of moles of analyte per total moles in solution; 4) Molar concentration (molarity)—number of moles of analyte contained in 1 L solution, designated by M; 5) Molal concentration (molality)—number of moles of analyte dissolved in 1 kg solvent; 6) Normality—number of equivalents (see ¶ c below) of analyte dissolved and diluted to a 1-L volume, designated by N. Also see ¶ d below. c. Equivalency: The unit milligram-equivalents per liter, or millequivalents per liter (me/L), can be valuable for making analytical and water-treatment calculations and for checking analyses by anion-cation balance. Table 1050:III presents factors for converting concentrations of common ions from milligrams per liter to milliequivalents per liter, and vice versa. The term “milliequivalent” represents 0.001 of an equivalent weight, which is defined as the weight of the ion (sum of the atomic weights of the atoms making up the ion) divided by the number of charges normally associated with the particular ion. The factors for converting results from milligrams per liter to milliequivalents per liter were computed by dividing the ion charge by weight of the ion. Conversely, factors for converting results from milliequivalents per liter to milligrams per liter were calculated by dividing the weight of the ion by the ion charge. To define equivalents,5 chemical reactions must be broken down into several types. 1) Neutralization reactions6—The equivalent weight of a substance in a neutralization reaction is the weight of acid that will furnish, react with, or be chemically equivalent to, one mole of hydrogen ion in the reaction in which it occurs. For example, the equivalent weight of an acid is its molar mass (molecular weight) 2. Mass a. Mass concentrations: Mass concentrations can be expressed in terms of weight/volume (w/v), weight/weight (w/w), and volume/volume (v/v).3 Those commonly used in Standard Methods are listed in Table 1050:I. Most analysis results are reported in terms of w/v, but some may be expressed in terms of w/w. Since analyses usually are performed for analytes in solution it may be necessary to report results in reference terms other than solution volume, such as milligrams per kilogram, parts per million, or percent by weight. These terms are computed as follows: mg/kg ⫽ mg/L ppm by weight ⫽ % by weight ⫽ mg/L mg/L 10 000 ⫻ where is the density of the sample solution or solid mixture being measured, in kilograms per liter (often reported as the same numeric value as grams per milliliter). The relative density of water can be computed from Table 1050:II4 using the densities at t°C and 4°C. Weight per unit weight (w/w) is defined as mass of analyte per total (wet) mass of solution or mixture. In some cases, the weight fraction results may be relative to total dry mass rather than total wet mass. The units would be multiples of (g analyte/kg dry). This is referred to as “moisture-free basis”; report concentration of total solids at the same time. * Reviewed by Standard Methods Committee, 2006. Joint Task Group: L. Malcolm Baker (chair), Stephen W. Johnson, David F. Parkhurst. 1 EXPRESSION OF RESULTS (1050)/Units TABLE 1050:II. DENSITY OF WATER FREE FROM DISSOLVED ATMOSPHERIC GASES, AT A PRESSURE OF 101.325 PA Density of Water at Given Temperature kg/m3 Temperature, t68 °C 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 999.8426 999.9015 999.9429 999.9672 999.9750 999.9668 999.9430 999.9043 999.8509 999.7834 999.7021 999.6074 999.4996 999.3792 999.2464 999.1016 998.9450 998.7769 998.5976 998.4073 998.2063 997.9948 997.7730 997.5412 997.2994 997.0480 996.7870 996.5166 996.2371 995.9486 995.6511 995.3450 995.0302 994.7071 994.3756 994.0359 993.6883 993.3328 992.9695 992.5987 992.2204 8493 9065 9461 9687 9748 9651 9398 8996 8448 7759 6932 5972 4882 3665 2325 0864 9287 7595 5790 3877 1856 9731 7503 5174 2747 0223 7604 4891 2087 9192 6209 3139 9983† 6743 3420 0015 6531 2968 9328 5612 8558 9112 9491 9700 9746 9632 9365 8948 8385 7682 6842 5869 4766 3536 2184 0712 9123 7419 5604 3680 1649 9513 7275 4936 2499 9965† 7337 4615 1801 8898 5906 2827 9663† 6414 3083 9671† 6178 2607 8960 5236 8622 9158 9519 9712 9742 9612 9330 8898 8321 7604 6751 5764 4648 3407 2042 0558 8957 7243 5416 3481 1440 9294 7045 4697 2250 9707† 7069 4337 1515 8603 5602 2514 9342† 6085 2745 9325† 5825 2246 8591 4860 8683 9202 9546 9722 9736 9591 9293 8847 8256 7525 6658 5658 4530 3276 1899 0403 8791 7065 5228 3282 1230 9073 6815 4456 2000 9447† 6800 4059 1228 8306 5297 2201 9020† 5755 2407 8978† 5470 1884 8221 4483 8743 9244 9571 9731 9728 9568 9255 8794 8189 7444 6564 5551 4410 3143 1755 0247 8623 6886 5038 3081 1019 8852 6584 4215 1749 9186† 6530 3780 0940 8009 4991 1887 8697† 5423 2068 8631† 5115 1521 7850 4105 8801 9284 9595 9738 9719 9544 9216 8740 8121 7362 6468 5443 4289 3010 1609 0090 8455 6706 4847 2880 0807 8630 6351 3973 1497 8925† 6259 3500 0651 7712 4685 1572 8373† 5092 1728 8283† 4759 1157 7479 3726 8857 9323 9616 9743 9709 9518 9175 8684 8051 7279 6372 5333 4167 2875 1463 9932† 8285 6525 4655 2677 0594 8406 6118 3730 1244 8663† 5987 3219 0361 7413 4377 1255 8049† 4759 1387 7934† 4403 0793 7107 3347 8912 9360 9636 9747 9696 9490 9132 8627 7980 7194 6274 5222 4043 2740 1315 9772† 8114 6343 4462 2474 0380 8182 5883 3485 0990 8399† 5714 2938 0070 7113 4069 0939 7724† 4425 1045 7585† 4045 0428 6735 2966 8964 9395 9655 9749 9683 9461 9088 8569 7908 7108 6174 5110 3918 2602 1166 9612† 7942 6160 4268 2269 0164 7957 5648 3240 0735 8135† 5441 2655 9778* 6813 3760 0621 7397† 4091 0703 7234† 3687 0062 6361 2586 * t68 represents temperature according to International Practical Temperature Scale 1968. † The leading figure decreases by 1.0. Source: MARSH, K.N., ed. 1987. Recommended Reference Materials for the Realization of Physicochemical Properties, Blackwell Scientific Publications, Oxford, U.K. H2SO4 ⫹ 2NaOH 3 Na2SO4 ⫹ 2H2O divided by the number of hydrogen atoms (equivalent number) contained in the acid. Thus, for acids that have reacted completely, the equivalent weight of HCl and HNO3 would equal the molar mass; the equivalent weight of H2SO4 and H2CO3 would equal the molar mass divided by 2 and the equivalent weight for H3PO4 would equal molar mass divided by 3. If the chemical formula is written out for the reaction of an acid with a base, the equivalent amounts for each compound in the reaction are obtained by dividing each stoichiometric coefficient in the equation by the equivalent number.7 For example, in the equation: dividing by the equivalent number, 2, yields: H2SO4 2NaOH Na2SO4 2H2O 2 2 2 2 and reducing to lowest terms yields: H2SO4 NaOH Na2SO4 H2O 2 1 2 1 2 EXPRESSION OF RESULTS (1050)/Units TABLE 1050:III. CONVERSION FACTORS* (Milligrams per Liter – Milliequivalents per Liter) Ion (Cation) me/L ⫽ mg/L⫻ Ion (Anion) mg/L ⫽ me/L⫻ 3⫹ Al B3⫹ Ba2⫹ Ca2⫹ Cr3⫹ 0.111 2 0.277 5 0.014 56 0.049 90 0.057 70 8.994 3.604 68.66 20.04 17.33 Cu2⫹ Fe2⫹ Fe3⫹ H⫹ K⫹ 0.031 47 0.035 81 0.053 72 0.992 1 0.025 58 31.77 27.92 18.62 1.008 39.10 Li⫹ Mg2⫹ Mn2⫹ Mn4⫹ Na⫹ NH4⫹ Pb2⫹ Sr2⫹ Zn2⫹ 0.144 1 0.082 29 0.036 40 0.072 81 0.043 50 0.055 44 0.009 653 0.022 83 0.030 59 6.941 12.15 27.47 13.73 22.99 18.04 103.6 43.81 32.70 me/L ⫽ mg/L⫻ mg/L ⫽ me/L⫻ 0.023 36 0.012 52 0.028 21 0.033 33 0.017 24 0.052 64 0.016 39 0.020 84 0.010 31 0.030 24 0.012 33 0.010 30 0.007 880 0.021 74 0.016 13 0.058 80 0.031 59 0.062 37 0.026 29 0.024 98 0.020 82 42.81 79.90 35.45 30.00 58.00 19.00 61.02 47.99 96.99 33.07 81.07 97.07 126.9 46.01 62.00 17.01 31.66 16.03 38.04 40.03 48.03 ⫺ BO2 Br⫺ Cl⫺ CO32⫺ CrO42⫺ F⫺ HCO3⫺ HPO42⫺ H2PO4⫺ HS⫺ HSO3⫺ HSO4⫺ I⫺ NO2⫺ NO3⫺ OH⫺ PO43⫺ S2⫺ SiO32⫺ SO32⫺ SO42⫺ * Factors are based on ion charge and not on redox reactions that may be possible for certain of these ions. Cations and anions are listed separately in alphabetical order. Thus, one-half formula weight of H2SO4 and Na2SO4, and one formula weight of NaOH and H2O are equivalent weights in this case. In some cases, the reaction may not go to completion and the following equivalent entities are obtained. BaCl2 2 NaHCO3 1 ⫺ AgNO3 2KCN Ag(CN) 2 KNO3 K⫹ 1 1 1 1 1 2AgNO3 ⫹ 2KCN 3 2AgCN ⫹ 2KNO3 H2 O 1 If the cation of the base is multivalent, the number of equivalents is calculated similarly. For example: 3HCl ⫹ Al(OH)3 3 AlCl3 ⫹ 3H2O HCl 1 Al(OH)3 3 AlCl3 3 NaCl 1 ⫹ AgNO3 ⫹ 2KCN 3 Ag(CN)⫺ 2 ⫹ KNO3 ⫹ K Because only one hydrogen was used, the equivalent number is 1. NaOH 1 BaSO4 2 Examples of reactions with complexes are: H2CO3 ⫹ NaOH 3 NaHCO3 ⫹ H2O H2CO3 1 Na2SO4 2 2AgNO3 2 2KCN 2 2AgCN 2 2KNO3 2 AgNO3 1 KCN 1 AgCN 1 KNO3 1 ⫹ NiSO4 ⫹ 4KCN 3 Ni(CN)2⫺ 4 ⫹ K2SO4 ⫹ 2K H2O 1 Thus, the equivalent number is equal to the number of hydroxyl groups in a compound. 2) Precipitation and complex reactions—In these reactions the equivalents are equal to the weight of substance that will furnish, react with, or be chemically equivalent to one mole of univalent cation in the precipitate or complex formed. An example of a precipitation reaction is: NiSO4 2 4KCN 2 Ni(CN)2⫺ 4 2 K2SO4 2 2K⫹ 2 NiSO4 2 2KCN 1 Ni(CN)2⫺ 4 2 K2SO4 2 K⫹ 1 3) Oxidation–reduction reactions—In these reactions the equivalent weight of a substance is the weight that will furnish, react with, or be chemically equivalent to one electron transferred. First, determine the number of electrons transferred in a redox reaction by writing out a balanced equation and the reduction and oxidation half-reactions. For example: BaCl2 ⫹ Na2SO4 3 BaSO4 ⫹ 2NaCl 3 EXPRESSION OF RESULTS (1050)/Units 5Na2C2O4 ⫹ 2KMnO4 ⫹ 8H2SO4 3 5Na2SO4 ⫹ 2MnSO4 ⫹ z 1V 1M 1 ⫽ z 2V 2M 2 K2SO4 ⫹ 10CO2 ⫹ 8H2O where: 5Na2C2O4 ⫹ 5H2SO4 3 5Na2SO4 ⫹ 10CO2 ⫹ 10H⫹ ⫹ 10e ⫺ z ⫽ equivalent number and M ⫽ molarity of substance in solution. 2KMnO4 ⫹ 3H2SO4 ⫹ 10H⫹ ⫹ 10e ⫺ 3 2MnSO4 ⫹ K2SO4 ⫹ 8H2O This relationship can be used to compute molarities or volumes of reagents being compared with one another. For example, the above equation could be reduced to the following for potassium permanganate (1) and sodium oxalate (2). Then divide the stoichiometric equation coefficients of the complete balanced equation by the number of electrons transferred in the half-reaction equations (in this case, 10) and reduce the fractions to yield: 5V 1M 1 ⫽ 2V 2M 2 Na2C2O4 KMnO4 4H2SO4 Na2SO4 MnSO4 K2SO4 CO2 4H2O 2 5 5 2 5 10 1 5 3. p Functions With these quantities, only one electron is transferred, and thus the amounts represent the equivalents. In this example the equivalent weights of these compounds in the reaction can be computed as follows: Equivalent weight ⫽ Concentrations can be reported in the form of p functions (e.g., pH). This form is used for expressing values when the analyte concentration varies over orders of magnitude. The pX concept is defined8,9 in terms of analyte activity, rather than concentration, as follows: molecular weight g/mole equivalent number Equivalent weight KMnO4 ⫽ pX ⫽ ⫺log (a x) KMnO4 158.04 g/mole ⫽ 5 5 where: ⫽ 31.61 g/equivalent ax ⫽ activity of analyte x in solution. Na2C2O4 134.00 g/mole Equivalent weight Na2C2O4 ⫽ ⫽ 2 2 Activity coefficients relate activity to concentration: ⫽ 67.00 g/equivalent pX ⫽ ⫺log (c x␥ x/c o) where: d. Normality: Normality, as defined in ¶ b above, may be calculated as follows: Normality ⫽ cx ⫽ molar concentration of x, mole/L ␥x ⫽ activity coefficient, dimensionless, and c° ⫽ molar concentration at standard state, mole/L. weight of substance (equivalent weight of substance)(liters of solution) With the development of electrode measurement technology, pH and other p functions have become more commonly used, especially for very low concentrations of analytes. However, the electrode measures only activity, ax, and not concentration, cx, directly. The Debye-Hückel equation provides a way to estimate the activity coefficients in low-ionic-strength solutions of 0.001M or less. This equation,4,10 –12 which relates concentration to activity coefficient, is: The normality concept allows substances to be compared with one another stoichiometrically through the relationship. V 1N 1 ⫽ V 2N 2 where: V ⫽ volume, mL or L, N ⫽ normality, and 1, 2 ⫽ compound 1 or 2. ⫺log ␥ i ⫽ If one knows the values of three of the variables, then the fourth can be computed. There is a trend to move from the normality6 concept because of possible ambiguities in determining normalities. These ambiguities arise because a substance being compared might have more than one computed normality concentration (e.g., potassium cyanide when it reacts with silver ion) and still be in the same concentration. The above equation relating volume and normality of one substance to those of another can be written using the following equation in terms of molarity: Az i2I 1/2 1 ⫹ B I 1/2 where: ␥I ⫽ analyte ion activity coefficient, zi ⫽ charge on ion species, I ⫽ ionic strength of all the ions in solution ⫽ 1⁄2 ⌺cizi2, mole/L, ⫽ radius of ionic atmosphere, picometers (see Table 1050:IV for values), and A, B ⫽ Debye-Hückel equation constants (see Table 1050:V for values given by temperature and molar or molal concentration). 4 EXPRESSION OF RESULTS (1050)/Units TABLE 1050:IV. EFFECTIVE HYDRATED RADIUS Type and Charge of Ion FOR COMMON IONS Ion Size, pm Ion Inorganic, ⫾1 H⫹ 900 800 700 600 500 Li⫹ Na⫹, CdCl⫹, ClO2⫺, IO3⫺, HCO3⫺,H2PO4⫺, HSO3⫺, H2AsO4⫺, Co(NH3)4(NO2)2⫹ 450 400 OH⫺, F⫺, SCN⫺, OCN⫺, HS⫺, ClO3⫺, ClO4⫺, BrO3⫺, IO4⫺, MnO4⫺ ⫹ K , Cl⫺, Br⫺, I⫺, CN⫺, NO2⫺, NO3⫺ Rb⫹, Cs⫹, NH4⫹, Tl⫹, Ag⫹ Inorganic, ⫾2 350 300 250 Mg2⫹, Be2⫹ 800 700 600 500 450 400 Ca2⫹, Cu2⫹, Zn2⫹, Sn2⫹, Mn2⫹, Fe2⫹, Ni2⫹, Co2⫹ Sr2⫹, Ba2⫹, Cd2⫹, Hg2⫹ , S2⫺ , S2O42⫺, WO42⫺ Pb2⫹, CO32⫺, SO32⫺, MoO42⫺, Co(NH3)5Cl2⫹, Fe(CN)5NO2⫹ Hg22⫹, SO42⫺, S2O32⫺, S2O62⫺, S2O82⫺, SeO42⫺, CrO42⫺, HPO42⫺ Inorganic, ⫾3 Al3⫹, Fe3⫹, Cr3⫹, Sc3⫹, Y3⫹, In3⫹, lanthanides* 900 500 400 PO43⫺, Fe(CN)63⫺, Cr(NH3)63⫹, Co(NH3)63⫹, Co(NH3)5H2O3⫹ Inorganic, ⫾4 Th4⫹, Zr4⫹, Ce4⫹, Sn4⫹ Fe(CN)64⫺ Organic, ⫾1 HCOO⫺, H2citrate⫺, CH3NH3⫹, (CH3)2NH2⫹ NH3⫹CH2COOH, (CH3)3NH⫹, C2H5NH3⫹ CH3COO⫺, CH2ClCOO⫺, (CH3)4N⫹, (C2H5)2NH2⫹, NH2CH2COO⫺ CHCl2COO⫺, CCl3COO⫺, ((C2H5)3NH⫹, (C3H7)NH3⫹ C6H5COO⫺, C6H4OHCOO⫺, C6H4ClCOO⫺, C6H5CH2COO⫺, CH2 ⫽ CHCH2COO⫺, (CH3)2CHCHCOO⫺, (C2H5)4N⫹, (C3H7)2NH2⫹ [OC6H2(NO3)3]⫺, (C3H7)3NH⫹, CH3OC6H4COO⫺ (C6H5)2CHCOO⫺, (C3H7)4N⫹ 350 400 Organic, ⫾2 (COO)22⫺, Hcitrate2⫺ H2C(COO)22⫺, (CH2COO)22⫺, (CHOHCOO)22⫺ C6H4(COO)22⫺, H2C(CH2COO)22⫺, CH2CH2COO22⫺ [OOC(CH2)5COO]2⫺, [OOC(CH2)6COO]2⫺, Congo red anion2⫺ 450 500 600 700 Organic, ⫾3 Citrate3⫺ 500 1100 500 450 500 600 700 800 * Elements 57–71 in the periodic table. SOURCE: Data from Kielland, J. 1937. Individual activity coefficients of ions in aqueous solutions. J. Amer. Chem. Soc., 59:1675; table in Harris, D.C. 1982. Quantitative Chemical Analysis. W.H. Freeman & Co., New York, N.Y. Example: Compute the hydrogen ion concentration of a solution of pure water with sufficient potassium chloride to make it 0.1M. The measured pH is 6.98 at 25°C. The ionic strength of the solution is I ⫽ 1/2 I ⫽ 1⁄2 [(0.1)(1)2 ⫹ (0.1)(1)2] ⫽ 0.1 mole/L Using the computed I and the values from Tables 1050:IV and 1050:V for (900 pm), A(0.5092), and B(0.003286), the DebyeHückel equation yields 冘 c iz i2 I ⫽ 1/2 [(c K⫹)(1)2 ⫹ (c Cl ⫺)(1)2 ⫹ (c H⫹)(1)2 ⫹ (c OH ⫺)(1)2] log ␥ H⫹ ⫽ The hydrogen ion and the hydroxyl ion concentrations (estimated to be about 10-7 mole/L) are insignificant in this example. Thus, ⫺(0.5092)(1)2 (0.1)0.5 1 ⫹ (0.003286)(900)(0.1)0.5 ␥H⫹ ⫽ 0.8256 5 ⫽ ⫺0.8321 EXPRESSION OF RESULTS (1050)/Units TABLE 1050:V. VALUES OF A B AND FROM 0 TO 100°C FOR DEBYE-H ÜCKEL EQUATION12 Temperature °C A (Abs. Units) in Terms of Unit Volume of Solution A⬘ (Abs. Units) in Terms of Unit Weight of Solvent B (cm⫺1 ⫻ 108) in Terms of Unit Volume of Solution B⬘ (cm⫺1 ⫻ 108) in Terms of Unit Weight of Solvent 0 5 10 15 18 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0.4883 0.4921 0.4961 0.5002 0.5028 0.5046 0.5092 0.5141 0.5190 0.5241 0.5296 0.5351 0.5410 0.5471 0.5534 0.5599 0.5668 0.5739 0.5814 0.5891 0.5972 0.6056 0.4883 0.4921 0.4960 0.5000 0.5025 0.5042 0.5085 0.5130 0.5175 0.5221 0.5270 0.5319 0.5371 0.5425 0.5480 0.5537 0.5596 0.5658 0.5722 0.5788 0.5857 0.5929 0.3241 0.3249 0.3258 0.3267 0.3273 0.3276 0.3286 0.3297 0.3307 0.3318 0.3330 0.3341 0.3353 0.3366 0.3379 0.3392 0.3406 0.3420 0.3434 0.3450 0.3466 0.3482 0.3241 0.3249 0.3258 0.3266 0.3271 0.3273 0.3281 0.3290 0.3297 0.3305 0.3314 0.3321 0.3329 0.3338 0.3346 0.3354 0.3363 0.3372 0.3380 0.3390 0.3399 0.3409 SOURCE: MANOV, G.C., R.G. BATES, W.J. HAMER & S.F. ACRES. 1943. Values of the constants in the Debye-Hückel equation for activity coefficients. J. Amer. Chem. Soc. 65:1765s. Since These stoichiometric factors provide for taking the analysis data in the form analyzed first, dividing the mass concentration by the analyte’s formula weight, and multiplying by the formula weight of the reported substance desired. For example, pH ⫽ –log a H⫹ at the measured pH of 6.98, mg/L chloride as NaCl ⫽ a H⫹ ⫽ 10⫺6.98 ⫽ 1.047 ⫻ 10⫺7 mole/L mg Cl⫺ formula weight of sodium chloride ⫻ L formula weight of chlorine Be sure that the formula weights of the substance to be reported and of the analyte include the same number of atoms of the common element. Some analyses involve multiple reactions. For example, listed below are the reactions involved in the Winkler determination.13 Then, c H⫹ ⫽ (a H⫹)/(␥H⫹) cH ⫹ ⫽ 1.047 ⫻ 10⫺7/0.8256 ⫽ 1.268 ⫻ 10⫺7 mole/L Note that the hydrogen ion concentration is about 21% greater than was indicated by the hydrogen activity given by the pH electrode. The same type of computation can be made for determining the ionic concentration from the measured electrode activity for fluoride, silver, and other substances. This is important if the true concentration of the ion is to be determined. 2MnSO4 ⫹ 4KOH 3 2Mn(OH)2 ⫹ 2K2SO4 2 Mn(OH)2 ⫹ O2 3 2MnO(OH)2 2 MnO(OH)2 ⫹ 4KI ⫹ 4H2SO4 3 2I2 ⫹ 2MnSO4 ⫹ 2K2SO4 ⫹ 6H2O 2I2 ⫹ 4Na2S2O3 3 4NaI ⫹ 2Na2S4O6 In the last equation, one mole of sodium thiosulfate is equivalent to one mole of elemental iodine. The number of moles of elemental iodine is equivalent to half the moles of manganese oxide hydroxide [MnO(OH)2] in the second and third equations. Then the number of moles of oxygen is half the number of moles of MnO(OH)2, and thus the equivalent weight of elemental oxygen is one-fourth that of elemental iodine or sodium thiosulfate (32 g/4 ⫽ 8g O2 / equivalent weight). This type of reasoning 4. Stoichiometric Factors Some analyses are reported as concentrations of other substances. Conversion factors to accomplish this are called stoichiometric factors. Examples are: alkalinity or acidity reported as CaCO3, total hardness as CaCO3, magnesium hardness as CaCO3, ammonia as nitrogen, nitrate as nitrogen, nitrite as nitrogen, and phosphate as phosphorus. 6 EXPRESSION OF RESULTS (1050)/Significant Figures must be used to determine the analytical relationship between the titrant and the analyte being sought. 2. 5. Molality 3. It is sometimes convenient to relate molality to molarity. That relationship is as follows.14 4. M⫽ 1000m (1000 ⫹ W Bm) 5. 6. where: M m WB ⫽ ⫽ ⫽ ⫽ 7. molarity, mole/L, density of solution, g/cm3, molality, mole/kg, and molecular mass of solute, g. 8. The molality in terms of molarity is given as 9. m⫽ 10. 1000M 1000 ⫺ MW B 11. Many electrochemical measurements are made in terms of molality rather than molarity. 12. 6. References 13. 1. AMERICAN SOCIETY FOR TESTING AND MATERIALS. 1993. Standard Practice for Use of the International System of Units (SI) (The 14. Modernized Metric System). E380-93, American Soc. Testing & Materials, Philadelphia, Pa. BUREAU INTERNATIONAL DES POIDS ET MEASURES. 1991. The International System of Units (SI), 6th ed. Bur. International Poids & Measures, Sevres, France. MILLS, I., T. CVITAS, K. HOMANN, N. KALLEY & K. KUCHITSU. 1993. Quantities, Units and Symbols in Physical Chemistry, 2nd ed. Blackwell Scientific Publications, Oxford, U.K. MARSH, K.M., ed. 1987. Recommended Reference Materials for the Realization of Physicochemical Properties, Blackwell Scientific Publications, Oxford, U.K. CVITAS, T. & I. MILLS. 1994. Replacing gram equivalents and normalities. Chem. Internat. 16:123. AYRES, G.H. 1958. Quantitative Chemical Analysis. Harper & Row Publishers, New York, N.Y. KENNER, C.T. & K.W. BUSCH. 1979. Quantitative Analysis. Harper & Row Publishers, New York, N.Y. FREISER, H. & G.H. NANCOLLAS. 1987. Compendium of Analytical Nomenclature, Definitive Rules ⫺ 1987, Blackwell Scientific Publications, London, U.K. GOLD, V., K.L. LOENING, A.D. MCNAUGHT & P. SEHMI. 1987. Compendium of Chemical Technology ⫺ IUPAC Recommendations. Blackwell Scientific Publications, Oxford, U.K. HARRIS, D.C. 1982. Quantitative Chemical Analysis. W.H. Freeman & Co., New York, N.Y. SKOOG, D.A. & D.M. WEST. 1976. Fundamentals of Analytical Chemistry, 3rd ed. Holt, Rinehart & Winston, New York, N.Y. MANOV, G.C., R.G. BATES, W.J. HAMER & S.F. ACRES. 1943. Values of the constants in the Debye-Hückel equation for activity coefficients. J. Amer. Chem. Soc. 65:1765. HACH COMPANY. 1984. Chemical Procedures Explained. Hach Co., Loveland, Colo. WEAST, R.C. & D.R. LIDE. 1989. Handbook of Chemistry and Physics, 70th ed. CRC Press, Inc., Boca Raton, Fla. 1050 B. Significant Figures 1. Reporting Requirements 2. Rounding Off To avoid ambiguity in reporting results or in presenting directions for a procedure, it is customary to use “significant figures.” All digits in a reported result are expected to be known definitely, except for the last digit, which may be in doubt. Such a number is said to contain only significant figures. If more than a single doubtful digit is reported, the extra digit or digits are not significant. This is an important distinction. Extra digits should be carried in calculation (see ¶ 2 below). If an analytical result is reported as “75.6 mg/L,” the analyst should be quite certain of the “75,” but may be uncertain as to whether the “.6” should be .5 or .7, or even .4 or .8, because of unavoidable uncertainty in the analytical procedure. If the standard deviation were known from previous work to be 2 mg/L , the analyst would have, or should have, rounded off the result to “76 mg/L” before reporting it. On the other hand, if the method was so good that a result of “75.64 mg/L” could have been conscientiously reported, then the analyst should not have rounded it off to 75.6 mg/L. Report only such figures as are justified by the accuracy of the work. Do not follow the common practice of requiring that quantities listed in a column have the same number of figures to the right of the decimal point. Round off by dropping digits that are not significant. If the digit 6, 7, 8, or 9 is dropped, increase preceding digit by one unit; if the digit 0, 1, 2, 3, or 4 is dropped, do not alter preceding digit. If the digit 5 is dropped, round off preceding digit to the nearest even number: thus 2.25 becomes 2.2 and 2.35 becomes 2.4. When making calculations, perform all computations before rounding off results. Repeated rounding off can result in changing the value of a reported result. For example, taking the measured value of 77.46 and rounding off to three significant figures yields 77.5. If the latter number were rounded a second time to two significant figures, the result would be 78. This is clearly a different result from a rounding of the original value of 77.46 to two significant figures (77). 3. Ambiguous Zeros The digit 0 may record a measured value of zero or it may serve merely as a spacer to locate the decimal point. If the result of a sulfate determination is reported as 420 mg/L, the 7 EXPRESSION OF RESULTS (1050)/Significant Figures 56 ⫻ 0.003 462 ⫻ 43.22 1.684 report recipient may be in doubt whether the zero is significant or not, because the zero cannot be deleted. If an analyst calculates a total residue of 1146 mg/L, but realizes that the 4 is somewhat doubtful and that therefore the 6 has no significance, the answer should be rounded off to 1150 mg/L and so reported but here, too, the report recipient will not know whether the zero is significant. Although the number could be expressed as a power of 10 (e.g., 11.5 ⫻ 102 or 1.15 ⫻ 103), this form is not used generally because it would not be consistent with the normal expression of results and might be confusing. In most other cases, there will be no doubt as to the sense in which the digit 0 is used. It is obvious that the zeros are significant in such numbers as 104 and 40.08. In a number written as 5.000, it is understood that all the zeros are significant, or else the number could have been rounded off to 5.00, 5.0, or 5, whichever was appropriate. Whenever zero is ambiguous, it is advisable to accompany the result with an estimate of its uncertainty. Sometimes, significant zeros are dropped without good cause. If a buret is read as “23.60 mL,” it should be so recorded, and not as “23.6 mL.” The first number indicates that the analyst took the trouble to estimate the second decimal place; “23.6 mL” would indicate an imprecise reading of the buret. A ten-place calculator yields an answer of “4.975 740 998.” If the number 56 is an exact number (a count or a mathematical constant such as ), it has no error associated with it and is considered to have unlimited significant figures. In that case, round off the result of the calculation to “4.976” because other numbers have only four significant figures. However, if 56 is an approximate measurement with uncertainly associated beyond the second figure, round off the result to “5.0” because 56 has only two significant figures. When numbers are added or subtracted, the number that has the least precision in its last significant digit limits the number of places that can justifiably be carried in the sum or difference. It is acceptable practice to carry one more digit beyond the least precise significant digit. Example: The following numbers are to be added: 0.0072 12.02 4.0078 25.9 4886 4. Standard Deviation Suppose that a set of potential results is normally distributed with a standard deviation of 100 mg/L and that a calculated value turns out to be 1450 mg/L. Then that 1450 is the best estimate available of this particular value, and from a Bayesian point of view, there would only be about a 31% chance that the true value was 1400 or lower or that the true value was 1500 or higher. It does not make sense to round the 1450 value to 1400. The arbitrary subtracting of half a standard deviation leaves us with a value that does not represent our best estimate well. The standard deviation should influence the last significant figures of a calculation only by ⫾ 0.51 and if more, the number of significant figures cannot be justified. When reporting numbers in the form x ⫾ y, always state whether y represents standard deviation, standard error, confidence limit, or an estimate of maximum bias. Standard deviations and standard errors often should be reported with extra digits (compared with single measurements) because they are calculated from variances and because they are square roots (see 1050B.5). When interpreting a quantity such as 1480 ⫾ 40, be aware that this notation seldom indicates a belief that the true value lies anywhere in the range from 1440 to 1520 with equal probability; instead, the probability is concentrated near the central value (1480). The number “4886” is the least precise number (decimal place). Round each number in the sum to one or more digits beyond the least precise number and compute the sum. 0.0 12.0 4.0 25.9 4886. 4927.9 Round the result to the precision of the least precise number in the sum, 4928. Some calculators or computers round off numbers by a different rule that tends to bias results toward the larger digit in the last significant figure. Before using such a device, determine which rounding techniques are programmed and if an incorrect rounding method is being used, reprogram to follow the correct rounding method. If it is not possible to reprogram, take unrounded number and manually round off using the correct scientific roundoff method. Interpret the foregoing guidelines flexibly. For example, consider a series of measurements of variable u, v, and w, and a derived variable y ⫽ uv/w that is calculated for each case. Suppose measurements give y ⫽ 9.90972 and y ⫽ 10.11389 for two cases with similar measured values. According to the guidelines, the first y should be rounded to 9.91; the final digit here is about 0.1% of the overall result. For comparable precision, round the y value for the second case to 10.11 (not 10.1) because the fourth digit is also about 0.1% of the number. To generalize, more significant digits should at times be provided for quantities having 1, 2, or 3 (say) as leading digits than for quantities beginning with 7, 8, or 9. 5. Calculations As a starting point, round off the results of any calculation in which several numbers are multiplied and divided to as few significant figures as are present in the factor with the fewest significant figures.2 However, several potential reasons to modify that guideline are noted below. Example: Assume that the following calculation must be made to obtain the results of an analysis: 8 EXPRESSION OF RESULTS (1050)/Other Considerations 6. General Arithmetic Functions Flexibility is also needed for the average of several numbers. The standard deviation (standard error) of a mean of N numbers is only 1/公N as large as the standard deviation of the individual numbers. Thus a mean of 100 numbers like d.dd is known to a precision of d.ddd. Even if fewer values than 100 are averaged, showing an extra digit may be justifiable. Because variances are averages of squared deviations, they too are more precise than individual deviations. Therefore, reporting variances with an extra digit or two is often justifiable. These guidelines refer only to final reported values. When performing any series of mathematical or statistical calculations, do not round measurements or other numbers until the very end of the analysis. Keep two or three extra digits for all intermediate calculations, to reduce round-off errors, which can be substantial. In analytical calculations, it may be necessary to use functions other than simple arithmetic, such as logarithmic, exponential, or trigonometric functions. A detailed treatment of significant figures in such cases is available.3 7. References 1. SCARBOROUGH, J.B. 1955. Numerical Mathematical Analysis, 3rd ed. Johns Hopkins Press, Baltimore, Md. 2. AMERICAN SOCIETY FOR TESTING AND MATERIALS. 1993. Standard Practice for Using Significant Digits for Test Data to Determine Conformance with Specifications. E29-93, American Soc. Testing & Materials, Philadelphia, Pa. 3. GRAHAM, D.M. 1989. Significant figure rules for general arithmetic functions. J. Chem. Ed. 66:573. 1050 C. Other Considerations 1. Scientific and Engineering Notation ment, temperature of drying, and temperature of conductance measurement. Other conditions, such as total metals or dissolved metals, are defined by filtration method. Table 1050:VI gives examples of a number of analytes that may be expressed in other terms. Include complete information on the analyte, because many times the data may be used in other databases. The user may not always be able to assume the chemical form or special physical conditions under which the analyte was analyzed and reported. Scientific notation is defined by placing a number, N, in the format N ⫽ Q ⫻ 10r where: Q ⫽ mantissa from 1 to 9.999, and r ⫽ integer exponent. 3. Propagation of Error Engineering notation1,2 is defined by placing the number, N, in the format For information on types and sources of error, see Section 1030. Frequently, an analytical procedure consists of a number of steps in which one measurement is used in computing another measurement. Each measurement has an associated error. The greater the number of measurements used to obtain a computed result, the greater the possible associated error. This process is known as propagation of error.3 When errors are compounded, they usually increase, but never decrease. In the propagation of error, we are considering the value of the measurement as well as its associated uncertainty. All measurements have an associated value of uncertainty. The only measurements that have no uncertainty are numbers of events (counts), when properly defined, and mathematical constants. Further information on propagation of error is available.4 – 6 N ⫽ P ⫻ 103t where: P ⫽ mantissa from 1 to 999.999, and 3t ⫽ exponent in integer multiples of 3. Engineering notation is a convenient form for reporting results when using SI units. It allows easy selection of the proper SI prefix name. 2. Specifying Analyte Nomenclature When reporting the concentration of an analyte, include not only the name of the analyte but its equivalent if the concentration units are in terms of the equivalent. A report giving only the analyte name is assumed to include only that chemical entity. Chemical equivalents such as nitrate nitrogen are used to ease accounting of nitrogen chemical forms by allowing them to be added. In the case of organic nitrogen, the analyst may not know the exact organic form of nitrogen analyzed, so for convenience the compound is reported in terms of nitrogen rather than the actual organic compound. Other concepts such as alkalinity are expressed in terms of calcium carbonate because the true form is not always known. Include other information such as temperature of pH measure- 4. References 1. TEXAS INSTRUMENTS. 1977. Personal Programming. Texas Instruments, Dallas, Tex. 2. HEWLETT PACKARD. 1989. Advanced Scientific Calculator – Owner’s Manual – HP-285. Hewlett Packard Co., Corvallis, Ore. 3. HARRIS, D.C. 1982. Quantitative Chemical Analysis. W.H. Freeman & Co., New York, N.Y. 4. SCARBOROUGH, J.G. 1955. Numerical Mathematical Analysis, 3rd ed. Johns Hopkins Press, Baltimore, Md. 9 EXPRESSION OF RESULTS (1050)/Other Considerations TABLE 1050:VI. EXAMPLES Analyte OF ALTERNATIVE EXPRESSIONS Units Alkalinity Ammonia, un-ionized Ammonium Bicarbonate Calcium Carbonate Conductivity Hydrogen sulfide Magnesium Nitrate Nitrite PCB mg/L mg/L mg/L mg/L mg/L mg/L s/m mg/L mg/L mg/L mg/L mg/L pH Silicon Sulfide Total dissolved solids Uranium — mg/L mg/L mg/L mg/L pCi/L Zinc mg/L mg/kg OF ANALYTICAL RESULTS Possible Reporting Nomenclature mg/L as CaCO3 or mg/L as HCO3⫺ mg/L as N or mg/L as NH3 mg/L as N or mg/L as NH4⫹ mg/L as HCO3⫺ or mg/L as CaCO3 mg/L as Ca2⫹ or mg/L as CaCO3 mg/L as CO32⫺ or mg/L as CaCO3 S/m at 25°C or S/m at t°C mg/L as H2S or mg/L as S2⫺ mg/L as Mg2⫹ or mg/L as CaCO3 mg/L as N or mg/L as NO3⫺ mg/L as N or mg/L as NO2⫺ mg/L as PCB or mg/L as decachlorobiphenyl or mg/L as Aroclor mixture pH at 25°C or pH at t°C mg/L as Si or mg/L as SiO2 mg/L as S or mg/L as H2S mg/L at 180°C or mg/L at t°C mg/L as U or mg/L as U3O8 pCi/L as natural isotopic abundance or pCi/L as specified isotopic abundance mg/L Zn as total or mg/L Zn as dissolved mg Zn/kg wet or mg Zn/kg dry 5. GUEDENS, W.J., J. YPERMAN, J. MULLENS & L.C. VANPOUCKE. 1993. Statistical analysis of errors: A practical approach for an undergraduate chemistry lab. Part I. The concepts. J. Chem. Edu. 70:776. 6. GUEDENS, W.J., J. YPERMAN, J. MULLENS & L.C. VANPOUCKE. 1998. Statistical analysis of errors: A practical approach for an undergraduate chemistry lab. Part 2. Some worked examples. J. Chem. Edu. 70:838. 10