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Yogesh
Indian Institute of Technology (IIT) Bombay, Mumbai
Chemical Engineering

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APPENDIX I Elements, their Atomic Number and Molar Mass Symbol Element Actinium Aluminium Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Bohrium Boron Bromine Cadmium Caesium Calcium Californium Carbon Cerium Chlorine Chromium Cobalt Copper Curium Dubnium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Hassium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Meitneium Mendelevium Atomic Number Molar mass/ (g mol 1) Element Ac Al Am Sb Ar As At Ba Bk Be Bi Bh B Br Cd Cs Ca Cf C Ce Cl Cr Co Cu Cm Db Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf Hs He Ho H In I Ir Fe Kr La Lr Pb Li Lu Mg Mn Mt Md 89 13 95 51 18 33 85 56 97 4 83 107 5 35 48 55 20 98 6 58 17 24 27 29 96 105 66 99 68 63 100 9 87 64 31 32 79 72 108 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 109 101 227.03 26.98 (243) 121.75 39.95 74.92 210 137.34 (247) 9.01 208.98 (264) 10.81 79.91 112.40 132.91 40.08 251.08 12.01 140.12 35.45 52.00 58.93 63.54 247.07 (263) 162.50 (252) 167.26 151.96 (257.10) 19.00 (223) 157.25 69.72 72.61 196.97 178.49 (269) 4.00 164.93 1.0079 114.82 126.90 192.2 55.85 83.80 138.91 (262.1) 207.19 6.94 174.96 24.31 54.94 (268) 258.10 Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Rutherfordium Samarium Scandium Seaborgium Selenium Silicon Silver Sodium Strontium Sulphur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Ununbium Ununnilium Unununium Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Symbol Atomic Number Hg Mo Nd Ne Np Ni Nb N No Os O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Rf Sm Sc Sg Se Si Ag Na Sr S Ta Tc Te Tb Tl Th Tm Sn Ti W Uub Uun Uuu U V Xe Yb Y Zn Zr 80 42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 104 62 21 106 34 14 47 11 38 16 73 43 52 65 81 90 69 50 22 74 112 110 111 92 23 54 70 39 30 40 Molar mass/ (g mol 1) 200.59 95.94 144.24 20.18 (237.05) 58.71 92.91 14.0067 (259) 190.2 16.00 106.4 30.97 195.09 (244) 210 39.10 140.91 (145) 231.04 (226) (222) 186.2 102.91 85.47 101.07 (261) 150.35 44.96 (266) 78.96 28.08 107.87 22.99 87.62 32.06 180.95 (98.91) 127.60 158.92 204.37 232.04 168.93 118.69 47.88 183.85 (277) (269) (272) 238.03 50.94 131.30 173.04 88.91 65.37 91.22 The value given in parenthesis is the molar mass of the isotope of largest known half-life. 261 Appendix APPENDIX II Some Useful Conversion Factors Common Unit of Mass and Weight 1 pound = 453.59 grams Common Units of Length 1 inch = 2.54 centimetres (exactly) 1 pound = 453.59 grams = 0.45359 kilogram 1 kilogram = 1000 grams = 2.205 pounds 1 gram = 10 decigrams = 100 centigrams = 1000 milligrams 1 gram = 6.022 1023 atomic mass units or u 1 atomic mass unit = 1.6606 10 24 gram 1 metric tonne = 1000 kilograms = 2205 pounds 1 mile = 5280 feet = 1.609 kilometres 1 yard = 36 inches = 0.9144 metre 1 metre = 100 centimetres = 39.37 inches = 3.281 feet = 1.094 yards 1 kilometre = 1000 metres = 1094 yards = 0.6215 mile 1 Angstrom = 1.0 10 8 centimetre = 0.10 nanometre = 1.0 10 10 metre = 3.937 10 9 inch Common Unit of Volume 1 quart = 0.9463 litre 1 litre = 1.056 quarts 1 litre = 1 cubic decimetre = 1000 cubic centimetres = 0.001 cubic metre 1 millilitre = 1 cubic centimetre = 0.001 litre = 1.056 10-3 quart 1 cubic foot = 28.316 litres = 29.902 quarts = 7.475 gallons Common Units of Energy 1 joule = 1 107 ergs 1 thermochemical calorie** = 4.184 joules = 4.184 107 ergs = 4.129 10 2 litre-atmospheres = 2.612 1019 electron volts 1 ergs = 1 10 7 joule = 2.3901 10 8 calorie 1 electron volt = 1.6022 10 19 joule = 1.6022 10 12 erg = 96.487 kJ/mol 1 litre-atmosphere = 24.217 calories = 101.32 joules = 1.0132 109 ergs 1 British thermal unit = 1055.06 joules = 1.05506 1010 ergs = 252.2 calories Common Units of Force* and Pressure 1 atmosphere = 760 millimetres of mercury = 1.013 105 pascals = 14.70 pounds per square inch 1 bar = 105 pascals 1 torr = 1 millimetre of mercury 1 pascal = 1 kg/ms2 = 1 N/m2 Temperature SI Base Unit: Kelvin (K) K = -273.15 C K = C + 273.15 F = 1.8( C) + 32 C = F 32 1.8 Force: 1 newton (N) = 1 kg m/s 2 , i.e.,the force that, when applied for 1 second, gives a 1-kilogram mass a velocity of 1 metre per second. ** The amount of heat required to raise the temperature of one gram of water from 14.50C to 15.50C. Note that the other units are per particle and must be multiplied by 6.022 1023 to be strictly comparable. * Chemistry 262 APPENDIX III Standard potentials at 298 K in electrochemical order Reduction half-reaction E J /V Reduction half-reaction E J /V H4XeO6 + 2H+ + 2e XeO3 + 3H2O F2 + 2e 2F O3 + 2H+ + 2e O2 + H2O 2 2 S2O8 + 2e 2SO4 Ag+ + e Ag+ Co3+ + e Co2+ H2O2 + 2H+ + 2e 2H2O Au+ + e Au Pb4+ + 2e Pb2+ 2HClO + 2H+ + 2e Cl2 + 2H2O Ce4+ + e Ce3+ 2HBrO + 2H+ + 2e Br2 + 2H2O MnO4 + 8H+ + 5e Mn2+ + 4H2O Mn3+ + e Mn2+ Au3+ + 3e Au Cl2 + 2e 2Cl 2 Cr2O 7 + 14H+ + 6e 2Cr3+ + 7H2O O3 + H2O + 2e O2 + 2OH O2 + 4H+ + 4e 2H2O + ClO 4 + 2H +2e ClO3 + 2H2O MnO2 + 4H+ + 2e Mn2+ + 2H2O Pt2+ + 2e Pt Br2 + 2e 2Br Pu4+ + e Pu3+ + NO 3 + 4H + 3e NO + 2H2O 2+ 2Hg + 2e Hg 2+ 2 ClO + H2O + 2e Cl + 2OH Hg2+ + 2e Hg + NO 3 + 2H + e NO2 + H2O + Ag + e Ag Hg 2+ 2 +2e 2Hg 3+ Fe + e Fe2+ BrO + H2O + 2e Br + 2OH 2 Hg2SO4 +2e 2Hg + SO4 2 MnO4 + 2H2O + 2e MnO2 + 4OH 2 MnO 4 + e MnO4 I2 + 2e 2I I3 + 2e 3I +3.0 +2.87 +2.07 +2.05 +1.98 +1.81 +1.78 +1.69 +1.67 +1.63 +1.61 +1.60 +1.51 +1.51 +1.40 +1.36 +1.33 +1.24 +1.23 +1.23 +1.23 +1.20 +1.09 +0.97 +0.96 +0.92 +0.89 +0.86 +0.80 +0.80 +0.79 +0.77 +0.76 +0.62 +0.60 +0.56 +0.54 +0.53 Cu+ + e Cu NiOOH + H2O + e Ni(OH)2 + OH 2 Ag2CrO4 + 2e 2Ag + CrO4 O2 + 2H2O + 4e 4OH ClO 4 + H2O + 2e ClO3 + 2OH 3 4 [Fe(CN)6] + e [Fe(CN)6] Cu2+ + 2e Cu Hg2Cl2 + 2e 2Hg + 2Cl AgCl + e Ag + Cl Bi3+ + 3e Bi 2 + 4H+ + 2e H2SO3 + H2O SO4 2+ Cu + e Cu+ Sn4+ + 2e Sn2+ AgBr + e Ag + Br Ti4+ + e Ti3+ 2H+ + 2e H2 +0.52 +0.49 +0.45 +0.40 +0.36 +0.36 +0.34 +0.27 +0.27 +0.20 +0.17 +0.16 +0.15 +0.07 0.00 0.0 by definition 0.04 0.08 0.13 0.14 0.14 0.15 0.23 0.26 0.28 0.34 0.34 0.36 0.37 0.40 0.40 0.41 0.44 0.44 0.48 0.49 0.61 0.74 0.76 Fe3+ + 3e Fe O2 + H2O + 2e HO 2 + OH 2+ Pb + 2e Pb In+ + e In Sn2+ + 2e Sn AgI + e Ag + I Ni2+ + 2e Ni V3+ + e V2+ Co2+ + 2e Co In3+ + 3e In Tl+ + e Tl PbSO4 + 2e Pb + SO2 4 Ti3+ + e Ti2+ Cd2+ + 2e Cd In2+ + e In+ Cr3+ + e Cr2+ Fe2+ + 2e Fe In3+ + 2e In+ S + 2e S2 In3+ + e In2+ U4+ + e U3+ Cr3+ + 3e Cr Zn2+ + 2e Zn (continued) 263 Appendix APPENDIX III CONTINUED Reduction half-reaction EJ /V Reduction half-reaction EJ /V Cd(OH)2 + 2e Cd + 2OH 2H2O + 2e H2 + 2OH Cr2+ + 2e Cr Mn2+ + 2e Mn V2+ + 2e V Ti2+ + 2e Ti Al3+ + 3e Al U3+ + 3e U Sc3+ + 3e Sc Mg2+ + 2e Mg Ce3+ + 3e Ce 0.81 0.83 0.91 1.18 1.19 1.63 1.66 1.79 2.09 2.36 2.48 La3+ + 3e La Na+ + e Na Ca2+ + 2e Ca Sr2+ + 2e Sr Ba2+ + 2e Ba Ra2+ + 2e Ra Cs+ + e Cs Rb+ + e Rb K+ +e K Li+ + e Li 2.52 2.71 2.87 2.89 2.91 2.92 2.92 2.93 2.93 3.05 Chemistry 264 APPENDIX IV Logarithms Sometimes, a numerical expression may involve multiplication, division or rational powers of large numbers. For such calculations, logarithms are very useful. They help us in making difficult calculations easy. In Chemistry, logarithm values are required in solving problems of chemical kinetics, thermodynamics, electrochemistry, etc. We shall first introduce this concept, and discuss the laws, which will have to be followed in working with logarithms, and then apply this technique to a number of problems to show how it makes difficult calculations simple. We know that 23 = 8, 32 = 9, 53 = 125, 70 = 1 In general, for a positive real number a, and a rational number m, let am = b, where b is a real number. In other words the mth power of base a is b. Another way of stating the same fact is logarithm of b to base a is m. If for a positive real number a, a 1 am = b, we say that m is the logarithm of b to the base a. b We write this as lo g a = m , log being the abbreviation of the word logarithm . Thus, we have log 2 8 = 3, Since 2 log 3 9 = 2, log 125 Since 3 = 3, 3 =8 2 =9 3 = 125 0 =1 Since 5 5 log 7 1 = 0, Since 7 Laws of Logarithms In the following discussion, we shall take logarithms to any base a, (a > 0 and a 1) First Law: loga (mn) = logam + logan Proof: Suppose that logam = x and logan = y Then ax= m, ay = n Hence mn = ax.ay = ax+y It now follows from the definition of logarithms that loga (mn) = x + y = loga m loga n m = loga m logan n Second Law: loga Proof: Let logam = x, logan = y 265 Appendix Then ax = m, ay = n Hence m = n a a x y =a x y Therefore log a m n = x y = log a m log a n Third Law : loga(mn) = n logam Proof : As before, if logam = x, then ax = m Then m n ( ) = a x n =a nx giving loga(mn) = nx = n loga m Thus according to First Law: the log of the product of two numbers is equal to the sum of their logs. Similarly, the Second Law says: the log of the ratio of two numbers is the difference of their logs. Thus, the use of these laws converts a problem of multiplication / division into a problem of addition/ subtraction, which are far easier to perform than multiplication/division. That is why logarithms are so useful in all numerical computations. Logarithms to Base 10 Because number 10 is the base of writing numbers, it is very convenient to use logarithms to the base 10. Some examples are: since 101 = 10 log10 10 = 1, since 102 = 100 log10 100 = 2, log10 10000 = 4, since 104 = 10000 since 10 2 = 0.01 log10 0.01 = 2, since 10 3 = 0.001 log10 0.001 = 3, and log101 = 0 since 100 = 1 The above results indicate that if n is an integral power of 10, i.e., 1 followed by several zeros or 1 preceded by several zeros immediately to the right of the decimal point, then log n can be easily found. If n is not an integral power of 10, then it is not easy to calculate log n. But mathematicians have made tables from which we can read off approximate value of the logarithm of any positive number between 1 and 10. And these are sufficient for us to calculate the logarithm of any number expressed in decimal form. For this purpose, we always express the given decimal as the product of an integral power of 10 and a number between 1 and 10. Standard Form of Decimal We can express any number in decimal form, as the product of (i) an integral power of 10, and (ii) a number between 1 and 10. Here are some examples: (i) 25.2 lies between 10 and 100 25.2 25.2 = 1 10 = 2.52 10 10 (ii) 1038.4 lies between 1000 and 10000. 1 0 3 8 .4 = 1 0 3 8 .4 10 3 = 1 .0 3 8 4 1 0 3 1000 (iii) 0.005 lies between 0.001 and 0.01 3 3 0.005 = (0.005 1000) 10 = 5.0 10 (iv) 0.00025 lies between 0.0001 and 0.001 0.00025 = (0.00025 10000) 10 4 = 2.5 10 4 Chemistry 266 In each case, we divide or multiply the decimal by a power of 10, to bring one non-zero digit to the left of the decimal point, and do the reverse operation by the same power of 10, indicated separately. Thus, any positive decimal can be written in the form p n = m 10 where p is an integer (positive, zero or negative) and 1< m < 10. This is called the standard form of n. Working Rule 1. Move the decimal point to the left, or to the right, as may be necessary, to bring one non-zero digit to the left of decimal point. p 2. (i) If you move p places to the left, multiply by 10 . (ii) If you move p places to the right, multiply by 10 p. (iii) If you do not move the decimal point at all, multiply by 100. (iv) Write the new decimal obtained by the power of 10 (of step 2) to obtain the standard form of the given decimal. Characteristic and Mantissa Consider the standard form of n n = m 10p, where 1 < m < 10 Taking logarithms to the base 10 and using the laws of logarithms log n = log m + log 10p = log m + p log 10 = p + log m Here p is an integer and as 1 < m < 10, so 0 < log m < 1, i.e., m lies between 0 and 1. When log n has been expressed as p + log m, where p is an integer and 0 log m < 1, we say that p is the characteristic of log n and that log m is the mantissa of log n. Note that characteristic is always an integer positive, negative or zero, and mantissa is never negative and is always less than 1. If we can find the characteristics and the mantissa of log n, we have to just add them to get log n. Thus to find log n, all we have to do is as follows: 1. Put n in the standard form, say n = m 10p, 1 < m <10 2. Read off the characteristic p of log n from this expression (exponent of 10). 3. Look up log m from tables, which is being explained below. 4. Write log n = p + log m If the characteristic p of a number n is say, 2 and the mantissa is .4133, then we have log n = 2 + .4133 which we can write as 2.4133. If, however, the characteristic p of a number m is say 2 and the mantissa is .4123, then we have log m = 2 + .4123. We cannot write this as 2.4123. (Why?) In order to avoid this confusion we write 2 for 2 and thus we write log m = 2.4 1 2 3 . Now let us explain how to use the table of logarithms to find mantissas. A table is appended at the end of this Appendix. Observe that in the table, every row starts with a two digit number, 10, 11, 12,... 97, 98, 99. Every column is headed by a one-digit number, 0, 1, 2, ...9. On the right, we have the section called Mean differences which has 9 columns headed by 1, 2...9. 0 1 2 3 4 .. .. .. .. .. .. 61 7853 7860 7868 7875 7882 62 7924 7931 7935 7945 7954 63 7993 8000 8007 8014 8021 . .Now suppose .. . . we wish .. .. to. . find log 5 6 7 .. .. .. 7889 7896 7803 7959 7966 7973 8028 8035 8041 .. .. .. (6.234). Then look 8 9 1 2 3 4 .. .. .. .. .. .. 7810 7817 1 1 2 3 7980 7987 1 1 2 3 8048 8055 1 1 2 3 . .the row . . starting . . . . with . . 62. .. into 5 6 7 8 9 .. .. .. .. .. 4 4 5 6 6 3 4 5 6 6 3 4 5 6 6 . . this . . .row, . . . look .. In 267 Appendix at the number in the column headed by 3. The number is 7945. This means that log (6.230) = 0.7945* But we want log (6.234). So our answer will be a little more than 0.7945. How much more? We look this up in the section on Mean differences. Since our fourth digit is 4, look under the column headed by 4 in the Mean difference section (in the row 62). We see the number 3 there. So add 3 to 7945. We get 7948. So we finally have log (6.234) = 0.7948. Take another example. To find log (8.127), we look in the row 81 under column 2, and we find 9096. We continue in the same row and see that the mean difference under 7 is 4. Adding this to 9096, and we get 9100. So, log (8.127) = 0.9100. Finding N when log N is given We have so far discussed the procedure for finding log n when a positive number n given. We now turn to its converse i.e., to find n when log n is given and give a method for this purpose. If log n = t, we sometimes say n = antilog t. Therefore our task is given t, find its antilog. For this, we use the readymade antilog tables. Suppose log n = 2.5372. To find n, first take just the mantissa of log n. In this case it is .5372. (Make sure it is positive.) Now take up antilog of this number in the antilog table which is to be used exactly like the log table. In the antilog table, the entry under column 7 in the row .53 is 3443 and the mean difference for the last digit 2 in that row is 2, so the table gives 3445. Hence, antilog (.5372) = 3.445 Now since log n = 2.5372, the characteristic of log n is 2. So the standard form of n is given by 2 n = 3.445 10 or n = 344.5 Illustration 1: If log x = 1.0712, find x. Solution: We find that the number corresponding to 0712 is 1179. Since characteristic of log x is 1, we have x = 1.179 101 = 11.79 Illustration 2: If log x = 2.1352, find x. Solution: From antilog tables, we find that the number corresponding to 1352 is 1366. Since the characteristic is 2 i.e., 2, so x = 1.366 10 2 = 0.01366 Use of Logarithms in Numerical Calculations Illustration 1: Find 6.3 1.29 Solution: Let x = 6.3 1.29 Then log x = log (6.3 1.29) = log 6.3 + log 1.29 Now, log 6.3 = 0.7993 log 1.29 = 0.1106 log x = 0.9099, Taking antilog * It should, however, be noted that the values given in the table are not exact. They are only approximate values, although we use the sign of equality which may give the impression that they are exact values. The same convention will be followed in respect of antilogarithm of a number. Chemistry 268 x = 8.127 Illustration 2: 1.5 Find (1.23) 11.2 23.5 3 Solution: Let x = (1.23)2 11.2 23.5 3 (1.23)2 Then log x = log = = 3 2 3 2 11.2 23.5 log 1.23 log (11.2 23.5) log 1.23 log 11.2 23.5 Now, log 1.23 = 0.0899 3 2 log 1.23 = 0.13485 log 11.2 = 1.0492 log 23.5 = 1.3711 log x = 0.13485 1.0492 1.3711 = 3.71455 x = 0.005183 Illustration 3: 5 Find (71.24) 56 7 (2.3) 21 5 (71.24) 56 Solution: Let x = Then log x = 7 (2.3) 21 1 2 = = (71.24)5 56 7 (2.3) 21 log 1 2 5 log 71.24 + 2 Now, using log tables log 71.24 = 1.8527 log 56 = 1.7482 log 2.3 = 0.3617 log 21 = 1.3222 log x = 5 log (1.8527) + 2 = 3.4723 x = 2967 7 56 log (2.3) log 21] [log (71.24)5 + log 1 4 1 4 log 56 (1.7482) 7 2 7 2 log 2.3 (0.3617) 1 4 1 4 log 21 (1.3222) 269 Appendix LOGARITHMS TABLE I N 0 2 3 4 10 0000 0043 1 0086 0128 0170 11 0414 0453 0492 0531 5 1 2 3 4 5 7 8 9 0253 0294 0334 0374 5 4 13 12 17 16 21 26 2O 24 30 28 34 38 32 36 0569 0607 0645 0682 0719 0755 4 4 8 7 12 11 16 15 20 18 23 22 27 26 31 35 29 33 0969 1004 1038 1072 1106 3 3 7 7 11 10 14 14 18 17 21 20 25 24 28 32 27 31 1303 1335 1367 1399 1430 3 3 6 7 10 10 13 13 16 16 19 19 23 22 26 29 25 29 1614 1644 1673 1703 1732 3 3 6 6 9 9 12 12 15 14 19 17 22 20 25 28 23 26 1903 1931 1959 1987 2014 3 3 6 6 9 8 11 11 14 14 17 17 20 19 23 26 22 25 2175 2201 2227 2253 2279 3 3 6 5 8 8 11 10 14 16 13 16 19 18 22 24 21 23 2430 2455 2480 2504 2529 3 3 5 5 8 8 10 10 13 12 15 15 18 17 20 23 20 22 2672 2695 2718 2742 2765 2 2 5 4 7 7 9 9 12 14 11 14 17 16 19 21 18 21 2900 2923 2945 2967 2989 2 2 4 4 7 6 9 8 11 11 13 13 16 15 18 20 17 19 0212 12 13 14 15 16 17 18 19 0792 0828 1139 1173 1461 1492 1761 1790 2041 2068 2304 2330 2553 2577 2788 2810 0864 1206 1523 1818 2095 2355 2601 2833 0899 1239 1553 1847 2122 2380 2625 2856 6 7 8 9 0934 1271 1584 1875 2148 2405 2648 2878 8 6 9 20 21 22 23 24 25 26 27 28 29 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 3032 3243 3444 3636 3820 3997 4166 4330 4487 4639 3054 3263 3464 3655 3838 4014 4183 4346 4502 4654 3075 3284 3483 3674 3856 4031 4200 4362 4518 4669 3096 3304 3502 3692 3874 4048 4216 4378 4533 4683 3118 3324 3522 3711 3892 4065 4232 4393 4548 4698 3139 3345 3541 3729 3909 4082 4249 4409 4564 4713 3160 3365 3560 3747 3927 4099 4265 4425 4579 4728 3181 3385 3579 3766 3945 4116 4281 4440 4594 4742 3201 3404 3598 3784 3962 4133 4298 4456 4609 4757 2 2 2 2 2 2 2 2 2 1 4 4 4 4 4 3 3 3 3 3 6 6 6 6 5 5 5 5 5 4 8 8 8 7 7 7 7 6 6 6 11 10 10 9 9 9 8 8 8 7 13 12 12 11 11 10 10 9 9 9 15 14 14 13 12 12 11 11 11 10 17 16 15 15 14 14 13 13 12 12 19 18 17 17 16 15 15 14 14 13 30 31 32 33 34 4771 4914 5051 5185 5315 4786 4928 5065 5198 5328 4800 4942 5079 5211 5340 4814 4955 5092 5224 5353 4829 4969 5105 5237 5366 4843 4983 5119 5250 5378 4857 4997 5132 5263 5391 4871 5011 5145 5276 5403 4886 5024 5159 5289 5416 4900 5038 5172 5302 5428 1 1 1 1 1 3 3 3 3 3 4 4 4 4 4 6 6 5 5 5 7 7 7 6 6 9 8 8 8 8 10 10 9 9 9 11 11 11 10 10 13 12 12 12 11 35 36 37 38 39 5441 5563 5682 5798 5911 5453 5575 5694 5809 5922 5465 5587 5705 5821 5933 5478 5599 5717 5832 5944 5490 5611 5729 5843 5955 5502 5623 5740 5855 5966 5514 5635 5752 5866 5977 5527 5647 5763 5877 5988 5539 5658 5775 5888 5999 5551 5670 5786 5899 6010 1 1 1 1 1 2 2 2 2 2 4 4 3 3 3 5 5 5 5 4 6 6 6 6 5 7 7 7 7 7 9 8 8 8 8 10 10 9 9 9 11 11 10 10 10 40 41 42 43 44 6021 6128 6232 6335 6435 6031 6138 6243 6345 6444 6042 6149 6253 6355 6454 6053 6160 6263 6365 6464 6064 6170 6274 6375 6474 6075 6180 6284 6385 6484 6085 6191 6294 6395 6493 6096 6201 6304 6405 6503 6107 6212 6314 6415 6513 6117 6222 6325 6425 6522 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 8 7 7 7 7 9 10 8 9 8 9 8 9 8 9 45 46 47 48 49 6532 6628 6721 6812 6902 6542 6637 6730 6821 6911 6551 6646 6739 6830 6920 6561 6656 6749 6839 6928 6471 6665 6758 6848 6937 6580 6675 6767 6857 6946 6590 6684 6776 6866 6955 6599 6693 6785 6875 6964 6609 6702 6794 6884 6972 6618 6712 6803 6893 6981 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 4 4 6 6 5 5 5 7 7 6 6 6 8 7 7 7 7 Chemistry 270 9 8 8 8 8 LOGARITHMS TABLE 1 (Continued) N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 50 51 52 53 54 6990 7076 7160 7243 7324 6998 7084 7168 7251 7332 7007 7093 7177 7259 7340 7016 7101 7185 7267 7348 7024 7110 7193 7275 7356 7033 7118 7202 7284 7364 7042 7126 7210 7292 7372 7050 7135 7218 7300 7380 7059 7143 7226 7308 7388 7067 7152 7235 7316 7396 1 1 1 1 1 2 2 2 2 2 3 3 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 6 6 8 8 7 7 7 55 56 57 58 59 7404 7482 7559 7634 7709 7412 7490 7566 7642 7716 7419 7497 7574 7649 7723 7427 7505 7582 7657 7731 7435 7513 7589 7664 7738 7443 7520 7597 7672 7745 7451 7528 7604 7679 7752 7459 7536 7612 7686 7760 7466 7543 7619 7694 7767 7474 7551 7627 7701 7774 1 1 1 1 1 2 2 2 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 60 61 62 63 64 7782 7853 7924 7993 8062 7789 7860 7931 8000 8069 7796 7768 7938 8007 8075 7803 7875 7945 8014 8082 7810 7882 7952 8021 8089 7818 7889 7959 8028 8096 7825 7896 7966 8035 8102 7832 7903 7973 8041 8109 7839 7910 7980 8048 8116 7846 7917 7987 8055 8122 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 5 5 6 6 6 6 6 65 66 67 68 69 8129 8195 8261 8325 8388 8136 8202 8267 8331 8395 8142 8209 8274 8338 8401 8149 8215 8280 8344 8407 8156 8222 8287 8351 8414 8162 8228 8293 8357 8420 8169 8235 8299 8363 8426 8176 8241 8306 8370 8432 8182 8248 8312 8376 8439 8189 8254 8319 8382 8445 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 2 3 3 3 3 3 4 4 4 4 4 5 5 5 4 4 5 5 5 5 5 6 6 6 6 6 70 71 72 73 74 8451 8513 8573 8633 8692 8457 8519 8579 8639 8698 8463 8525 8585 8645 8704 8470 8531 8591 8651 8710 8476 8537 8597 8657 8716 8482 8543 8603 8663 8722 8488 8549 8609 8669 8727 8494 8555 8615 8675 8733 8500 8561 8621 8681 8739 8506 8567 8627 8686 8745 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 6 5 5 5 5 75 76 77 78 79 8751 8808 8865 8921 8976 8756 8814 8871 8927 8982 8762 8820 8876 8932 8987 8768 8825 8882 8938 8993 8774 8831 8887 8943 8998 8779 8837 8893 8949 9004 8785 8842 8899 8954 9009 8791 8848 8904 8960 9015 8797 8854 8910 8965 9020 8802 8859 8915 8971 9025 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 5 5 4 4 4 5 5 5 5 5 80 81 82 83 84 9031 9085 9138 9191 9243 9036 9090 9143 9196 9248 9042 9096 9149 9201 9253 9047 9101 9154 9206 9258 9053 9106 9159 9212 9263 9058 9112 9165 9217 9269 9063 9117 9170 9222 9274 9069 9122 9175 9227 9279 9074 9128 9180 9232 9284 9079 9133 9186 9238 9289 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 85 86 87 88 89 9294 9345 9395 9445 9494 9299 9350 9400 9450 9499 9304 9355 9405 9455 9504 9309 9360 9410 9460 9509 9315 9365 9415 9465 9513 9320 9370 9420 9469 9518 9325 9375 9425 9474 9523 9330 9380 9430 9479 9528 9335 9385 9435 9484 9533 9340 9390 9440 9489 9538 1 1 0 0 0 1 1 1 1 1 2 2 1 1 1 2 2 2 2 2 3 3 2 2 2 3 3 3 3 3 4 4 3 3 3 4 4 4 4 4 5 5 4 4 4 90 91 92 93 94 9542 9590 9638 9685 9731 9547 9595 9643 9689 9736 9552 9600 9647 9694 9741 9557 9605 9652 9699 9745 9562 9609 9657 9703 9750 9566 9614 9661 9708 9754 9571 9619 9666 9713 9759 9576 9624 9671 9717 9763 9581 9628 9675 9722 9768 9586 9633 9680 9727 9773 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 95 96 97 98 99 9777 9823 9868 9912 9956 9782 9827 9872 9917 9961 9786 9832 9877 9921 9965 9791 9836 9881 9926 9969 9795 9841 9886 9930 9974 9800 9845 9890 9934 9978 9805 9850 9894 9939 9983 9809 9854 9899 9943 9987 9814 9859 9903 9948 9997 9818 9863 9908 9952 9996 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 3 4 4 4 4 4 271 Appendix ANTILOGARITHMS TABLE II N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 00 .01 .02 .03 .04 .05 .06 .07 .08 .09 1000 1023 1047 1072 1096 1122 1148 1175 1202 1230 1002 1026 1050 1074 1099 1125 1151 1178 1205 1233 1005 1028 1052 1076 1102 1127 1153 1180 1208 1236 1007 1030 1054 1079 1104 1130 1156 1183 1211 1239 1009 1033 1057 1081 1107 1132 1159 1186 1213 1242 1012 1035 1059 1084 1109 1135 1161 1189 1216 1245 1014 1038 1062 1086 1112 1138 1164 1191 1219 1247 1016 1040 1064 1089 1114 1140 1167 1194 1222 1250 1019 1042 1067 1091 1117 1143 1169 1197 1225 1253 1021 1045 1069 1094 1119 1146 1172 1199 1227 1256 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 1259 1288 1318 1349 1380 1413 1445 1479 1514 1549 1262 1291 1321 1352 1384 1416 1449 1483 1517 1552 1265 1294 1324 1355 1387 1419 1452 1486 1521 1556 1268 1297 1327 1358 1390 1422 1455 1489 1524 1560 1271 1300 1330 1361 1393 1426 1459 1493 1528 1563 1274 1303 1334 1365 1396 1429 1462 1496 1531 1567 1276 1306 1337 1368 1400 1432 1466 1500 1535 1570 1279 1309 1340 1371 1403 1435 1469 1503 1538 1574 1282 1312 1343 1374 1406 1439 1472 1507 1542 1578 1285 1315 1346 1377 1409 1442 1476 1510 1545 1581 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 .20 .21 .22 .23 .24 1585 1622 1660 1698 1738 1589 1626 1663 1702 1742 1592 1629 1667 1706 1746 1596 1633 1671 1710 1750 1600 1637 1675 1714 1754 1603 1641 1679 1718 1758 1607 1644 1683 1722 1762 1611 1648 1687 1726 1766 1614 1652 1690 1730 1770 1618 1656 1694 1734 1774 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 .25 .26 .27 .28 .29 1778 1820 1862 1905 1950 1782 1824 1866 1910 1954 1786 1828 1871 1914 1959 1791 1832 1875 1919 1963 1795 1837 1879 1923 1968 1799 1841 1884 1928 1972 1803 1845 1888 1932 1977 1807 1849 1892 1936 1982 1811 1854 1897 1941 1986 1816 1858 1901 1945 1991 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 1995 2042 2089 2138 2188 2239 2291 2344 2399 2455 2000 2046 2094 2143 2193 2244 2296 2350 2404 2460 2004 2051 2099 2148 2198 2249 2301 2355 2410 2466 2009 2056 2104 2153 2203 2254 2307 2360 2415 2472 2014 2061 2109 2158 2208 2259 2312 2366 2421 2477 2018 2065 2113 2163 2213 2265 2317 2371 2427 2483 2023 2070 2118 2168 2218 2270 2323 2377 2432 2489 2028 2075 2123 2173 2223 2275 2328 2382 2438 2495 2032 2080 2128 2178 2228 2280 2333 2388 2443 2500 2037 2084 2133 2183 2234 2286 2339 2393 2449 2506 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 4 4 4 4 5 5 5 5 5 5 .40 .41 .42 .43 .44 .45 .46 .47 .48 2512 2570 2630 2692 2754 2818 2884 2951 3020 2518 2576 2636 2698 2761 2825 2891 2958 3027 2523 2582 2642 2704 2767 2831 2897 2965 3034 2529 2588 2649 2710 2773 2838 2904 2972 3041 2535 2594 2655 2716 2780 2844 2911 2979 3048 2541 2600 2661 2723 2786 2851 2917 2985 3055 2547 2606 2667 2729 2793 2858 2924 2992 3062 2553 2612 2673 2735 2799 2864 2931 2999 3069 2559 2618 2679 2742 2805 2871 2938 3006 3076 2564 2624 2685 2748 2812 2877 2944 3013 3083 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 5 5 6 6 6 6 6 6 6 .49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155 1 1 2 3 3 4 5 6 6 Chemistry 272 ANTILOGARITHMS TABLE II (Continued) N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 .50 .51 .52 .53 .54 .55 .56 3162 3236 3311 3388 3467 3548 3631 3170 3243 3319 3396 3475 3556 3639 3177 3251 3327 3404 3483 3565 3648 3184 3258 3334 3412 3491 3573 3656 3192 3266 3342 3420 3499 3581 3664 3199 3273 3350 3428 3508 3589 3673 3206 3281 3357 3436 3516 3597 3681 3214 3289 3365 3443 3524 3606 3690 3221 3296 3373 3451 3532 3614 3698 3228 3304 3381 3459 3540 3622 3707 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 .57 .58 .59 3715 3724 3802 3811 3890 3899 3733 3819 3908 3741 3828 3917 3750 3837 3926 3758 3846 3936 3767 3776 3784 3793 3855 3864 3873 3882 3945 3954 3963 3972 1 1 1 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 8 .60 .61 .62 .63 .64 .65 .66 .67 .68 .69 3981 4074 4169 4266 4365 4467 4571 4677 4786 4898 3990 4083 4178 4276 4375 4477 4581 4688 4797 4909 3999 4093 4188 4285 4385 4487 4592 4699 4808 4920 4009 4102 4198 4295 4395 4498 4603 4710 4819 4932 4018 4111 4207 4305 4406 4508 4613 4721 4831 4943 4027 4121 4217 4315 4416 4519 4624 4732 4842 4955 4036 4130 4227 4325 4426 4529 4634 4742 4853 4966 4046 4140 4236 4335 4436 4539 4645 4753 4864 4977 4055 4150 4246 4345 4446 4550 4656 4764 4875 4989 4064 4159 42S6 4355 4457 4560 4667 4775 4887 5000 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 6 7 7 7 7 7 7 8 8 8 7 8 8 8 8 8 9 9 9 9 8 9 9 9 9 9 10 10 10 10 .70 5012 5023 .71 5129 5140 .72 5248 5260 .73 5370 5383 .74 5495 5508 .75 5623 5636 .76 5754 5768 .77 5888 5902 .78 6026 6039 .79 6166 6180 5035 5152 5272 5395 5521 5649 5781 5916 6053 6194 5047 5164 5284 5408 5534 5662 5794 5929 6067 6209 5058 5176 5297 5420 5546 5675 5808 5943 6081 6223 5070 5188 5309 5433 5559 5689 5821 5957 6095 6237 5082 5200 5321 5445 5572 5702 5834 5970 6109 6252 5093 5212 5333 5458 5585 5715 5848 5984 6124 6266 5105 5224 5346 5470 5598 5728 5861 5998 6138 6281 5117 5236 5358 5483 5610 5741 5875 6012 6152 6295 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 9 8 8 9 9 9 9 9 10 10 10 9 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 13 13 .80 .81 .82 .83 .84 6310 6457 6607 6761 6918 6324 6471 6622 6776 6934 6339 6486 6637 6792 6950 6353 6501 6653 6808 6966 6368 6516 6668 6823 6982 6383 6531 6683 6839 6998 6397 6546 6699 6855 7015 6412 6561 6714 6871 7031 6427 6577 6730 6887 7047 6442 6592 6745 6902 7063 1 2 2 2 2 3 3 3 3 3 4 5 5 5 5 6 6 6 6 6 7 8 8 8 8 9 9 9 9 10 10 11 11 11 11 12 13 12 14 12 14 1314 13 15 .85 .86 .87 .88 .89 7079 7244 7413 7586 7762 7096 7261 7430 7603 7780 7112 7278 7447 7621 7798 7129 7295 7464 7638 7816 7145 7311 7482 7656 7834 7161 7328 7499 7674 7852 7178 7345 7516 7691 7870 7194 7362 7534 7709 7889 7211 7379 7551 7727 7907 7228 7396 7568 7745 7925 2 2 2 2 2 3 3 3 4 4 5 5 5 5 5 7 7 7 7 7 8 8 9 9 9 10 10 10 11 11 12 12 12 12 13 13 13 14 14 14 15 15 16 16 16 .90 .91 .92 .93 .94 7943 8128 8318 8511 8710 7962 8147 8337 8531 8730 7980 8166 8356 8551 8750 7998 8185 8375 8570 8770 8017 8204 8395 8590 8790 8035 8222 8414 8610 8810 8054 8241 8433 8630 8831 8072 8260 8453 8650 8851 8091 8279 8472 8670 8872 8110 8299 8492 8690 8892 2 2 2 2 2 4 4 4 4 4 6 6 6 6 6 7 8 8 8 8 9 9 10 10 10 11 11 12 12 12 13 13 14 14 14 15 15 15 16 16 17 17 17 18 18 .95 .96 .97 .98 .99 8913 9120 9333 9550 9772 8933 9141 9354 9572 9795 8954 9162 9376 9594 9817 8974 9183 9397 9616 9840 8995 9204 9419 9638 9863 9016 9226 9441 9661 9886 9036 9247 9462 9683 9908 9057 9268 9484 9705 9931 9078 9290 9506 9727 9954 9099 9311 9528 9750 9977 2 2 2 2 2 4 4 4 4 5 6 6 7 7 7 8 8 9 9 9 10 11 11 11 11 12 13 13 13 14 15 15 15 16 16 17 17 17 18 18 19 19 20 20 20 273 Appendix

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