The value given in parenthesis is the molar mass of the isotope of largest known half-life. Element Symbol Atomic Molar Element Symbol Atomic Molar Number mass/ Number mass/ (g mol–1) (g mol–1) Actinium Ac 89 227.03 Mercury Hg 80 200.59 Aluminium Al 13 26.98 Molybdenum Mo 42 95.94 Americium Am 95 (243) Neodymium Nd 60 144.24 Antimony Sb 51 121.75 Neon Ne 10 20.18 Argon Ar 18 39.95 Neptunium Np 93 (237.05) Arsenic As 33 74.92 Nickel Ni 28 58.71 Astatine At 85  210 Niobium Nb 41 92.91 Barium Ba 56 137.34 Nitrogen N 7 14.0067 Berkelium Bk 97 (247) Nobelium No 102 (259) Beryllium Be 4 9.01 Osmium Os 76 190.2 Bismuth Bi 83 208.98 Oxygen O 8 16.00 Bohrium Bh 107 (264) Palladium Pd 46 106.4 Boron B 5 10.81 Phosphorus P 15 30.97 Bromine Br 35 79.91 Platinum Pt 78 195.09 Cadmium Cd 48 112.40 Plutonium Pu 94 (244) Caesium Cs 55 132.91 Polonium Po 84 210 Calcium Ca 20 40.08 Potassium K 19 39.10 Californium Cf 98 251.08 Praseodymium Pr 59 140.91 Carbon C 6 12.01 Promethium Pm 61 (145) Cerium Ce 58 140.12 Protactinium Pa 91 231.04 Chlorine Cl 17 35.45 Radium Ra 88 (226) Chromium Cr 24 52.00 Radon Rn 86 (222) Cobalt Co 27 58.93 Rhenium Re 75 186.2 Copper Cu 29 63.54 Rhodium Rh 45 102.91 Curium Cm 96 247.07 Rubidium Rb 37 85.47 Dubnium Db 105 (263) Ruthenium Ru 44 101.07 Dysprosium Dy 66 162.50 Rutherfordium Rf 104 (261) Einsteinium Es 99 (252) Samarium Sm 62 150.35 Erbium Er 68 167.26 Scandium Sc 21 44.96 Europium Eu 63 151.96 Seaborgium Sg 106 (266) Fermium Fm 100 (257.10) Selenium Se 34 78.96 Fluorine F 9 19.00 Silicon Si 14 28.08 Francium Fr 87 (223) Silver Ag 47 107.87 Gadolinium Gd 64 157.25 Sodium Na 11 22.99 Gallium Ga 31 69.72 Strontium Sr 38 87.62 Germanium Ge 32 72.61 Sulphur S 16 32.06 Gold Au 79 196.97 Tantalum Ta 73 180.95 Hafnium Hf 72 178.49 Technetium Tc 43 (98.91) Hassium Hs 108 (269) Tellurium Te 52 127.60 Helium He 2 4.00 Terbium Tb 65 158.92 Holmium Ho 67 164.93 Thallium Tl 81 204.37 Hydrogen H 1 1.0079 Thorium Th 90 232.04 Indium In 49 114.82 Thulium Tm 69 168.93 Iodine I 53 126.90 Tin Sn 50 118.69 Iridium Ir 77 192.2 Titanium Ti 22 47.88 Iron Fe 26 55.85 Tungsten W 74 183.85 Krypton Kr 36 83.80 Ununbium Uub 112 (277) Lanthanum La 57 138.91 Ununnilium Uun 110 (269) Lawrencium Lr 103 (262.1) Unununium Uuu 111 (272) Lead Pb 82 207.19 Uranium U 92 238.03 Lithium Li 3 6.94 Vanadium V 23 50.94 Lutetium Lu 71 174.96 Xenon Xe 54 131.30 Magnesium Mg 12 24.31 Ytterbium Yb 70 173.04 Manganese Mn 25 54.94 Yttrium Y 39 88.91 Meitneium Mt 109 (268) Zinc Zn 30 65.37 Mendelevium Md 101 258.10 Zirconium Zr 40 91.22 Common Unit of Mass and Weight 1 pound = 453.59 grams 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 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 Length 1 inch = 2.54 centimetres (exactly) 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 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 °F − 32°C = 1.8 * Force: 1 newton (N) = 1 kg m/s2, 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 Reduction half-reaction E⊖ /V Reduction half-reaction E⊖ /V H4XeO6 + 2H+ + 2e – ⎯→ XeO3 + 3H2O +3.0 Cu+ + e – ⎯→ Cu +0.52 F2 + 2e – ⎯→ 2F– +2.87 NiOOH + H2O + e – ⎯→ Ni(OH)2 + OH – +0.49 O3 + 2H+ + 2e – ⎯→ O2 + H2O +2.07 Ag2CrO4 + 2e – ⎯→ 2Ag + CrO2– 4 +0.45 S2O 2– 8 + 2e – ⎯→ 2SO 2– 4 +2.05 O2 + 2H2O + 4e – ⎯→ 4OH – +0.40 Ag+ + e – ⎯→ Ag+ Co3+ + e – ⎯→ Co2+ +1.98 +1.81 ClO – 4 + H2O + 2e – ⎯→ ClO – 3 + 2OH – [Fe(CN)6]3– + e – ⎯→ [Fe(CN)6]4– +0.36 +0.36 H2O2 + 2H+ + 2e – ⎯→ 2H2O Au+ + e – ⎯→ Au +1.78 +1.69 Cu2+ + 2e – ⎯→ Cu Hg2Cl2 + 2e – ⎯→ 2Hg + 2Cl – +0.34 +0.27 Pb4+ + 2e – ⎯→ Pb2+ 2HClO + 2H+ + 2e – ⎯→ Cl2 + 2H2O Ce4+ + e – ⎯→ Ce3+ 2HBrO + 2H+ + 2e – ⎯→ Br2 + 2H2O +1.67 +1.63 +1.61 +1.60 AgCl + e – ⎯→ Ag + Cl – Bi3+ + 3e – ⎯→ Bi SO4 2– + 4H+ + 2e – ⎯→ H2SO3 + H2O Cu2+ + e – ⎯→ Cu+ Sn4+ + 2e – ⎯→ Sn2+ +0.27 +0.20 +0.17 +0.16 +0.15 MnO – 4 + 8H+ + 5e – ⎯→ Mn2+ + 4H2O Mn3+ + e – ⎯→ Mn2+ +1.51 +1.51 AgBr + e – ⎯→ Ag + Br – Ti4+ + e – ⎯→ Ti3+ +0.07 0.00 Au3+ + 3e – ⎯→ Au Cl2 + 2e – ⎯→ 2Cl – +1.40 +1.36 2H+ + 2e– ⎯→ H2 0.0 by definition Cr2O 2– 7 + 14H+ + 6e – ⎯→ 2Cr3+ + 7H2O +1.33 Fe3+ + 3e – ⎯→ Fe –0.04 O3 + H2O + 2e – ⎯→ O2 + 2OH – +1.24 O2 + H2O + 2e – ⎯→ HO – 2 + OH – –0.08 O2 + 4H+ + 4e – ⎯→ 2H2O +1.23 Pb2+ + 2e – ⎯→ Pb –0.13 ClO – 4 + 2H+ +2e – ⎯→ ClO – 3 + 2H2O +1.23 In+ + e – ⎯→ In –0.14 MnO2 + 4H+ + 2e – ⎯→ Mn2+ + 2H2O +1.23 Sn2+ + 2e – ⎯→ Sn –0.14 Pt2+ + 2e – ⎯→ Pt +1.20 AgI + e – ⎯→ Ag + I – –0.15 Br2 + 2e – ⎯→ 2Br – +1.09 Ni2+ + 2e – ⎯→ Ni –0.23 Pu4+ + e – ⎯→ Pu3+ +0.97 V3+ + e – ⎯→ V2+ –0.26 NO – 3 + 4H+ + 3e – ⎯→ NO + 2H2O +0.96 Co2+ + 2e – ⎯→ Co –0.28 2Hg2+ + 2e – ⎯→ Hg2+ 2 +0.92 In3+ + 3e – ⎯→ In –0.34 ClO – + H2O + 2e – ⎯→ Cl – + 2OH – +0.89 Tl+ + e – ⎯→ Tl –0.34 Hg2+ + 2e – ⎯→ Hg +0.86 ⎯→ Pb + SO2– PbSO4 + 2e – 4 –0.36 NO – 3 + 2H+ + e – ⎯→ NO2 + H2O Ag+ + e – ⎯→ Ag Hg2+ 2 +2e – ⎯→ 2Hg Fe3+ + e – ⎯→ Fe2+ +0.80 +0.80 +0.79 +0.77 Ti3+ + e – ⎯→ Ti2+ Cd2+ + 2e – ⎯→ Cd In2+ + e – ⎯→ In+ Cr3+ + e – ⎯→ Cr2+ –0.37 –0.40 –0.40 –0.41 BrO – + H2O + 2e – ⎯→ Br – + 2OH – Hg2SO4 +2e – 2⎯→ 2Hg + SO2– 4 O + 2e – ⎯→ MnO2 + 4OH – + 2HMnO2– 4⎯→ MnO2– MnO – 4 + e – 4 +0.76 +0.62 +0.60 +0.56 Fe2+ + 2e – ⎯→ Fe In3+ + 2e – ⎯→ In+ S + 2e – ⎯→ S2– In3+ + e – ⎯→ In2+ U4+ + e – ⎯→ U3+ –0.44 –0.44 –0.48 –0.49 –0.61 I2 + 2e – ⎯→ 2I – +0.54 Cr3+ + 3e – ⎯→ Cr –0.74 I – 3 + 2e – ⎯→ 3I – +0.53 Zn2+ + 2e – ⎯→ Zn –0.76 (continued) Reduction half-reaction E⊖/V Reduction half-reaction E⊖/V Cd(OH)2 + 2e – ⎯→ Cd + 2OH – –0.81 La3+ + 3e – ⎯→ La –2.52 2H2O + 2e – ⎯→ H2 + 2OH – –0.83 Na+ + e – ⎯→ Na –2.71 Cr2+ + 2e – ⎯→ Cr –0.91 Ca2+ + 2e – ⎯→ Ca –2.87 Mn2+ + 2e – ⎯→ Mn –1.18 Sr2+ + 2e – ⎯→ Sr –2.89 V2+ + 2e – ⎯→ V –1.19 Ba2+ + 2e – ⎯→ Ba –2.91 Ti2+ + 2e – ⎯→ Ti –1.63 Ra2+ + 2e – ⎯→ Ra –2.92 Al3+ + 3e – ⎯→ Al –1.66 Cs+ + e – ⎯→ Cs –2.92 U3+ + 3e – ⎯→ U –1.79 Rb+ + e – ⎯→ Rb –2.93 Sc3+ + 3e – ⎯→ Sc –2.09 K+ +e – ⎯→ K –2.93 Mg2+ + 2e – ⎯→ Mg –2.36 Li+ + e – ⎯→ Li –3.05 Ce3+ + 3e – ⎯→ Ce –2.48 Chemistry 264 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. bWe write this as log = m, a “log” being the abbreviation of the word “logarithm”. Thus, we have 3 log 2 8 = 3, Since2 = 8 2 log 3 9 = 2, Since3 = 9 125 3 log = 3, Since5 = 125 5 0 log 7 1 = 0, Since7 = 1 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 y x+yHence mn = ax.a= aIt now follows from the definition of logarithms that loga (mn) = x + y = loga m – loga n  m  Second Law: log⎜ = logm – logna a a n  Proof: Let logam = x, logan = y Then ax = m, ay = n x ma x−yHence == a y n a Therefore  m  log = x − y = log m − log n a ⎜ aa n  Third Law : loga(mn) = n logam Proof : As before, if logam = x, then ax = m n n x nx Then m =(a )= a 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: log10 10 = 1, since 101 = 10 log10 100 = 2, since 102 = 100 log10 10000 = 4, since 104 = 10000 log10 0.01 = –2, since 10–2 = 0.01 log10 0.001 = –3, since 10–3 = 0.001 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 1× 10 = 2.52 × 10 10 25.2 = (ii) 1038.4 lies between 1000 and 10000. 1038 .4 ∴ 1038 .4 =× 10 3 = 1 .0 3 8 4 × 10 3 1000 (iii) 0.005 lies between 0.001 and 0.01 ∴ 0.005 = (0.005 × 1000) × 10–3 = 5.0 × 10–3 (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 n = m × 10p 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. 2. (i) If you move p places to the left, multiply by 10p. (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 for –2 and thus we write log m =22.4123 . 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 5 6 7 8 9 1 2 3 4 5 6 7 8 9 .. 61 62 63 .. .. 7853 7924 7993 .. .. 7860 7931 8000 .. .. 7868 7935 8007 .. .. 7875 7945 8014 .. .. 7882 7954 8021 .. .. 7889 7959 8028 .. .. 7896 7966 8035 .. .. 7803 7973 8041 .. .. 7810 7980 8048 .. .. 7817 7987 8055 .. .. 1 1 1 .. .. 1 1 1 .. .. 2 2 2 .. .. 3 3 3 .. .. 4 3 3 .. .. 4 4 4 .. .. 5 5 5 .. .. 6 6 6 .. .. 6 6 6 .. Now suppose we wish to find log (6.234). Then look into the row starting with 62. In this row, look 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 ready-made 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 byn = 3.445 × 102 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 i.e., –2, so2x = 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, * 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 Taking antilog x = 8.127 Illustration 2: 1.5(1.23) Find 11.2 × 23.5 3 (1.23)2 Solution: Let x = 11.2 × 23.5 3 (1.23)2 Then log x = log 11.2 × 23.5 3 = 2 log 1.23 – log (11.2 × 23.5) 3 = 2 log 1.23 – log 11.2 – 23.5 Now, log 1.23 = 0.0899 3 log 1.23 = 0.134852log 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(71.24) × 56 Solution: Let x = 7(2.3) × 21 Then log x = 1 2 log 5 7 (71.24) (2.3) × ×  ⎢ 21 ⎥ = 1 2 [log (71.24)5 + log 56 7log (2.3) − log − 21] = 5 log 71.24 + 1 log 56 − 7 log 2.3 − 1 log 21 2 4 2 4 Now, using log tables log 71.24 = 1.8527 log 56 = 1.7482 log 2.3 = 0.3617 log 21 = 1.3222 5 171 ∴ log x = log (1.8527) + (1.7482) − (0.3617) − (1.3222) 2 424 = 3.4723 ∴ x = 2967 LOGARITHMS TABLE I N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 5 4 9 8 13 12 17 16 21 2O 26 24 30 28 34 38 32 36 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 44 8 7 12 11 16 15 20 18 23 22 27 26 31 35 29 33 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 3 3 7 7 11 10 14 14 18 17 21 20 25 24 28 32 27 31 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 3 3 6 7 10 10 13 13 16 16 19 19 23 22 26 29 25 29 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 3 3 6 6 9 9 12 12 15 14 19 17 22 20 25 28 23 26 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 3 3 6 6 9 8 11 11 14 14 17 17 20 19 23 26 22 25 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 3 3 6 5 8 8 11 10 14 13 16 16 19 18 22 24 21 23 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 3 3 5 5 8 8 10 10 13 12 15 15 18 17 20 23 20 22 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2 2 5 4 7 7 9 9 12 11 14 14 17 16 19 21 18 21 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 2 2 4 4 7 6 9 8 11 11 13 13 16 15 18 20 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 6021 6128 6232 6335 6435 6532 6628 6721 6812 6902 3032 3243 3444 3636 3820 3997 4166 4330 4487 4639 4786 49285065 5198 5328 5453 5575 5694 5809 5922 6031 6138 6243 6345 6444 6542 6637 6730 6821 6911 3054 3263 3464 3655 3838 4014 4183 4346 4502 4654 4800 4942 5079 5211 5340 5465 5587 5705 5821 5933 6042 6149 6253 6355 6454 6551 6646 6739 6830 6920 3075 3284 3483 3674 3856 4031 4200 4362 4518 4669 4814 4955 5092 5224 5353 5478 5599 5717 5832 5944 6053 6160 6263 6365 6464 6561 6656 6749 6839 6928 3096 3304 3502 3692 3874 4048 4216 4378 4533 4683 4829 4969 5105 5237 5366 5490 5611 5729 5843 5955 6064 6170 6274 6375 6474 6471 6665 6758 6848 6937 3118 3324 3522 3711 3892 4065 4232 4393 4548 4698 4843 4983 5119 5250 5378 5502 5623 5740 5855 5966 6075 6180 6284 6385 6484 6580 6675 6767 6857 6946 3139 3345 3541 3729 3909 4082 4249 4409 4564 4713 4857 4997 5132 5263 5391 5514 5635 5752 5866 5977 6085 6191 6294 6395 6493 6590 6684 6776 6866 6955 3160 3365 3560 3747 3927 4099 4265 4425 4579 4728 4871 5011 5145 5276 5403 5527 5647 5763 5877 5988 6096 6201 6304 6405 6503 6599 6693 6785 6875 6964 3181 3385 3579 3766 3945 4116 4281 4440 4594 4742 4886 5024 5159 5289 5416 5539 5658 5775 5888 5999 6107 6212 6314 6415 6513 6609 6702 6794 6884 6972 3201 3404 3598 3784 3962 4133 4298 4456 4609 4757 4900 5038 5172 5302 5428 5551 5670 5786 5899 6010 6117 6222 6325 6425 6522 6618 6712 6803 6893 6981 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 3 3 3 3 3 3 33 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 11 10 10 9 9 9 8 8 8 7 7 7 7 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 4 4 13 12 12 11 11 10 10 9 9 9 9 8 8 8 8 7 7 7 7 7 6 6 6 6 6 6 6 5 5 5 15 14 14 13 12 12 11 11 11 10 10 10 9 9 9 9 8 8 8 8 8 7 7 7 7 7 7 6 6 6 17 19 16 18 15 17 15 17 14 16 14 15 13 15 13 14 12 14 12 13 11 13 11 12 11 12 10 12 10 11 10 11 10 11 9 10 9 10 9 10 9 10 8 9 8 9 8 9 8 9 8 9 7 8 7 8 7 8 7 8 Chemistry 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 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 8 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6 61 7853 7860 7768 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 2 3 3 4 5 6 6 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 2 3 4 4 5 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 1 1 2 2 3 3 4 4 5 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 0 1 1 2 2 3 3 4 4 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 4 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4 99 9956 9961 9965 9969 9974 9978 9983 9987 9997 9996 0 1 1 2 2 3 3 3 4 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 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 0 0 1 1 1 1 2 2 2 .01 1023 1026 1028 1030 1033 1035 1038 1040 1042 1045 0 0 1 1 1 1 2 2 2 .02 1047 1050 1052 1054 1057 1059 1062 1064 1067 1069 0 0 1 1 1 1 2 2 2 .03 1072 1074 1076 1079 1081 1084 1086 1089 1091 1094 0 0 1 1 1 1 2 2 2 .04 1096 1099 1102 1104 1107 1109 1112 1114 1117 1119 0 1 1 1 1 2 2 2 2 .05 1122 1125 1127 1130 1132 1135 1138 1140 1143 1146 0 1 1 1 1 2 2 2 2 .06 1148 1151 1153 1156 1159 1161 1164 1167 1169 1172 0 1 1 1 1 2 2 2 2 .07 1175 1178 1180 1183 1186 1189 1191 1194 1197 1199 0 1 1 1 1 2 2 2 2 .08 1202 1205 1208 1211 1213 1216 1219 1222 1225 1227 0 1 1 1 1 2 2 2 3 .09 1230 1233 1236 1239 1242 1245 1247 1250 1253 1256 0 1 1 1 1 2 2 2 3 .10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285 0 1 1 1 1 2 2 2 3 .11 1288 1291 1294 1297 1300 1303 1306 1309 1312 1315 0 1 1 1 2 2 2 2 3 .12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346 0 1 1 1 2 2 2 2 3 .13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377 0 1 1 1 2 2 2 3 3 .14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409 0 1 1 1 2 2 2 3 3 .15 1413 1416 1419 1422 1426 1429 1432 1435 1439 1442 0 1 1 1 2 2 2 3 3 .16 1445 1449 1452 1455 1459 1462 1466 1469 1472 1476 0 1 1 1 2 2 2 3 3 .17 1479 1483 1486 1489 1493 1496 1500 1503 1507 1510 0 1 1 1 2 2 2 3 3 .18 1514 1517 1521 1524 1528 1531 1535 1538 1542 1545 0 1 1 1 2 2 2 3 3 .19 1549 1552 1556 1560 1563 1567 1570 1574 1578 1581 0 1 1 1 2 2 3 3 3 .20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618 0 1 1 1 2 2 3 3 3 .21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656 0 1 1 2 2 2 3 3 3 .22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694 0 1 1 2 2 2 3 3 3 .23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734 0 1 1 2 2 2 3 3 4 .24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774 0 1 1 2 2 2 3 3 4 .25 1778 1782 1786 1791 1795 1799 1803 1807 1811 1816 0 1 1 2 2 2 3 3 4 .26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858 0 1 1 2 2 3 3 3 4 .27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901 0 1 1 2 2 3 3 3 4 .28 1905 1910 1914 1919 1923 1928 1932 1936 1941 1945 0 1 1 2 2 3 3 4 4 .29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991 0 1 1 2 2 3 3 4 4 .30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 0 1 1 2 2 3 3 4 4 .31 2042 2046 2051 2056 2061 2065 2070 2075 2080 2084 0 1 1 2 2 3 3 4 4 .32 2089 2094 2099 2104 2109 2113 2118 2123 2128 2133 0 1 1 2 2 3 3 4 4 .33 2138 2143 2148 2153 2158 2163 2168 2173 2178 2183 0 1 1 2 2 3 3 4 4 .34 2188 2193 2198 2203 2208 2213 2218 2223 2228 2234 1 1 2 2 3 3 4 4 5 .35 2239 2244 2249 2254 2259 2265 2270 2275 2280 2286 1 1 2 2 3 3 4 4 5 .36 2291 2296 2301 2307 2312 2317 2323 2328 2333 2339 1 1 2 2 3 3 4 4 5 .37 2344 2350 2355 2360 2366 2371 2377 2382 2388 2393 1 1 2 2 3 3 4 4 5 .38 2399 2404 2410 2415 2421 2427 2432 2438 2443 2449 1 1 2 2 3 3 4 4 5 .39 2455 2460 2466 2472 2477 2483 2489 2495 2500 2506 1 1 2 2 3 3 4 5 5 .40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 1 1 2 2 3 4 4 5 5 .41 2570 2576 2582 2588 2594 2600 2606 2612 2618 2624 1 1 2 2 3 4 4 5 5 .42 2630 2636 2642 2649 2655 2661 2667 2673 2679 2685 1 1 2 2 3 4 4 5 6 .43 2692 2698 2704 2710 2716 2723 2729 2735 2742 2748 1 1 2 3 3 4 4 5 6 .44 2754 2761 2767 2773 2780 2786 2793 2799 2805 2812 1 1 2 3 3 4 4 5 6 .45 2818 2825 2831 2838 2844 2851 2858 2864 2871 2877 1 1 2 3 3 4 5 5 6 .46 2884 2891 2897 2904 2911 2917 2924 2931 2938 2944 1 1 2 3 3 4 5 5 6 .47 2951 2958 2965 2972 2979 2985 2992 2999 3006 3013 1 1 2 3 3 4 5 5 6 .48 3020 3027 3034 3041 3048 3055 3062 3069 3076 3083 1 1 2 3 3 4 5 6 6 .49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155 1 1 2 3 3 4 5 6 6 Chemistry 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 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 1 1 2 3 4 4 5 6 7 .51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304 1 2 2 3 4 5 5 6 7 .52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381 1 2 2 3 4 5 5 6 7 .53 3388 3396 3404 3412 3420 3428 3436 3443 3451 3459 1 2 2 3 4 5 6 6 7 .54 3467 3475 3483 3491 3499 3508 3516 3524 3532 3540 1 2 2 3 4 5 6 6 7 .55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622 1 2 2 3 4 5 6 7 7 .56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707 1 2 3 3 4 5 6 7 8 .57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793 1 2 3 3 4 5 6 7 8 .58 3802 3811 3819 3828 3837 3846 3855 3864 3873 3882 1 2 3 4 4 5 6 7 8 .59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 1 2 3 4 5 5 6 7 8 .60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 1 2 3 4 5 6 6 7 8 .61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4159 1 2 3 4 5 6 7 8 9 .62 4169 4178 4188 4198 4207 4217 4227 4236 4246 42S6 1 2 3 4 5 6 7 8 9 .63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 1 2 3 4 5 6 7 8 9 .64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457 1 2 3 4 5 6 7 8 9 .65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 1 2 3 4 5 6 7 8 9 .66 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667 1 2 3 4 5 6 7 9 10 .67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775 1 2 3 4 5 7 8 9 10 .68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887 1 2 3 4 6 7 8 9 10 .69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000 1 2 3 5 6 7 8 9 10 .70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 1 2 4 5 6 7 8 9 11 .71 5129 5140 5152 5164 5176 5188 5200 5212 5224 5236 1 2 4 5 6 7 8 10 11 .72 5248 5260 5272 5284 5297 5309 5321 5333 5346 5358 1 2 4 5 6 7 9 10 11 .73 5370 5383 5395 5408 5420 5433 5445 5458 5470 5483 1 3 4 5 6 8 9 10 11 .74 5495 5508 5521 5534 5546 5559 5572 5585 5598 5610 1 3 4 5 6 8 9 10 12 .75 5623 5636 5649 5662 5675 5689 5702 5715 5728 5741 1 3 4 5 7 8 9 10 12 .76 5754 5768 5781 5794 5808 5821 5834 5848 5861 5875 1 3 4 5 7 8 9 11 12 .77 5888 5902 5916 5929 5943 5957 5970 5984 5998 6012 1 3 4 5 7 8 10 11 12 .78 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152 1 3 4 6 7 8 10 11 13 .79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295 1 3 4 6 7 9 10 11 13 .80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 1 3 4 6 7 9 10 12 13 .81 6457 6471 6486 6501 6516 6531 6546 6561 6577 6592 2 3 5 6 8 9 11 12 14 .82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745 2 3 5 6 8 9 11 12 14 .83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902 2 3 5 6 8 9 11 1314 .84 6918 6934 6950 6966 6982 6998 7015 7031 7047 7063 2 3 5 6 8 10 11 13 15 .85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228 2 3 5 7 8 10 12 13 15 .86 7244 7261 7278 7295 7311 7328 7345 7362 7379 7396 2 3 5 7 8 10 12 13 15 .87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568 2 3 5 7 9 10 12 14 16 .88 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745 2 4 5 7 9 11 12 14 16 .89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925 2 4 5 7 9 11 13 14 16 .90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 2 4 6 7 9 11 13 15 17 .91 8128 8147 8166 8185 8204 8222 8241 8260 8279 8299 2 4 6 8 9 11 13 15 17 .92 8318 8337 8356 8375 8395 8414 8433 8453 8472 8492 2 4 6 8 10 12 14 15 17 .93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690 2 4 6 8 10 12 14 16 18 .94 8710 8730 8750 8770 8790 8810 8831 8851 8872 8892 2 4 6 8 10 12 14 16 18 .95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 15 17 19 .96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19 .97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528 2 4 7 9 11 13 15 17 20 .98 9550 9572 9594 9616 9638 9661 9683 9705 9727 9750 2 4 7 9 11 13 16 18 20 .99 9772 9795 9817 9840 9863 9886 9908 9931 9954 9977 2 5 7 9 11 14 16 18 20 Appendix

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