Chapter 2 Basic Tools of analytical Chemistry n the chapters that follow we will learn about the specifics of analytical chemistry. In the process we will ask and answer questic such as"How do we treat experimental data? ""How do we ensure that our results are accurate? "" How do we obtain a representative sample?"and"How do we select an appropriate analytical technique? Before we look more closely at these and other questions, we will first review some basic numerical and experimental tools of importance to analytical chemists
Chapter 2 11 Basic Tools of Analytical Chemistry In the chapters that follow we will learn about the specifics of analytical chemistry. In the process we will ask and answer questions such as “How do we treat experimental data?” “How do we ensure that our results are accurate?” “How do we obtain a representative sample?” and “How do we select an appropriate analytical technique?” Before we look more closely at these and other questions, we will first review some basic numerical and experimental tools of importance to analytical chemists. 1400-CH02 9/8/99 3:47 PM Page 11
Modern Analytical Chemistry Numbers in Analytical chemist Analytical chemistry is inherently a quantitative science. Whether determining the concentration of a species in a solution, evaluating an equilibrium constant,mea- suring a reaction rate, or drawing a correlation between a compounds structure and its reactivity, analytical chemists make measurements and perform calculations. In this section we briefly review several important topics involving the use of num- bers in analytical chemistry 2A. I Fundamental Units of Measure Imagine that you find the following instructions in a laboratory procedure: Trans r 1.5 of your sample to a 100 volumetric flask, and dilute to volume. " How do you do this? Clearly these instructions are incomplete since the units of measurement are not stated. Compare this with a complete instruction: Transfer 1.5 g of your sample to a 100-mL volumetric flask, and dilute to volume. " This is an instruction that you can easily follow Measurements usually consist of a unit and a number expressing the quantity of that unit. Unfortunately, many different units may be used to express the same physical measurement. For example, the mass of a sample weighing 1.5 g also may be expressed as 0.0033 Ib or 0.053 oz. For consistency, and to avoid confusion, sci- entists use a common set of fundamental units. several of which are listed in Table 2. 1. These units are called SI units after the Systeme International d'Unites. Other Stands for Systeme International d Unites. measurements are defined using these fundamental SI units. For example,we mea sure the quantity of heat produced during a chemical reaction in joules, (), where 1J=1 Table 2.2 provides a list of other important derived SI units, as well as a few com monly used non-SI unit Chemists frequently work with measurements that are very large or very small. A mole, for example, contains 602, 213, 670,000,000,000,000,000 particles, and some analytical techniques can detect as little as 0.000000000000001 g of a compound notation For simplicity, we express these measurements using scientific notation; thus, a hand method for expressing very very small numbers by mole contains 6.0221367 x 102 particles, and the stated mass is 1 x 10-5 g. Some- times it is preferable to express measurements without the exponential term, replac l000isl×10 ing it with a prefix. A mass of 1 x 10-5 g is the same as 1 femtogram. Table 2.3 lists other common prefixes. Table 2.1 Fundamental SI Units measurement mass kilogram volume distance current ampere amount of substance mole mol
12 Modern Analytical Chemistry 2A Numbers in Analytical Chemistry Analytical chemistry is inherently a quantitative science. Whether determining the concentration of a species in a solution, evaluating an equilibrium constant, measuring a reaction rate, or drawing a correlation between a compound’s structure and its reactivity, analytical chemists make measurements and perform calculations. In this section we briefly review several important topics involving the use of numbers in analytical chemistry. 2A.1 Fundamental Units of Measure Imagine that you find the following instructions in a laboratory procedure: “Transfer 1.5 of your sample to a 100 volumetric flask, and dilute to volume.” How do you do this? Clearly these instructions are incomplete since the units of measurement are not stated. Compare this with a complete instruction: “Transfer 1.5 g of your sample to a 100-mL volumetric flask, and dilute to volume.” This is an instruction that you can easily follow. Measurements usually consist of a unit and a number expressing the quantity of that unit. Unfortunately, many different units may be used to express the same physical measurement. For example, the mass of a sample weighing 1.5 g also may be expressed as 0.0033 lb or 0.053 oz. For consistency, and to avoid confusion, scientists use a common set of fundamental units, several of which are listed in Table 2.1. These units are called SI units after the Système International d’Unités. Other measurements are defined using these fundamental SI units. For example, we measure the quantity of heat produced during a chemical reaction in joules, (J), where Table 2.2 provides a list of other important derived SI units, as well as a few commonly used non-SI units. Chemists frequently work with measurements that are very large or very small. A mole, for example, contains 602,213,670,000,000,000,000,000 particles, and some analytical techniques can detect as little as 0.000000000000001 g of a compound. For simplicity, we express these measurements using scientific notation; thus, a mole contains 6.0221367 × 1023 particles, and the stated mass is 1 × 10–15 g. Sometimes it is preferable to express measurements without the exponential term, replacing it with a prefix. A mass of 1 × 10–15 g is the same as 1 femtogram. Table 2.3 lists other common prefixes. 1 J = 1 m kg 2 s2 Table 2.1 Fundamental SI Units Measurement Unit Symbol mass kilogram kg volume liter L distance meter m temperature kelvin K time second s current ampere A amount of substance mole mol scientific notation A shorthand method for expressing very large or very small numbers by indicating powers of ten; for example, 1000 is 1 × 103. SI units Stands for Système International d’Unités. These are the internationally agreed on units for measurements. 1400-CH02 9/8/99 3:47 PM Page 12
Chapter 2 Basic Tools of Analytical Chemistry 3 Table 2.2 Other SI and Non-SI Units Measurement Symbol Equivalent sl units 1A=1×101 newton N 1N=1m·kgs2 pressure pascal 1Pa=1Nm2=1kg(m·s2) atmosphere tm 1 atm= 101, 325 Pa energy, work, heat joule 1J=1N·m=1m2·kg/s power W1W=1J=1m2·kg/s3 charge coulomb 1c=1A·s potential 1V=1WA=1m2·kg/(s3·A) temperature C=K-273.15 degree Fahrenheit°F °F=18(K-273.15)+32 Table 2. 3 Common Prefixes for Exponential Notation Symbol erd G d 10-9 2A.2 Significant Figure Recording a measurement provides information about both its magnitude and un certainty. For example, if we weigh a sample on a bala nce an 1. 2637 g, we assume that all digits, except the last, are known exactly. We assume at least +0.0001g, or a relative uncertainty of at leasr ving an absolute uncertain ±000g×100=±009% 1.2637 Significant figures are a reflection of a measurement's uncertainty. The num- significant figures ber of significant figures is equal to the number of digits in the measurement, with The digits in a measured quantity, the exception that a zero(O)used to fix the location of a decimal point is not con- one digit (the last) whose quantity is including all digits known exactly and idered significant. This definition can be ambiguous. For example, how many sig- uncertain nificant figures are in the number 100? If measured to the nearest hundred, then there is one significant figure. If measured to the nearest ten, however, then two
2A.2 Significant Figures Recording a measurement provides information about both its magnitude and uncertainty. For example, if we weigh a sample on a balance and record its mass as 1.2637 g, we assume that all digits, except the last, are known exactly. We assume that the last digit has an uncertainty of at least ±1, giving an absolute uncertainty of at least ±0.0001 g, or a relative uncertainty of at least Significant figures are a reflection of a measurement’s uncertainty. The number of significant figures is equal to the number of digits in the measurement, with the exception that a zero (0) used to fix the location of a decimal point is not considered significant. This definition can be ambiguous. For example, how many significant figures are in the number 100? If measured to the nearest hundred, then there is one significant figure. If measured to the nearest ten, however, then two ± × =± 0 0001 1 2637 100 0 0079 . . . % g g Chapter 2 Basic Tools of Analytical Chemistry 13 Table 2.2 Other SI and Non-SI Units Measurement Unit Symbol Equivalent SI units length angstrom Å 1 Å = 1 × 10–10 m force newton N 1 N = 1 m ⋅ kg/s2 pressure pascal Pa 1 Pa = 1 N/m2 = 1 kg/(m ⋅ s2) atmosphere atm 1 atm = 101,325 Pa energy, work, heat joule J 1 J = 1 N ⋅ m = 1 m2 ⋅ kg/s2 power watt W 1 W = 1 J/s = 1 m2 ⋅ kg/s3 charge coulomb C 1 C = 1 A ⋅ s potential volt V 1 V = 1 W/A = 1 m2 ⋅ kg/(s3 ⋅ A) temperature degree Celsius °C °C = K – 273.15 degree Fahrenheit °F °F = 1.8(K – 273.15) + 32 Table 2.3 Common Prefixes for Exponential Notation Exponential Prefix Symbol 1012 tera T 109 giga G 106 mega M 103 kilo k 10–1 deci d 10–2 centi c 10–3 milli m 10–6 micro µ 10–9 nano n 10–12 pico p 10–15 femto f 10–18 atto a significant figures The digits in a measured quantity, including all digits known exactly and one digit (the last) whose quantity is uncertain. 1400-CH02 9/8/99 3:47 PM Page 13
Modern Analytical Chemistry significant figures are included. To avoid ambiguity we use scientific notation. Thus, 1 x 10 has one significant figure, whereas 1.0 x 10- has two significant figures For measurements using logarithms, such as pH, the number of significant figures is equal to the number of digits to the right of the decimal, including all zeros. Digits to the left of the decimal are not included as significant figures since they only indicate the power of 10. A ph of 2.45, therefore, contains two signifi- cant figures Exact numbers such as the stoichiometric coefficients in a chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures A mole of CaCl2, for example, contains exactly two moles of chloride and one mole of calcium. In the equal both numbers have an infinite number of significant figures Recording a measurement to the correct number of significant figures is im- ortant because it tells others about how precisely you made your measurement. For example, suppose you weigh an object on a balance capable of measuring mass to the nearest +0. I mg, but record its mass as 1. 762 g instead of 1.7620 g By failing to record the trailing zero, which is a significant figure, you suggest to others that the mass was determined using a balance capable of weighing to only the nearest +l mg. Similarly, a buret with scale markings every 0. 1 mL can be read to the nearest +0.01 mL. The digit in the hundredths place is the least sig nificant figure since we must estimate its value. Reporting a volume of 12.241 mL implies that your burets scale is more precise than it actually is, with di sions every 0.01 mL. Significant figures are also important because they guide us in reporting the re- sult of an analysis. When using a measurement in a calculation, the result of that calculation can never be more certain than that measurement,s uncertainty. Simply put, the result of an analysis can never be more certain than the least certain mea surement included in the analysi As a general rule, mathematical operations involving addition and subtraction are carried out to the last digit that is significant for all numbers included in the cal- culation. Thus, the sum of 135.621,0.33, and 21 2163 is 157. 17 since the last digi that is significant for all three numbers is in the hundredth's place. 135.621+0.33+21.2163=157.1673=157.17 When multiplying and dividing, the general rule is that the answer contains the same number of significant figures as that number in the calculation having the fewest significant figures. Thus, 22.91×0.52 =0.21361=0.214 It is important to remember, however, that these rules are generalizations What is conserved is not the number of significant figures, but absolute uncertainty when adding or subtracting, and relative uncertainty when multiplying or dividing For example, the following calculation reports the answer to the correct number of significant figures, even though it violates the general rules outlined earlier
significant figures are included. To avoid ambiguity we use scientific notation. Thus, 1 × 102 has one significant figure, whereas 1.0 × 102 has two significant figures. For measurements using logarithms, such as pH, the number of significant figures is equal to the number of digits to the right of the decimal, including all zeros. Digits to the left of the decimal are not included as significant figures since they only indicate the power of 10. A pH of 2.45, therefore, contains two significant figures. Exact numbers, such as the stoichiometric coefficients in a chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures. A mole of CaCl2, for example, contains exactly two moles of chloride and one mole of calcium. In the equality 1000 mL = 1 L both numbers have an infinite number of significant figures. Recording a measurement to the correct number of significant figures is important because it tells others about how precisely you made your measurement. For example, suppose you weigh an object on a balance capable of measuring mass to the nearest ±0.1 mg, but record its mass as 1.762 g instead of 1.7620 g. By failing to record the trailing zero, which is a significant figure, you suggest to others that the mass was determined using a balance capable of weighing to only the nearest ±1 mg. Similarly, a buret with scale markings every 0.1 mL can be read to the nearest ±0.01 mL. The digit in the hundredth’s place is the least significant figure since we must estimate its value. Reporting a volume of 12.241 mL implies that your buret’s scale is more precise than it actually is, with divisions every 0.01 mL. Significant figures are also important because they guide us in reporting the result of an analysis. When using a measurement in a calculation, the result of that calculation can never be more certain than that measurement’s uncertainty. Simply put, the result of an analysis can never be more certain than the least certain measurement included in the analysis. As a general rule, mathematical operations involving addition and subtraction are carried out to the last digit that is significant for all numbers included in the calculation. Thus, the sum of 135.621, 0.33, and 21.2163 is 157.17 since the last digit that is significant for all three numbers is in the hundredth’s place. 135.621 + 0.33 + 21.2163 = 157.1673 = 157.17 When multiplying and dividing, the general rule is that the answer contains the same number of significant figures as that number in the calculation having the fewest significant figures. Thus, It is important to remember, however, that these rules are generalizations. What is conserved is not the number of significant figures, but absolute uncertainty when adding or subtracting, and relative uncertainty when multiplying or dividing. For example, the following calculation reports the answer to the correct number of significant figures, even though it violates the general rules outlined earlier. 101 99 = 1 02 . 22 91 0 152 16 302 0 21361 0 214 . . . . . × = = 14 Modern Analytical Chemistry 1400-CH02 9/8/99 3:48 PM Page 14
Chapter 2 Basic Tools of Analytical Chemistry Since the relative uncertainty in both measurements is roughly 1%(101 +1, 99+1) the relative uncertainty in the final answer also must be roughly 1%. Reporting the answer to only two significant figures(1.0), as required by the general rules, implies a relative uncertainty of 10%. The correct answer, with three significant figures ields the expected relative uncertainty. Chapter 4 presents a more thorough treat ment of uncertainty and its importance in reporting the results of an analysis Finally, to avoid"round-off" errors in calculations, it is a good idea to retain at least one extra significant figure throughout the calculation. This is the practice adopted in this textbook. Better yet, invest in a good scientific calculator that allows ou to perform lengthy calculations without recording intermediate values. When the calculation is complete, the final answer can be rounded to the correct number of significant figures using the following simple rules. 1. Retain the least significant figure if it and the digits that follow are less than halfway to the next higher digit; thus, rounding 12.442 to the nearest tenth gives 12. 4 since 0.442 is less than halfway between 0.400 and 0.500 2. Increase the least significant figure by I if it and the digits that follow are more than halfway to the next higher digit; thus, rounding 12. 476 to the nearest tenth gives 12. 5 since 0.476 is more than halfway between 0.400 and 0.500 3. If the least significant figure and the digits that follow are exactly halfway to the next higher digit, then round the least significant figure to the nearest even number; thus, rounding 12. 450 to the nearest tenth gives 12. 4, but rounding 12.550 to the nearest tenth ways rounding up or down anner prevents us res 12.6. Rounding in this m from introducing a bias by alwa 2B Units for Expressing Concentration Concentration is a general measurement unit stating the amount of solute present a known amount of solution An expression stating the relative amount of solute per unit volume or oncentration s amount of solute unit mass of solution 2.1 amount of solution Although the terms"solute"and"solution"are often associated with liquid sam ples, they can be extended to gas-phase and solid-phase samples as well. The actual units for reporting concentration depend on how the amounts of solute and solt tion are measured. table 2.4 lists the most common units of concentration 2B. l Molarity and Formalin Both molarity and formality express concentration as moles of solute per liter of solu tion. There is, however, a subtle difference between molarity and formality. Molarity is the concentration of a particular chemical species in solution. Formality, on the per of moles of solute per liter other hand, is a substances total concentration in solution without regard to its spe- cific chemical form. There is no difference between a substance's molarity and for- mality if it dissolves without dissociating into ions. The molar concentration of a so- formality lution of glucose, for example, is the same as its formality The number of moles of solute gardless of chemical form, per liter of For substances that ionize in solution, such as NaCl, molarity and formality are solution(F) different. For example, dissolving 0 I mol of NaCl in I L of water gives a solution containing 0.1 mol of Na+ and 0.1 mol of Ch. The molarity of NaCl, therefore, is zero since there is essentially no undissociated Nacl in solution. The solution
Chapter 2 Basic Tools of Analytical Chemistry 15 Since the relative uncertainty in both measurements is roughly 1% (101 ±1, 99 ±1), the relative uncertainty in the final answer also must be roughly 1%. Reporting the answer to only two significant figures (1.0), as required by the general rules, implies a relative uncertainty of 10%. The correct answer, with three significant figures, yields the expected relative uncertainty. Chapter 4 presents a more thorough treatment of uncertainty and its importance in reporting the results of an analysis. Finally, to avoid “round-off ” errors in calculations, it is a good idea to retain at least one extra significant figure throughout the calculation. This is the practice adopted in this textbook. Better yet, invest in a good scientific calculator that allows you to perform lengthy calculations without recording intermediate values. When the calculation is complete, the final answer can be rounded to the correct number of significant figures using the following simple rules. 1. Retain the least significant figure if it and the digits that follow are less than halfway to the next higher digit; thus, rounding 12.442 to the nearest tenth gives 12.4 since 0.442 is less than halfway between 0.400 and 0.500. 2. Increase the least significant figure by 1 if it and the digits that follow are more than halfway to the next higher digit; thus, rounding 12.476 to the nearest tenth gives 12.5 since 0.476 is more than halfway between 0.400 and 0.500. 3. If the least significant figure and the digits that follow are exactly halfway to the next higher digit, then round the least significant figure to the nearest even number; thus, rounding 12.450 to the nearest tenth gives 12.4, but rounding 12.550 to the nearest tenth gives 12.6. Rounding in this manner prevents us from introducing a bias by always rounding up or down. 2B Units for Expressing Concentration Concentration is a general measurement unit stating the amount of solute present in a known amount of solution 2.1 Although the terms “solute” and “solution” are often associated with liquid samples, they can be extended to gas-phase and solid-phase samples as well. The actual units for reporting concentration depend on how the amounts of solute and solution are measured. Table 2.4 lists the most common units of concentration. 2B.1 Molarity and Formality Both molarity and formality express concentration as moles of solute per liter of solution. There is, however, a subtle difference between molarity and formality. Molarity is the concentration of a particular chemical species in solution. Formality, on the other hand, is a substance’s total concentration in solution without regard to its specific chemical form. There is no difference between a substance’s molarity and formality if it dissolves without dissociating into ions. The molar concentration of a solution of glucose, for example, is the same as its formality. For substances that ionize in solution, such as NaCl, molarity and formality are different. For example, dissolving 0.1 mol of NaCl in 1 L of water gives a solution containing 0.1 mol of Na+ and 0.1 mol of Cl–. The molarity of NaCl, therefore, is zero since there is essentially no undissociated NaCl in solution. The solution, Concentration amount of solute amount of solution = molarity The number of moles of solute per liter of solution (M). formality The number of moles of solute, regardless of chemical form, per liter of solution (F). concentration An expression stating the relative amount of solute per unit volume or unit mass of solution. 1400-CH02 9/8/99 3:48 PM Page 15