Measurement and Problem Solving PDF

2: MEASUREMENT AND PROBLEM SOLVING  CHAPTER OVERVIEW 2: Measurement and Problem Solving Chemistry, like all sciences, is quantitative. It concerns quantities, things that have amounts and units. Dealing with quantities and relating them to one another is very important in chemistry. In this chapte...

2: MEASUREMENT AND PROBLEM SOLVING  CHAPTER OVERVIEW 2: Measurement and Problem Solving Chemistry, like all sciences, is quantitative. It concerns quantities, things that have amounts and units. Dealing with quantities and relating them to one another is very important in chemistry. In this chapter, we will discuss how we deal with numbers and units, including how they are combined and manipulated. 2.1: Taking Measurements 2.2: Scientific Notation - Writing Large and Small Numbers 2.3: Significant Figures - Writing Numbers to Reflect Precision 2.4: Significant Figures in Calculations 2.5: The Basic Units of Measurement 2.6: Problem Solving and Unit Conversions 2.7: Solving Multi-step Conversion Problems 2.8: Units Raised to a Power 2.9: Density 2.E: Measurement and Problem Solving (Exercises) 2: Measurement and Problem Solving is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 1 2.1: TAKING MEASUREMENTS  LEARNING OBJECTIVES Express quantities properly, using a number and a unit. A coffee maker’s instructions tell you to fill the coffee pot with 4 cups of water and to use 3 scoops of coffee. When you follow these instructions, you are measuring. When you visit a doctor’s office, a nurse checks your temperature, height, weight, and perhaps blood pressure (Figure 2.1.1 ); the nurse is also measuring. Figure 2.1.1: Measuring Blood Pressure. A nurse or a doctor measuring a patient’s blood pressure is taking a measurement. (GFDL; Pia von Lützau). Chemists measure the properties of matter and express these measurements as quantities. A quantity is an amount of something and consists of a number and a unit. The number tells us how many (or how much), and the unit tells us what the scale of measurement is. For example, when a distance is reported as “5 kilometers,” we know that the quantity has been expressed in units of kilometers and that the number of kilometers is 5. If you ask a friend how far they walk from home to school, and the friend answers “12” without specifying a unit, you do not know whether your friend walks 12 kilometers, 12 miles, 12 furlongs, or 12 yards. Both a number and a unit must be included to express a quantity properly. To understand chemistry, we need a clear understanding of the units chemists work with and the rules they follow for expressing numbers. The next two sections examine the rules for expressing numbers.  EXAMPLE 2.1.1 Identify the number and the unit in each quantity. a. one dozen eggs b. 2.54 centimeters c. a box of pencils d. 88 meters per second SOLUTION a. The number is one, and the unit is a dozen eggs. b. The number is 2.54, and the unit is centimeter. c. The number 1 is implied because the quantity is only a box. The unit is box of pencils. d. The number is 88, and the unit is meters per second. Note that in this case the unit is actually a combination of two units: meters and seconds. KEY TAKE AWAY Identify a quantity properly with a number and a unit. 2.1: Taking Measurements is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.1.1 https://chem.libretexts.org/@go/page/47446 2.2: Scientific Notation - Writing Large and Small Numbers  Learning Objectives Express a large number or a small number in scientific notation. Carry out arithmetical operations and express the final answer in scientific notation Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator would require a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes. In scientific notation, these numbers are expressed in the form n N × 10 where N is greater than or equal to 1 and less than 10 (1 ≤ N < 10), and n is a positive or negative integer (100 = 1). The number 10 is called the base because it is this number that is raised to the power n. Although a base number may have values other than 10, the base number in scientific notation is always 10. A simple way to convert numbers to scientific notation is to move the decimal point as many places to the left or right as needed to give a number from 1 to 10 (N). The magnitude of n is then determined as follows: If the decimal point is moved to the left n places, n is positive. If the decimal point is moved to the right n places, n is negative. Another way to remember this is to recognize that as the number N decreases in magnitude, the exponent increases and vice versa. The application of this rule is illustrated in Example 2.2.1.  Example 2.2.1: Expressing Numbers in Scientific Notation Convert each number to scientific notation. a. 637.8 b. 0.0479 c. 7.86 d. 12,378 e. 0.00032 f. 61.06700 g. 2002.080 h. 0.01020 Solution Solutions to Example 2.2.1 Explanation Answer To convert 637.8 to a number from 1 to 10, we move the decimal point two places to the left: a 637.8 6.378 × 10 2 Because the decimal point was moved two places to the left, n = 2. To convert 0.0479 to a number from 1 to 10, we move the decimal point two places to the right: b 0.0479 4.79 × 10 −2 Because the decimal point was moved two places to the right, n = −2. This is usually expressed simply as 7.86. c 7.86 × 10 0 (Recall that 100 = 1.) Because the decimal point was moved four d 1.2378 × 10 4 places to the left, n = 4. Because the decimal point was moved four e −4 3.2 × 10 places to the right, n = −4. 2.2.1 https://chem.libretexts.org/@go/page/47447  Explanation Answer Because the decimal point was moved one f 6.106700 × 10 1 place to the left, n = 1. Because the decimal point was moved three g 2.002080 × 10 3 places to the left, n = 3. Because the decimal point was moved two h 1.020 × 10 −2 places to the right, n = -2. Addition and Subtraction Before numbers expressed in scientific notation can be added or subtracted, they must be converted to a form in which all the exponents have the same value. The appropriate operation is then carried out on the values of N. Example 2.2.2 illustrates how to do this.  Example 2.2.2: Expressing Sums and Differences in Scientific Notation Carry out the appropriate operation and then express the answer in scientific notation. a. (1.36 × 10 ) + (4.73 × 10 ) 2 3 b. (6.923 × 10 ) − (8.756 × 10 −3 −4 ) Solution Solutions to Example 2.2.2. A n s Explanation w e r Both exponents must have the same value, so these numbers are converted to either 2 2 (1.36 × 10 ) + (47.3 × 10 ) = 2 2 (1.36 + 47.3) × 10 = 48.66 × 10 or a 3 (0.136 × 10 ) + (4.73 × 10 ) = 3 4.87 × 3 (0.136 + 4.73) × 10 ) = 4.87 × 10. 3 Choosing either alternative gives the same answer, reported to two decimal places. In converting 48.66 × 102 to scientific notation, n has become more positive by 1 because the value of N has decreased. Converting the exponents to the same value gives either −3 −3 (6.923 × 10 ) − (0.8756 × 10 ) = −3 (6.923 − 0.8756) × 10 or b −4 −4 6.047 (69.23 × 10 ) − (8.756 × 10 ) = (69.23 − 8.756) × 10 −4 = 60.474 × 10. −4 In converting 60.474 × 10-4 to scientific notation, n has become more positive by 1 because the value of N has decreased. Multiplication and Division When multiplying numbers expressed in scientific notation, we multiply the values of N and add together the values of n. Conversely, when dividing, we divide N in the dividend (the number being divided) by N in the divisor (the number by which we are dividing) and then subtract n in the divisor from n in the dividend. In contrast to addition and subtraction, the exponents do not have to be the same in multiplication and division. Examples of problems involving multiplication and division are shown in Example 2.2.3. 2.2.2 https://chem.libretexts.org/@go/page/47447   Example 2.2.3: Expressing Products and Quotients in Scientific Notation Perform the appropriate operation and express your answer in scientific notation. a. (6.022 × 10 23 −2 )(6.42 × 10 ) −24 1.67 × 10 b. −28 9.12 × 10 −34 (6.63 × 10 )(6.0 × 10) c. −2 8.52 × 10 Solution Solution to Example 2.2.3 Explanation In multiplication, we add the exponents: 23 −2 [23+(−2)] 21 (6.022 × 10 )(6.42 × 10 ) = (6.022)(6.42) × 10 = 38.7 × 10 a In converting 38.7 × 10 to scientific notation, 21 n has become more positive by 1 because the 22 3.87 × 10 value of N has decreased. In division, we subtract the exponents: − 24 1.67×10 = − 28 9.12×10 1.67 [−24−(−28)] 4 × 10 = 0.183 × 10 b 9.12 1.83 × 10 3 In converting 0.183 × 10 to scientific 4 notation, n has become more negative by 1 because the value of N has increased. This problem has both multiplication and division: − 34 (6.63×10 )(6.0×10) c −2 = (8.52×10 ) −31 39.78 [−34+1−(−2)] 4.7 × 10 × 10 8.52 2.2: Scientific Notation - Writing Large and Small Numbers is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar- Agnew & Henry Agnew. 2.2.3 https://chem.libretexts.org/@go/page/47447 2.3: Significant Figures - Writing Numbers to Reflect Precision  Learning Objectives Identify the number of significant figures in a reported value. The significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit. In the ruler illustration below, the bottom ruler gave a length with 2 significant figures, while the top ruler gave a length with 3 significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. With either ruler, it would not be possible to report the length at 2.553 cm as there is no possible way that the thousandths digit could be estimated. The 3 is not significant and would not be reported. Figure 2.3.1 : Measurement with two different rulers. Ruler A's measurement can be rounded to 2.55, with 2 certain digits, while Ruler B's measurement of 2.5 has 1 certain digit Measurement Uncertainty Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. While some balances are capable of measuring masses only to the nearest 0.1 g, other highly sensitive balances are capable of measuring to the nearest 0.001 g or even better. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement. Figure 2.3.1 shows two rulers making the same measurement of an object (indicated by the blue arrow). With either ruler, it is clear that the length of the object is between 2 and 3 cm. The bottom ruler contains no millimeter markings. With that ruler, the tenths digit can be estimated and the length may be reported as 2.5 cm. However, another person may judge that the measurement is 2.4 cm or perhaps 2.6 cm. While the 2 is known for certain, the value of the tenths digit is uncertain. The top ruler contains marks for tenths of a centimeter (millimeters). Now the same object may be measured as 2.55 cm. The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. Again, another measurer may report the length to be 2.54 cm or 2.56 cm. In this case, there are two certain digits (the 2 and the 5), with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.  Example 2.3.1: Reporting Measurements to the Proper Number of Significant Figures Use each diagram to report a measurement to the proper number of significant figures. a. b. 2.3.1 https://chem.libretexts.org/@go/page/47448  Ruler measuring a rectangle in units of centimeters, with the rectangle's edge between 1.2 and 1.3 cm marks Solutions Solutions to Example 2.3.1 Explanation Answer The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it’s at least 0.3. We will have to estimate the last place. It looks like about a. one-third of the way across the space, so let 4.33 psi us estimate the hundredths place as 3. The symbol psi stands for “pounds per square inch” and is a unit of pressure, like air in a tire. The measurement is reported to three significant figures. The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle’s width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant b. digit. We have to estimate the next place 1.25 cm because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. The measurement is reported to three significant figures.  Exercise 2.3.1 What would be the reported width of this rectangle? Answer 1.25 cm When you look at a reported measurement, it is necessary to be able to count the number of significant figures. The table below details the rules for determining the number of significant figures in a reported measurement. For the examples in the table, assume 2.3.2 https://chem.libretexts.org/@go/page/47448 that the quantities are correctly reported values of a measured quantity. Table 2.3.1 : Significant Figure Rules Rule Examples 237 has three significant figures. 1. All nonzero digits in a measurement are significant. 1.897 has four significant figures. 2. Zeros that appear between other nonzero digits (middle zeros) are 39,004 has five significant figures. always significant. 5.02 has three significant figures. 3. Zeros that appear in front of all of the nonzero digits are called 0.008 has one significant figure. leading zeros. Leading zeros are never significant. 0.000416 has three significant figures. 4. Zeros that appear after all nonzero digits are called trailing zeros. A 1400 is ambiguous. number with trailing zeros that lacks a decimal point may or may not be 1.4 × 10 has two significant figures. 3 significant. Use scientific notation to indicate the appropriate 1.40 × 10 three significant figures. 3 number of significant figures. 1.400 × 10 has four significant figures. 3 5. Trailing zeros in a number with a decimal point are significant. This 620.0 has four significant figures. is true whether the zeros occur before or after the decimal point. 19.000 has five significant figures. Exact Numbers Integers obtained either by counting objects or from definitions are exact numbers, which are considered to have infinitely many significant figures. If we have counted four objects, for example, then the number 4 has an infinite number of significant figures (i.e., it represents 4.000…). Similarly, 1 foot (ft) is defined to contain 12 inches (in), so the number 12 in the following equation has infinitely many significant figures:  Example 2.3.2 Give the number of significant figures in each. Identify the rule for each. a. 5.87 b. 0.031 c. 52.90 d. 00.2001 e. 500 f. 6 atoms Solution Solution to Example 2.3.2 Explanation Answer a All three numbers are significant (rule 1). 5.87, three significant figures The leading zeros are not significant (rule 3). b 0.031, two significant figures The 3 and the 1 are significant (rule 1). The 5, the 2 and the 9 are significant (rule 1). c 52.90, four significant figures The trailing zero is also significant (rule 5). The leading zeros are not significant (rule 3). d The 2 and the 1 are significant (rule 1) and 00.2001, four significant figures the middle zeros are also significant (rule 2). The number is ambiguous. It could have one, e 500, ambiguous two or three significant figures. The 6 is a counting number. A counting f 6, infinite number is an exact number. 2.3.3 https://chem.libretexts.org/@go/page/47448   Exercise 2.3.2 Give the number of significant figures in each. a. 36.7 m b. 0.006606 s c. 2,002 kg d. 306,490,000 people e. 3,800 g Answer a three significant figures Answer b four significant figures Answer c four significant figures Answer d infinite (exact number) Answer e Ambiguous, could be two, three or four significant figures. Accuracy and Precision Measurements may be accurate, meaning that the measured value is the same as the true value; they may be precise, meaning that multiple measurements give nearly identical values (i.e., reproducible results); they may be both accurate and precise; or they may be neither accurate nor precise. The goal of scientists is to obtain measured values that are both accurate and precise. The video below demonstrates the concepts of accuracy and precision. What's the difference between accurac… accurac… Video 2.3.1 : Difference between precision and accuracy.  Example 2.3.3 The following archery targets show marks that represent the results of four sets of measurements. 2.3.4 https://chem.libretexts.org/@go/page/47448  Which target shows a. a precise, but inaccurate set of measurements? b. a set of measurements that is both precise and accurate? c. a set of measurements that is neither precise nor accurate? Solution a. Set (a) is precise, but inaccurate. b. Set (c) is both precise and accurate. c. Set (d) is neither precise nor accurate. Summary Uncertainty exists in all measurements. The degree of uncertainty is affected in part by the quality of the measuring tool. Significant figures give an indication of the certainty of a measurement. Rules allow decisions to be made about how many digits to use in any given situation. 2.3: Significant Figures - Writing Numbers to Reflect Precision is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew, Henry Agnew, Sridhar Budhi, & Sridhar Budhi. 2.3.5 https://chem.libretexts.org/@go/page/47448 2.4: Significant Figures in Calculations  Learning Objectives Use significant figures correctly in arithmetical operations. Rounding Before dealing with the specifics of the rules for determining the significant figures in a calculated result, we need to be able to round numbers correctly. To round a number, first decide how many significant figures the number should have. Once you know that, round to that many digits, starting from the left. If the number immediately to the right of the last significant digit is less than 5, it is dropped and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1. Consider the measurement 207.518 m. Right now, the measurement contains six significant figures. How would we successively round it to fewer and fewer significant figures? Follow the process as outlined in Table 2.4.1. Table 2.4.1 : Rounding examples Number of Significant Figures Rounded Value Reasoning 6 207.518 All digits are significant 5 207.52 8 rounds the 1 up to 2 4 207.5 2 is dropped 3 208 5 rounds the 7 up to 8 2 210 8 is replaced by a 0 and rounds the 0 up to 1 1 200 1 is replaced by a 0 Notice that the more rounding that is done, the less reliable the figure is. An approximate value may be sufficient for some purposes, but scientific work requires a much higher level of detail. It is important to be aware of significant figures when you are mathematically manipulating numbers. For example, dividing 125 by 307 on a calculator gives 0.4071661238… to an infinite number of digits. But do the digits in this answer have any practical meaning, especially when you are starting with numbers that have only three significant figures each? When performing mathematical operations, there are two rules for limiting the number of significant figures in an answer—one rule is for addition and subtraction, and one rule is for multiplication and division. In operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. An answer is no more precise than the least precise number used to get the answer. Multiplication and Division For multiplication or division, the rule is to count the number of significant figures in each number being multiplied or divided and then limit the significant figures in the answer to the lowest count. An example is as follows: The final answer, limited to four significant figures, is 4,094. The first digit dropped is 1, so we do not round up. Scientific notation provides a way of communicating significant figures without ambiguity. You simply include all the significant figures in the leading number. For example, the number 450 has two significant figures and would be written in scientific notation 2.4.1 https://chem.libretexts.org/@go/page/47449 as 4.5 × 102, whereas 450.0 has four significant figures and would be written as 4.500 × 102. In scientific notation, all significant figures are listed explicitly.  Example 2.4.1 Write the answer for each expression using scientific notation with the appropriate number of significant figures. a. 23.096 × 90.300 b. 125 × 9.000 Solution a Explanation Answer The calculator answer is 2,085.5688, but we need to round it to five significant figures. Because the first digit to be dropped (in the tenths 2.0856 × 10 3 place) is greater than 5, we round up to 2,085.6. b Explanation Answer The calculator gives 1,125 as the answer, but we limit it to three 3 1.13 × 10 significant figures. Addition and Subtraction How are significant figures handled in calculations? It depends on what type of calculation is being performed. If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common. For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column. We therefore limit our answer to the tenths column. We drop the last digit—the 1—because it is not significant to the final answer. The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater, and rounded down if the first dropped digit is less than 5.  Example 2.4.2 a. 13.77 + 908.226 b. 1,027 + 611 + 363.06 2.4.2 https://chem.libretexts.org/@go/page/47449  Solution a Explanation Answer The calculator answer is 921.996, but because 13.77 has its farthest- right significant figure in the hundredths place, we need to round the final answer to the hundredths position. Because the first digit to be 922.00 = 9.2200 × 10 2 dropped (in the thousandths place) is greater than 5, we round up to 922.00 b Explanation Answer The calculator gives 2,001.06 as the answer, but because 611 and 1027 has its farthest-right significant figure in the ones place, the 2, 001.06 = 2.001 × 10 3 final answer must be limited to the ones position.  Exercise 2.4.2 Write the answer for each expression using scientific notation with the appropriate number of significant figures. a. 217 ÷ 903 b. 13.77 + 908.226 + 515 c. 255.0 − 99 d. 0.00666 × 321 Answer a: −1 0.240 = 2.40 × 10 Answer b: 3 1, 437 = 1.437 × 10 Answer c: 2 156 = 1.56 × 10 Answer d: 0 2.14 = 2.14 × 10 Remember that calculators do not understand significant figures. You are the one who must apply the rules of significant figures to a result from your calculator. Calculations Involving Multiplication/Division and Addition/Subtraction In practice, chemists generally work with a calculator and carry all digits forward through subsequent calculations. When working on paper, however, we often want to minimize the number of digits we have to write out. Because successive rounding can compound inaccuracies, intermediate rounding needs to be handled correctly. When working on paper, always round an intermediate result so as to retain at least one more digit than can be justified and carry this number into the next step in the calculation. The final answer is then rounded to the correct number of significant figures at the very end. 2.4.3 https://chem.libretexts.org/@go/page/47449  Signi cant gures in mixed operations Video 2.4.1 : Significant figures in mixed operations (https://www.youtube.com/watch?v=yBntMndXQWA). Signi cant Figures in Combined Operati… Operati… Video 2.4.2 : https://www.youtube.com/watch?v=__csP0NtlGI In the worked examples in this text, we will often show the results of intermediate steps in a calculation. In doing so, we will show the results to only the correct number of significant figures allowed for that step, in effect treating each step as a separate calculation. This procedure is intended to reinforce the rules for determining the number of significant figures, but in some cases it may give a final answer that differs in the last digit from that obtained using a calculator, where all digits are carried through to the last step.  Example 2.4.3 a. 2(1.008 g) + 15.99 g b. 137.3 s + 2(35.45 s) 118.7g c. 2 − 35.5g Solution a. Explanation Answer 2.4.4 https://chem.libretexts.org/@go/page/47449  2(1.008 g) + 15.99 g = Perform multiplication first. 2 (1.008 g 4 sig figs) = 2.016 g 4 sig figs The number with the least number of significant figures is 1.008 g; the number 2 is an exact number and therefore has an infinite number of significant figures. 18.01 g (rounding up) Then, perform the addition. 2.016 g thousandths place + 15.99 g hundredths place (least precise) = 18.006 g Round the final answer. Round the final answer to the hundredths place since 15.99 has its farthest right significant figure in the hundredths place (least precise). b. Explanation Answer 137.3 s + 2(35.45 s) = Perform multiplication first. 2(35.45 s 4 sig figs) = 70.90 s 4 sig figs The number with the least number of significant figures is 35.45; the number 2 is an exact number and therefore has an infinite number of significant figures. 208.2 s Then, perform the addition. 137.3 s tenths place (least precise) + 70.90 s hundredths place = 208.20 s Round the final answer. Round the final answer to the tenths place based on 137.3 s. c. Explanation Answer 118.7g 2 − 35.5g = Perform division first. 118.7g 2 4 sig figs = 59.35 g 4 sig figs The number with the least number of significant figures is 118.7 g; the number 2 is an exact number and therefore has an infinite number of significant figures. 23.9 g (rounding up) Perform subtraction next. 59.35 g hundredths place − 35.5 g tenths place (least precise) = 23.85 g Round the final answer. Round the final answer to the tenths place based on 35.5 g.  Exercise 2.4.3 Complete the calculations and report your answers using the correct number of significant figures. a. 5(1.008s) - 10.66 s b. 99.0 cm+ 2(5.56 cm) 2.4.5 https://chem.libretexts.org/@go/page/47449  Answer a -5.62 s Answer b 110.2 cm Summary Rounding If the number to be dropped is greater than or equal to 5, increase the number to its left by 1 (e.g. 2.9699 rounded to three significant figures is 2.97). If the number to be dropped is less than 5, there is no change (e.g. 4.00443 rounded to four significant figures is 4.004). The rule in multiplication and division is that the final answer should have the same number of significant figures as there are in the number with the fewest significant figures. The rule in addition and subtraction is that the answer is given the same number of decimal places as the term with the fewest decimal places. 2.4: Significant Figures in Calculations is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.4.6 https://chem.libretexts.org/@go/page/47449 2.5: The Basic Units of Measurement  Learning Objectives State the different measurement systems used in chemistry. Describe how prefixes are used in the metric system and identify how the prefixes milli-, centi-, and kilo- compare to the base unit. How long is a yard? It depends on whom you ask and when you asked the question. Today we have a standard definition of the yard, which you can see marked on every football field. If you move the ball ten yards, you get a first down and it does not matter whether you are playing in Los Angeles, Dallas, or Green Bay. But at one time that yard was arbitrarily defined as the distance from the tip of the king's nose to the end of his outstretched hand. Of course, the problem there is simple: new king, new distance (and then you have to re-mark all of those football fields). Figure 2.5.1 : Meter standard (left) and Kilogram standard (right). SI Base Units All measurements depend on the use of units that are well known and understood. The English system of measurement units (inches, feet, ounces, etc.) are not used in science because of the difficulty in converting from one unit to another. The metric system is used because all metric units are based on multiples of 10, making conversions very simple. The metric system was originally established in France in 1795. The International System of Units is a system of measurement based on the metric system. The acronym SI is commonly used to refer to this system and stands for the French term, Le Système International d'Unités. The SI was adopted by international agreement in 1960 and is composed of seven base units in Table 2.5.1. Table 2.5.1 : SI Base Units of Measurement Quantity SI Base Unit Symbol Length meter m Mass kilogram kg Temperature kelvin K Time second s Amount of a Substance mole mol Electric Current ampere A Luminous Intensity candela cd The first units are frequently encountered in chemistry. All other measurement quantities, such as volume, force, and energy, can be derived from these seven base units.  Unfortunately, the Metric System is Not Ubiquitous The map below shows the adoption of the SI units in countries around the world. The United States has legally adopted the metric system for measurements, but does not use it in everyday practice. Great Britain and much of Canada use a combination of metric and imperial units. 2.5.1 https://chem.libretexts.org/@go/page/47451  Figure 2.5.1 : Areas of world using metric system (in green). Only a few countries are slow or resistant to adoption including the United States. Prefix Multipliers Conversions between metric system units are straightforward because the system is based on powers of ten. For example, meters, centimeters, and millimeters are all metric units of length. There are 10 millimeters in 1 centimeter and 100 centimeters in 1 meter. Metric prefixes are used to distinguish between units of different size. These prefixes all derive from either Latin or Greek terms. For example, mega comes from the Greek word μεγας , meaning "great". Table 2.5.2 lists the most common metric prefixes and their relationship to the central unit that has no prefix. Length is used as an example to demonstrate the relative size of each prefixed unit. Table 2.5.2 : SI Prefixes Prefix Unit Abbreviation Meaning Example giga G 1,000,000,000 1 gigameter (Gm) = 10 9 m mega M 1,000,000 1 megameter (Mm) = 10 6 m kilo k 1,000 1 kilometer (km) = 1, 000 m hecto h 100 1 hectometer (hm) = 100 m deka da 10 1 dekameter (dam) = 10 m 1 1 meter (m) deci d 1/10 1 decimeter (dm) = 0.1 m centi c 1/100 1 centimeter (cm) = 0.01 m milli m 1/1,000 1 millimeter (mm) = 0.001 m micro μ 1/1,000,000 1 micrometer (μm) = 10 −6 m nano n 1/1,000,000,000 1 nanometer (nm) = 10 −9 m pico p 1/1,000,000,000,000 1 picometer (pm) = 10 −12 m There are a couple of odd little practices with the use of metric abbreviations. Most abbreviations are lowercase. We use "m" for meter and not "M ". However, when it comes to volume, the base unit "liter" is abbreviated as "L" and not "l". So we would write 3.5 milliliters as 3.5 mL. As a practical matter, whenever possible you should express the units in a small and manageable number. If you are measuring the weight of a material that weighs 6.5 kg, this is easier than saying it weighs 6500 g or 0.65 dag. All three are correct, but the kg units in this case make for a small and easily managed number. However, if a specific problem needs grams instead of kilograms, go with the grams for consistency. 2.5.2 https://chem.libretexts.org/@go/page/47451   Example 2.5.1: Unit Abbreviations Give the abbreviation for each unit and define the abbreviation in terms of the base unit. a. kiloliter b. microsecond c. decimeter d. nanogram Solutions Solutions to Example 2.5.1 Explanation Answer The prefix kilo means “1,000 ×,” so 1 kL a kL equals 1,000 L. The prefix micro implies 1/1,000,000th of a b µs unit, so 1 µs equals 0.000001 s. The prefix deci means 1/10th, so 1 dm equals c dm 0.1 m. The prefix nano means 1/1000000000, so a d ng nanogram is equal to 0.000000001 g.  Exercise 2.5.1 Give the abbreviation for each unit and define the abbreviation in terms of the base unit. a. kilometer b. milligram c. nanosecond d. centiliter Answer a: km Answer b: mg Answer c: ns Answer d: cL Summary Metric prefixes derive from Latin or Greek terms. The prefixes are used to make the units manageable. The SI system is based on multiples of ten. There are seven basic units in the SI system. Five of these units are commonly used in chemistry. 2.5: The Basic Units of Measurement is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.5.3 https://chem.libretexts.org/@go/page/47451 2.6: PROBLEM SOLVING AND UNIT CONVERSIONS  LEARNING OBJECTIVES To convert a value reported in one unit to a corresponding value in a different unit using conversion factors. During your studies of chemistry (and physics also), you will note that mathematical equations are used in many different applications. Many of these equations have a number of different variables with which you will need to work. You should also note that these equations will often require you to use measurements with their units. Algebra skills become very important here! CONVERTING BETWEEN UNITS WITH CONVERSION FACTORS A conversion factor is a factor used to convert one unit of measurement into another. A simple conversion factor can convert meters into centimeters, or a more complex one can convert miles per hour into meters per second. Since most calculations require measurements to be in certain units, you will find many uses for conversion factors. Always remember that a conversion factor has to represent a fact; this fact can either be simple or more complex. For instance, you already know that 12 eggs equal 1 dozen. A more complex fact is that the speed of light is 1.86 × 10 miles/sec. Either one of these can be used as a conversion factor depending on what type of calculation you are working 5 with (Table 2.6.1 ). Table 2.6.1: Conversion Factors from SI units to English Units English Units Metric Units Quantity 1 ounce (oz) 28.35 grams (g) *mass 1 fluid once (oz) 29.6 mL volume 2.205 pounds (lb) 1 kilogram (kg) *mass 1 inch (in) 2.54 centimeters (cm) length 0.6214 miles (mi) 1 kilometer (km) length 1 quarter (qt) 0.95 liters (L) volume *Pounds and ounces are technically units of force, not mass, but this fact is often ignored by the non-scientific community. Of course, there are other ratios which are not listed in Table 2.6.1. They may include: Ratios embedded in the text of the problem (using words such as per or in each, or using symbols such as / or %). Conversions in the metric system, as covered earlier in this chapter. Common knowledge ratios (such as 60 seconds = 1 minute). If you learned the SI units and prefixes described, then you know that 1 cm is 1/100th of a meter. 1 −2 1 cm = m = 10 m 100 or 100 cm = 1 m Suppose we divide both sides of the equation by 1m (both the number and the unit): 100 cm 1 m = 1 m 1 m As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1: 6 100 cm 1000 mm 1 × 10 μm = = = 1 1 m 1 m 1 m We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units. PERFORMING DIMENSIONAL ANALYSIS Dimensional analysis is amongst the most valuable tools that physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are 2.6.1 https://chem.libretexts.org/@go/page/51021 more accurate or easier to find than others. The use of units in a calculation to ensure that we obtain the final proper units is called dimensional analysis. Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as 100 cm and multiply: 1 m 100 cm 3.55 m × 1 m The 3.55 m can be thought of as a fraction with a 1 in the denominator. Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out: 3.55 m 100 cm × 1 1 m The final step is to perform the calculation that remains once the units have been canceled: 3.55 100 cm × = 355 cm 1 1 In the final answer, we omit the 1 in the denominator. Thus, by a more formal procedure, we find that 3.55 m equals 355 cm. A generalized description of this process is as follows: quantity (in old units) × conversion factor = quantity (in new units) You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. In later studies, the conversion problems you encounter will not always be so simple. If you master the technique of applying conversion factors, you will be able to solve a large variety of problems. 100 cm 1 m In the previous example, we used the fraction as a conversion factor. Does the conversion factor also equal 1? Yes, it 1 m 100 cm does; it has the same quantity in the numerator as in the denominator (except that they are expressed in different units). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten: 2 1 m m 3.55 m × = 0.0355 100 cm cm For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out. Figure 2.6.1 shows a concept map for constructing a proper conversion. Figure 2.6.1: A Concept Map for Conversions. This is how you construct a conversion factor to convert from one unit to another. Converting meters to centimeters: the unit you are converting is on the bottom of the fraction, and the unit to convert to is on top  GENERAL STEPS IN PERFORMING DIMENSIONAL ANALYSIS 1. Identify the "given" information in the problem. Look for a number with units to start this problem with. 2. What is the problem asking you to "find"? In other words, what unit will your answer have? 3. Use ratios and conversion factors to cancel out the units that aren't part of your answer, and leave you with units that are part of your answer. 4. When your units cancel out correctly, you are ready to do the math. You are multiplying fractions, so you multiply the top numbers and divide by the bottom numbers in the fractions. 2.6.2 https://chem.libretexts.org/@go/page/51021 SIGNIFICANT FIGURES IN CONVERSIONS How do conversion factors affect the determination of significant figures? Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact. Exact numbers are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.) Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact. In contrast, conversion factors that come from measurements (such as density, as we will see shortly) or that are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer.  EXAMPLE 2.6.1 Steps for Problem Solving for Example 2.6.1 and 2.6.2 Example 2.6.1 Example 2.6.2 The average volume of blood in an adult male is 4.7 L. A hummingbird can flap its wings once in 18 ms. How Steps for Problem Solving What is this volume in milliliters? many seconds are in 18 ms? Identify the "given" information and what the Given: 4.7 L Given: 18 ms problem is asking you to "find." Find: mL Find: s List other known quantities. 1 mL = 10 −3 L 1 ms = 10 −3 s Prepare a concept map and use the proper conversion factor. 1 mL 4.7 L × = 4, 700 mL −3 −3 10 L 10 s 18 ms × = 0.018 s or 1 ms Cancel units and calculate. 1, 000 mL or 4.7 L × = 4, 700 mL 1 s 1 L 18 ms × = 0.018 s or 1, 000 ms 4.7 x 103 2SF, not ambiguous The amount in mL should be 1000 times larger than the The amount in s should be 1/1000 the given amount in Think about your result. given amount in L. ms.  EXERCISE 2.6.1 Perform each conversion. a. 101,000 ns to seconds b. 32.08 kg to grams c. 1.53 grams to cg Answer a: −4 1.01000x10 s Answer b: 4 3.208x10 g Answer c: 2 1.53x10 cg SUMMARY Conversion factors are used to convert one unit of measurement into another. Dimensional analysis (unit conversions) involves the use of conversion factors that will cancel unwanted units and produce the appropriate units. 2.6: Problem Solving and Unit Conversions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Marisa Alviar- Agnew & Henry Agnew. 2.6.3 https://chem.libretexts.org/@go/page/51021 2.7: SOLVING MULTI-STEP CONVERSION PROBLEMS MULTIPLE CONVERSIONS Sometimes you will have to perform more than one conversion to obtain the desired unit. For example, suppose you want to convert 54.7 km into millimeters. We will set up a series of conversion factors so that each conversion factor produces the next unit in the sequence. We first convert the given amount in km to the base unit, which is meters. We know that 1,000 m =1 km. Then we convert meters to mm, remembering that 1 mm = 10 −3 m. CONCEPT MAP Convert kilometers to meters to millimeters: use conversion factors 1000 meters per 1 kilometer and 1 millimeter per 0.001 meter CALCULATION 1, 000 m 1 mm 54.7 km × × = 54, 700, 000 mm 1 km −3 10 m 7 = 5.47 × 10 mm In each step, the previous unit is canceled and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left.  EXAMPLE 2.7.1: UNIT CONVERSION Convert 58.2 ms to megaseconds in one multi-step calculation. SOLUTION Steps for Problem Solving Unit Conversion Identify the "given" information and what the problem is Given: 58.2 ms asking you to "find." Find: Ms −3 1ms = 10 s List other known quantities 6 1M s = 10 s Prepare a concept map. Convert milliseconds to seconds to microseconds: use conversion factors 0.001 second per millisecond and 1 microsecond per 1 million seconds −3 10 s 1 Ms 58.2 ms × × = 0.0000000582 Ms 1 ms 1, 000, 000 s Calculate. −8 = 5.82 × 10 Ms Neither conversion factor affects the number of significant figures in the final answer. 2.7.1 https://chem.libretexts.org/@go/page/47453  EXAMPLE 2.7.2: UNIT CONVERSION How many seconds are in a day? Solution Steps for Problem Solving Unit Conversion Identify the "given" information and what the Given: 1 day problem is asking you to "find." Find: s 1 day = 24 hours List other known quantities. 1 hour = 60 minutes 1 minute = 60 seconds Prepare a concept map. Convert day to hour to minute to second: use conversion factors 24 hours per day, 60 minutes per hour, and 60 seconds per minute 24 hr 60 min 60 s Calculate. 1 d × × × = 86, 400 s 1 d 1 hr 1 min  EXERCISE 2.7.1 Perform each conversion in one multi-step calculation. a. 43.007 ng to kg b. 1005 in to ft c. 12 mi to km Answer a −11 4.3007 × 10 kg Answer b 83.75 ft Answer c 19 km  CAREER FOCUS: PHARMACIST A pharmacist dispenses drugs that have been prescribed by a doctor. Although that may sound straightforward, pharmacists in the United States must hold a doctorate in pharmacy and be licensed by the state in which they work. Most pharmacy programs require four years of education in a specialty pharmacy school. Pharmacists must know a lot of chemistry and biology so they can understand the effects that drugs (which are chemicals, after all) have on the body. Pharmacists can advise physicians on the selection, dosage, interactions, and side effects of drugs. They can also advise patients on the proper use of their medications, including when and how to take specific drugs properly. Pharmacists can be found in drugstores, hospitals, and other medical facilities. Curiously, an outdated name for pharmacist is chemist, which was used when pharmacists formerly did a lot of drug preparation, or compounding. In modern times, pharmacists rarely compound their own drugs, but their knowledge of the sciences, including chemistry, helps them provide valuable services in support of everyone’s health. 2.7.2 https://chem.libretexts.org/@go/page/47453  A woman consulting with a pharmacist. (Public Domain; Rhoda Baer via National Cancer Institute, an agency that is part of the National Institutes of Health.) SUMMARY In multi-step conversion problems, the previous unit is canceled for each step and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left. 2.7: Solving Multi-step Conversion Problems is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Marisa Alviar- Agnew & Henry Agnew. 2.7.3 https://chem.libretexts.org/@go/page/47453 2.8: Units Raised to a Power  Learning Objectives To convert a value reported in one unit raised to a power of 10, to a corresponding value in a different unit raised to the same power of 10, using conversion factors. Conversion factors for area and volume can also be produced by the dimensional analysis method. Just remember that if a quantity is raised to a power of 10, both the number and the unit must be raised to the same power of 10. For example, to convert 1500 cm 2 to m , we need to start with the relationship between centimeter and meter. We know that 1 cm = 10-2 m or 100 cm =1 m, but since 2 we are given the quantity in 1500 cm2, then we have to use the relationship: 2 −2 2 −4 2 1 cm = (10 m) = 10 m CONCEPT MAP To convert centimeters squared to meters squared, use the conversion factor 0.01 meters per 1 centimeter, squared overall CALCULATION 2 −2 2 10 m 2 1500 cm ×( ) = 0.15 m 1 cm or 2 2 1 m 2 1500 cm ×( ) = 0.15 m 100 cm or 2 2 1 m 2 1500 cm × = 0.15 m 2 10, 000 cm  Example 2.8.1: Volume of a Sphere What is the volume of a sphere (radius 4.30 inches) in cubic cm (cm3)? Solution Solution for Example 2.8.1 What is the volume of a sphere (radius 4.30 inches) in cubic cm Steps for Problem Solving (cm3)? Identify the "given” information and what the problem is asking you Given: radius = 4.30 in to "find." Find: cm3 (volume) 4 Volume of a sphere: V = 3 × π× r 3 Determine other known quantities. = 4 × 3.1416 × (4.3 0in ) 3 3 – = 333.04in 3 – 2.8.1 https://chem.libretexts.org/@go/page/47455  What is the volume of a sphere (radius 4.30 inches) in cubic cm Steps for Problem Solving (cm3)? Prepare a concept map. To convert inches cubed to centimeters cubed, use conversion factor 2.54 centimeters per 1 inch, cubed overall 3 2.54cm Calculate. 33 3.04 in 3 ( ) 3 = 5.46 × 10 cm 3 – 1 in A centimeter is a smaller unit than an inch, so the answer in cubic Think about your result. centimeters is larger than the given value in cubic inches.  Exercise 2.8.1 Lake Tahoe has a surface area of 191 square miles. What is the area in square km (km2)? Answer 495 km2 Contributions & Attributions 2.8: Units Raised to a Power is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.8.2 https://chem.libretexts.org/@go/page/47455 2.9: Density  Learning Objectives Define density. Use density as a conversion factor. Density (ρ) is a physical property found by dividing the mass of an object by its volume. Regardless of the sample size, density is always constant. For example, the density of a pure sample of tungsten is always 19.25 grams per cubic centimeter. This means that whether you have one gram or one kilogram of the sample, the density will never vary. The equation, as we already know, is as follows: Mass Density = Volume or just m ρ = (2.9.1) V Based on this equation, it's clear that density can, and does, vary from element to element and substance to substance due to differences in the relationship of mass and volume. Pure water, for example, has a density of 0.998 g/cm3 at 25° C. The average densities of some common substances are in Table 2.9.1. Notice that corn oil has a lower mass to volume ratio than water. This means that when added to water, corn oil will “float.” Table 2.9.1 : Densities of Common Substances Substance Density at 25°C (g/cm3) blood 1.035 body fat 0.918 whole milk 1.030 corn oil 0.922 mayonnaise 0.910 honey 1.420 Density can be measured for all substances—solids, liquids and gases. For solids and liquids, density is often reported using the units of g/cm3. Densities of gases, which are significantly lower than the densities of solids and liquids, are often given using units of g/L.  Example 2.9.1: Ethyl Alcohol Calculate the density of a 30.2 mL sample of ethyl alcohol with a mass of 23.71002 g Solution This is a direct application of Equation 2.9.1: 23.71002 g ρ = = 0.785 g/mL 30.2 mL  Exercise 2.9.1 a. Find the density (in kg/L) of a sample that has a volume of 36.5 L and a mass of 10.0 kg. b. If you have a 2.130 mL sample of acetic acid with mass 0.002234 kg, what is the density in kg/L? Answer a 2.9.1 https://chem.libretexts.org/@go/page/47457  0.274 kg/L Answer b 1.049 kg/L Density as a Conversion Factor Conversion factors can also be constructed for converting between different kinds of units. For example, density can be used to convert between the mass and the volume of a substance. Consider mercury, which is a liquid at room temperature and has a density of 13.6 g/mL. The density tells us that 13.6 g of mercury have a volume of 1 mL. We can write that relationship as follows: 13.6 g mercury = 1 mL mercury This relationship can be used to construct two conversion factors: 13.6 g =1 1 mL and 1 mL =1 13.6 g Which one do we use? It depends, as usual, on the units we need to cancel and introduce. For example, suppose we want to know the mass of 2.0 mL of mercury. We would use the conversion factor that has milliliters on the bottom (so that the milliliter unit cancels) and grams on top, so that our final answer has a unit of mass: 13.6 g 2.0 mL × = 27.2 g = 27 g 1 mL In the last step, we limit our final answer to two significant figures because the volume quantity has only two significant figures; the 1 in the volume unit is considered an exact number, so it does not affect the number of significant figures. The other conversion factor would be useful if we were given a mass and asked to find volume, as the following example illustrates. Density can be used as a conversion factor between mass and volume.  Example 2.9.2: Mercury Thermometer Steps for Problem Solving A mercury thermometer for measuring a patient’s temperature contains 0.750 g of mercury. What is the volume of this mass of mercury? Solution Solution to Example 2.9.2 Steps for Problem Solving Unit Conversion Identify the "given" information and what the problem is asking you Given: 0.750 g to "find." Find: mL List other known quantities. 13.6 g/mL (density of mercury) Prepare a concept map. 1 mL 0.750 g × = 0.055147... mL ≈ 0.0551 mL Calculate. 13.6 g We have limited the final answer to three significant figures. 2.9.2 https://chem.libretexts.org/@go/page/47457   Exercise 2.9.2 What is the volume of 100.0 g of air if its density is 1.3 g/L? Answer 77 L Summary Density is defined as the mass of an object divided by its volume. Density can be used as a conversion factor between mass and volume. Contributions & Attributions 2.9: Density is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.9.3 https://chem.libretexts.org/@go/page/47457 2.E: Measurement and Problem Solving (Exercises) 2.1: Measuring Global Temperatures 2.2: Scientific Notation: Writing Large and Small Numbers 2.3: Significant Figures: Writing Numbers to Reflect Precision 1. Define significant figures. Why are they important? 2. Define the different types of zeros found in a number and explain whether or not they are significant. 3. How many significant figures are in each number? a. 140 b. 0.009830 c. 15,050 d. 221,560,000 e. 5.67 × 103 f. 2.9600 × 10−5 4. How many significant figures are in each number? a. 1.05 b. 9,500 c. 0.0004505 d. 0.00045050 e. 7.210 × 106 f. 5.00 × 10−6 5. Round each number to three significant figures. a. 34,705 b. 34,750 c. 34,570 2.4: Significant Figures in Calculations 2.5: The Basic Units of Measurement 2.6: Problem Solving and Unit Conversions 2.7: Solving Multi-step Conversion Problems 2.8: Units Raised to a Power 2.9: Density 2.10: Numerical Problem-Solving Strategies and the Solution Map 2.E: Measurement and Problem Solving (Exercises) is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar- Agnew & Henry Agnew. 2.E.1 https://chem.libretexts.org/@go/page/53686