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Chapter 2

Chapter 2. Scientific Measurements by Christopher Hamaker. Uncertainty in Measurements. A measurement is a number with a unit attached. It is not possible to make exact measurements, thus all measurements have uncertainty . We will generally use metric system units. These include:

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Chapter 2

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  1. Chapter 2 Scientific Measurements by Christopher Hamaker Chapter 2

  2. Uncertainty in Measurements • A measurement is a number with a unit attached. • It is not possible to make exact measurements, thus all measurements have uncertainty. • We will generally use metric system units. These include: • The meter, m, for length measurements • The gram, g, for mass measurements • The liter, L, for volume measurements Chapter 2

  3. Length Measurements • Let’s measure the length of a candy cane. • Ruler A has 1 cm divisions, so we can estimate the length to ± 0.1 cm. The length is 4.2 ± 0.1 cm. • Ruler B has 0.1 cm divisions, so we can estimate the length to ± 0.05 cm. The length is 4.25 ± 0.05 cm. Chapter 2

  4. Uncertainty in Length • Ruler A: 4.2 ± 0.1 cm; Ruler B: 4.25 ± 0.05 cm. • Ruler A has more uncertainty than Ruler B. • Ruler B gives a more precise measurement. Chapter 2

  5. Mass Measurements • The mass of an object is a measure of the amount of matter it possesses. • Mass is measured with a balance and is not affected by gravity. • Mass and weight are not interchangeable. Chapter 2

  6. Mass Versus Weight • Mass and weight are not the same. • Weight is the force exerted by gravity on an object. Chapter 2

  7. Volume Measurements • Volume is the amount of space occupied by a solid, a liquid, or a gas. • There are several instruments for measuring volume, including: • Graduated cylinder • Syringe • Buret • Pipet • Volumetric flask Chapter 2

  8. Significant Digits • Each number in a properly recorded measurement is a significant digit (or significant figure). • Significant digits express the uncertainty in the measurement. • When you count significant digits, start counting with the first nonzero number. • Let’s look at a reaction measured by three stopwatches. Chapter 2

  9. Significant Digits, Continued • Stopwatch A is calibrated to seconds (±1 s); Stopwatch B to tenths of a second (±0.1 s); and Stopwatch C to hundredths of a second (±0.01 s). • Stopwatch A reads 35 s; B reads 35.1 s; and C reads 35.08 s. • 35 s has one significant figure. • 35.1 s has two significant figures. • 35.08 has three significant figures. Chapter 2

  10. Significant Digits and Placeholders • If a number is less than 1, a placeholder zero is never significant. • Therefore, 0.5 cm, 0.05 cm, and 0.005 cm all have one significant digit. • If a number is greater than 1, a placeholder zero is usually not significant. • Therefore, 50 cm, 500 cm, and 5000 cm all have one significant digit. Chapter 2

  11. Exact Numbers • When we count something, it is an exact number. • Significant digit rules do not apply to exact numbers. • An example of an exact number: There are seven coins on this slide. Chapter 2

  12. Rounding Off Nonsignificant Digits • All numbers from a measurement are significant. However, we often generate nonsignificant digits when performing calculations. • We get rid of nonsignificant digits by rounding off numbers. • There are three rules for rounding off numbers. Chapter 2

  13. Rules for Rounding Numbers • If the first nonsignificant digit is less than 5, drop all nonsignificant digits. • If the first nonsignificant digit is greater than or equal to 5, increase the last significant digit by 1 and drop all nonsignificant digits. • If a calculation has two or more operations, retain all the nonsignificant digits until the final operation and then round off the answer. Chapter 2

  14. Rounding Examples • A calculator displays 12.846239 and 3 significant digits are justified. • The first nonsignificant digit is a 4, so we drop all nonsignificant digits and get 12.8 as the answer. • A calculator displays 12.856239 and 3 significant digits are justified. • The first nonsignificant digit is a 5, so the last significant digit is increased by one to 9. All the nonsignificant digits are dropped, and we get 12.9 as the answer. Chapter 2

  15. Rounding Off and Placeholder Zeros • Round the measurement 151 mL to two significant digits. • If we keep two digits, we have 15 mL, which is only about 10% of the original measurement. • Therefore, we must use a placeholder zero: 150 mL • Recall that placeholder zeros are not significant. • Round the measurement 2788 g to two significant digits. • We get 2800 g. • Remember, the placeholder zeros are not significant, and 28 grams is significantly less than 2800 grams. Chapter 2

  16. Adding and Subtracting Measurements • When adding or subtracting measurements, the answer is limited by the value with the most uncertainty. • Let’s add three mass measurements. • The measurement 106.7 g has the greatest uncertainty (±0.1 g). • The correct answer is 107.1 g. Chapter 2

  17. Multiplying and Dividing Measurements • When multiplying or dividing measurements, the answer is limited by the value with the fewest significant figures. • Let’s multiply two length measurements: (5.15 cm)(2.3 cm) = 11.845 cm2 • The measurement 2.3 cm has the fewest significant digits—two. • The correct answer is 12 cm2. Chapter 2

  18. Exponential Numbers • Exponents are used to indicate that a number has been multiplied by itself. • Exponents are written using a superscript; thus, (2)(2)(2) = 23. • The number 3 is an exponent and indicates that the number 2 is multiplied by itself 3 times. It is read “2 to the third power” or “2 cubed”. • (2)(2)(2) = 23 = 8 Chapter 2

  19. Powers of 10 • A power of 10 is a number that results when 10 is raised to an exponential power. • The power can be positive (number greater than 1) or negative (number less than 1). Chapter 2

  20. Scientific Notation • Numbers in science are often very large or very small. To avoid confusion, we use scientific notation. • Scientific notation utilizes the significant digits in a measurement followed by a power of 10. The significant digits are expressed as a number between 1 and 10. power of 10 D.DDx 10n significant digits Chapter 2

  21. Applying Scientific Notation • To use scientific notation, first place a decimal after the first nonzero digit in the number followed by the remaining significant digits. • Indicate how many places the decimal is moved by the power of 10. • A positive power of 10 indicates that the decimal moves to the left. • A negative power of 10 indicates that the decimal moves to the right. Chapter 2

  22. Scientific Notation, Continued There are 26,800,000,000,000,000,000,000 helium atoms in 1.00 L of helium gas. Express the number in scientific notation. • Place the decimal after the 2, followed by the other significant digits. • Count the number of places the decimal has moved to the left (22). Add the power of 10 to complete the scientific notation. x1022 atoms 2.68 Chapter 2

  23. Another Example The typical length between two carbon atoms in a molecule of benzene is 0.000000140 m. What is the length expressed in scientific notation? • Place the decimal after the 1, followed by the other significant digits. • Count the number of places the decimal has moved to the right (7). Add the power of 10 to complete the scientific notation. x10-7 m 1.40 Chapter 2

  24. Scientific Calculators • A scientific calculator has an exponent key (often “EXP” or “EE”) for expressing powers of 10. • If your calculator reads 7.45 E-17, the proper way to write the answer in scientific notation is 7.45 x 10-17. • To enter the number in your calculator, type 7.45, then press the exponent button (“EXP” or “EE”), and type in the exponent (17 followed by the +/– key). Chapter 2

  25. Unit Equations • A unit equation is a simple statement of two equivalent quantities. • For example: • 1 hour = 60 minutes • 1 minute = 60 seconds • Also, we can write: • 1 minute = 1/60 of an hour • 1 second = 1/60 of a minute Chapter 2

  26. Unit Factors • A unit conversion factor, or unit factor, is a ratio of two equivalent quantities. • For the unit equation 1 hour = 60 minutes, we can write two unit factors: 1 hour or 60 minutes 60 minutes 1 hour Chapter 2

  27. Unit Analysis Problem Solving • An effective method for solving problems in science is the unit analysis method. • It is also often called dimensional analysis or the factor-label method. • There are three steps to solving problems using the unit analysis method. Chapter 2

  28. Steps in the Unit Analysis Method • Write down the unit asked for in the answer. • Write down the given value related to the answer. • Apply a unit factor to convert the unit in the given value to the unit in the answer. Chapter 2

  29. Unit Analysis Problem How many days are in 2.5 years? • Step 1: We want days. • Step 2: We write down the given: 2.5 years. • Step 3: We apply a unit factor (1 year = 365 days) and round to two significant figures. Chapter 2

  30. Another Unit Analysis Problem A can of soda contains 12 fluid ounces. What is the volume in quarts (1 qt = 32 fl oz)? • Step 1: We want quarts. • Step 2: We write down the given: 12 fl oz. • Step 3: We apply a unit factor (1 qt = 12 fl oz) and round to two significant figures. Chapter 2

  31. Another Unit Analysis Problem, Continued A marathon is 26.2 miles. What is the distance in kilometers (1 km = 0.62 mi)? • Step 1: We want km. • Step 2: We write down the given: 26.2 mi. • Step 3: We apply a unit factor (1 km = 0.62 mi) and round to three significant figures. Chapter 2

  32. Critical Thinking: Units • When discussing measurements, it is critical that we use the proper units. • NASA engineers mixed metric and English units when designing software for the Mars Climate Orbiter. • The engineers used kilometers rather than miles. • 1 kilometer is 0.62 mile. • The spacecraft approached too close to the Martian surface and burned up in the atmosphere. Chapter 2

  33. The Percent Concept • A percent, %, expresses the amount of a single quantity compared to an entire sample. • A percent is a ratio of parts per 100 parts. • The formula for calculating percent is shown below: Chapter 2

  34. Calculating Percentages • Sterling silver contains silver and copper. If a sterling silver chain contains 18.5 g of silver and 1.5 g of copper, what is the percent of silver in sterling silver? Chapter 2

  35. Percent Unit Factors • A percent can be expressed as parts per 100 parts. • 25% can be expressed as 25/100 and 10% can be expressed as 10/100. • We can use a percent expressed as a ratio as a unit factor. • A rock is 4.70% iron, so Chapter 2

  36. Percent Unit Factor Calculation The Earth and Moon have a similar composition; each contains 4.70% iron. What is the mass of iron in a lunar sample that weighs 235 g? • Step 1: We want g iron. • Step 2: We write down the given: 235 g sample. • Step 3: We apply a unit factor (4.70 g iron = 100 g sample) and round to three significant figures. Chapter 2

  37. Chemistry Connection: Coins • A nickel coin contains 75.0 % copper metal and 25.0 % nickel metal, and has a mass of 5.00 grams. • What is the mass of nickel metal in a nickel coin? Chapter 2

  38. Chapter Summary • A measurement is a number with an attached unit. • All measurements have uncertainty. • The uncertainty in a measurement is dictated by the calibration of the instrument used to make the measurement. • Every number in a recorded measurement is a significant digit. Chapter 2

  39. Chapter Summary, Continued • Placeholding zeros are not significant digits. • If a number does not have a decimal point, all nonzero numbers and all zeros between nonzero numbers are significant. • If a number has a decimal place, significant digits start with the first nonzero number and all digits to the right are also significant. Chapter 2

  40. Chapter Summary, Continued • When adding and subtracting numbers, the answer is limited by the value with the most uncertainty. • When multiplying and dividing numbers, the answer is limited by the number with the fewest significant figures. • When rounding numbers, if the first nonsignificant digit is less than 5, drop the nonsignificant figures. If the number is 5 or more, raise the first significant number by 1, and drop all of the nonsignificant digits. Chapter 2

  41. Chapter Summary, Continued • Exponents are used to indicate that a number is multiplied by itself n times. • Scientific notation is used to express very large or very small numbers in a more convenient fashion. • Scientific notation has the form D.DD x 10n, where D.DD are the significant figures (and is between 1 and 10) and n is the power of ten. Chapter 2

  42. Chapter Summary, Continued • A unit equation is a statement of two equivalent quantities. • A unit factor is a ratio of two equivalent quantities. • Unit factors can be used to convert measurements between different units. • A percent is the ratio of parts per 100 parts. Chapter 2

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