Interlude Prerequisite Science Skills by Christopher G. Hamaker Illinois State University © 2014 Pearson Education, Inc. 1

Measurements A measurement is a number with a unit attached. Every measurement has a degree of inexactness, termed uncertainty. We will generally use metric system units. These include: The centimeter, cm, for length measurements The gram, g, for mass measurements The milliliter, mL, for volume measurements

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.

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.

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

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

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.

Significant Digits, Continued Stopwatch A is calibrated to seconds (0 s); Stopwatch B to tenths of a second (0.0 s); and Stopwatch C to hundredths of a second (0.00 s). Stopwatch A reads 35 s; B reads 35.1 s; and C reads 35.08 s. 35 s has two significant figures. 35.1 s has three significant figures. 35.08 has four significant figures.

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.

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 7 quarters on this slide.

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.

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 several multiplication or division operations, retain nonsignificant digits in your calculator until the last operation.

Rounding Examples A calculator displays 15.73849 and 3 significant digits are justified. The first nonsignificant digit is a 3, so we drop all nonsignificant digits and get 15.7 as the answer. A calculator displays 18.750019 and 3 significant digits are justified. The first nonsignificant digit is a 5, so the last significant digit is increased by one to 8. All the nonsignificant digits are dropped, and we get 18.8 as the answer.

Rounding Off and Placeholder Zeros Round the measurement 183 mL to two significant digits. If we keep two digits, we have 18 mL, which is only about 10% of the original measurement. Therefore, we must use a placeholder zero: 180 mL. Recall that placeholder zeros are not significant. Round the measurement 48,457 g to two significant digits. We get 48,000 g. Remember, the placeholder zeros are not significant, and 48 grams is significantly less than 48,000 grams.

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 114.3 g has the greatest uncertainty (± 0.1 g). The correct answer is 116.6 g. 114.3 g 0.75 + 0.581 115.631

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: (7.28 cm)(4.6 cm) = 33.488 cm2 The measurement 4.6 cm has the fewest significant digits—two. The correct answer is 33 cm2.

Exponential Numbers Exponents are used to indicate that a number has been multiplied by itself. Exponents are written using a superscript; thus, (4)(4)(4) = 43. The number 3 is an exponent and indicates that the number 4 is multiplied by itself three times. It is read “4 to the third power” or “4 cubed.” (4)(4)(4) = 43 = 64

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).

D.DD x 10n 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. D.DD x 10n power of 10 significant digits

Applying Scientific Notation Step 1: Place a decimal after the first nonzero digit in the number, followed by the remaining significant digits. Step 2: 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.

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. 2.68 x 1022 atoms

Another Example The typical length between a carbon and oxygen atom in a molecule of carbon dioxide is 0.000000116 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. 1.16 x 10-7 m

Scientific Calculators A scientific calculator has an exponent key (often EXP) 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, press the exponent button (EXP) and type in the exponent followed by the +/– key.

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 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 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 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 10.