INTRODUCTORY CHEMISTRY INTRODUCTORY CHEMISTRY Concepts and Critical Thinking Sixth Edition by Charles H. Corwin Chapter 2 1 © 2011 Pearson Education, Inc. Chapter 2 Scientific Measurements by Christopher Hamaker

2 Chapter 2 © 2011 Pearson Education, Inc. 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

3 Chapter 2 © 2011 Pearson Education, Inc. 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.

4 Chapter 2 © 2011 Pearson Education, Inc. 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.

5 Chapter 2 © 2011 Pearson Education, Inc. 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.

6 Chapter 2 © 2011 Pearson Education, Inc. Mass Versus Weight Mass and weight are not the same. –Weight is the force exerted by gravity on an object.

7 Chapter 2 © 2011 Pearson Education, Inc. 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

8 Chapter 2 © 2011 Pearson Education, Inc. 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.

9 Chapter 2 © 2011 Pearson Education, Inc. 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 s. –35 s has one significant figure. –35.1 s has two significant figures. –35.08 has three significant figures.

10 Chapter 2 © 2011 Pearson Education, Inc. 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 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.

11 Chapter 2 © 2011 Pearson Education, Inc. 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.

12 Chapter 2 © 2011 Pearson Education, Inc. 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.

13 Chapter 2 © 2011 Pearson Education, Inc. Rules for Rounding Numbers 1.If the first nonsignificant digit is less than 5, drop all nonsignificant digits. 2.If the first nonsignificant digit is greater than or equal to 5, increase the last significant digit by 1 and drop all nonsignificant digits. 3.If a calculation has two or more operations, retain all the nonsignificant digits until the final operation and then round off the answer.

14 Chapter 2 © 2011 Pearson Education, Inc. Rounding Examples A calculator displays 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 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.

15 Chapter 2 © 2011 Pearson Education, Inc. 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.

16 Chapter 2 © 2011 Pearson Education, Inc. 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 g has the greatest uncertainty (± 0.1 g). The correct answer is g g 0.25g g g

17 Chapter 2 © 2011 Pearson Education, Inc. 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) = cm 2 The measurement 2.3 cm has the fewest significant digits—two. The correct answer is 12 cm 2.

18 Chapter 2 © 2011 Pearson Education, Inc. 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) = 2 3. 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) = 2 3 = 8

19 Chapter 2 © 2011 Pearson Education, Inc. 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).

20 Chapter 2 © 2011 Pearson Education, Inc. 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 10 n power of 10 significant digits

21 Chapter 2 © 2011 Pearson Education, Inc. 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.

22 Chapter 2 © 2011 Pearson Education, Inc. 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 x atoms

23 Chapter 2 © 2011 Pearson Education, Inc. Another Example The typical length between two carbon atoms in a molecule of benzene is 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 x m

24 Chapter 2 © 2011 Pearson Education, Inc. 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 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).

25 Chapter 2 © 2011 Pearson Education, Inc. 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

26 Chapter 2 © 2011 Pearson Education, Inc. 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

27 Chapter 2 © 2011 Pearson Education, Inc. 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.

28 Chapter 2 © 2011 Pearson Education, Inc. Steps in the Unit Analysis Method 1.Write down the unit asked for in the answer. 2.Write down the given value related to the answer. 3.Apply a unit factor to convert the unit in the given value to the unit in the answer.

29 Chapter 2 © 2011 Pearson Education, Inc. 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.

30 Chapter 2 © 2011 Pearson Education, Inc. 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.

31 Chapter 2 © 2011 Pearson Education, Inc. 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.

32 Chapter 2 © 2011 Pearson Education, Inc. 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.

33 Chapter 2 © 2011 Pearson Education, Inc. 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:

34 Chapter 2 © 2011 Pearson Education, Inc. 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?

35 Chapter 2 © 2011 Pearson Education, Inc. 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

36 Chapter 2 © 2011 Pearson Education, Inc. 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.

37 Chapter 2 © 2011 Pearson Education, Inc. 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?

38 Chapter 2 © 2011 Pearson Education, Inc. 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.

39 Chapter 2 © 2011 Pearson Education, Inc. 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.

40 Chapter 2 © 2011 Pearson Education, Inc. 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.

41 Chapter 2 © 2011 Pearson Education, Inc. 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 10 n, where D.DD are the significant figures (and is between 1 and 10) and n is the power of ten.

42 Chapter 2 © 2011 Pearson Education, Inc. 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.