Uncertainty in Measurements A measurement is a number with a unit attached. It is not possible to make exact measurements, and 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

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

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

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

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

Volume Measurements Volume is the amount of space occupied by a solid, liquid, or gas. There are several instruments for measuring volume, including: graduated cylinder syringe buret pipet volumetric flask Chapter 2

Significant Digits Each number in a properly recorded measurement is a significant digit (or significant figure). The significant digits express the uncertainty in the measurement. When you count significant digits, start counting with the first non-zero number. Lets look at a reaction measured by three stopwatches. Chapter 2

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 2 sig fig 35.1 s has 3 sig figs 35.08 has 4 sig figs Chapter 2

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

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 3 coins on this slide. Chapter 2

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

Rule 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 nonsignificant digits until the final operation and then round off the answer. Chapter 2

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

Rounding Off & Placeholder Zeros Round the measurement 151 mL to two significant digits. If we keep 2 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

Adding & 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. 106.7 g 0.25 + 0.195 107.145 Chapter 2

Multiplying & 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

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

Powers of Ten 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

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 ten. The significant digits are expressed as a number between 1 and 10. Chapter 2

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

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 × 1022 atoms Chapter 2

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. 1.40 × 10-7 m Chapter 2

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 × 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

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

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

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

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

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

Another Unit Analysis Problem A can of Coca-Cola 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

Another Unit Analysis Problem 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 miles. Step 3: We apply a unit factor (1 km = 0.62 mi) and round to three significant figures. Chapter 2

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

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

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

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

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

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

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

Chapter Summary, continued Place-holding 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

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 one and drop all of the nonsignificant digits. Chapter 2

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 × 10n, where D.DD are the significant figures (and is between 1 and 10) and n is the power of ten. Chapter 2

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 ration of parts per 100 parts. Chapter 2