Reading Chapter Three: 3.1

Connecting to Your World On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Equipped with five scientific instruments and a rock abrasion tool (shown at left), Spirit was sent to examine the Martian surface around Gusev Crater, a wide basin that may have once held a lake. Each day of its mission, Spirit recorded measurements for analysis. This data helped scientists learn about the geology and climate on Mars. All measurements have some uncertainty. In the chemistry laboratory, you must strive for accuracy and precision in your measurements.



Key Concepts

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How do measurements relate to science?
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How do you evaluate accuracy and precision?
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Why must measurements be reported to the correct number of significant figures?
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How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?

Vocabulary

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measurement
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scientific notation
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accuracy
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precision
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accepted value
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experimental value
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error
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percent error
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significant figures

Reading Strategy

Building Vocabulary As you read, write a definition of each key term in your own words.

Using and Expressing Measurements

Your height (67 inches), your weight (134 pounds), and the speed you drive on the highway (65 miles/hour) are some familiar examples of measurements. A measurement is a quantity that has both a number and a unit. Everyone makes and uses measurements. For instance, you decide how to dress in the morning based on the temperature outside. If you were baking cookies, you would measure the volumes of the ingredients as indicated in the recipe.

Such everyday situations are similar to those faced by scientists. Measurements are fundamental to the experimental sciences. For that reason, it isimportant to be able to make measurements and to decide whether a measurement is correct. The units typically used in the sciences are those of the International System of Measurements(SI).

In chemistry, you will often encounter very large or very small numbers. A single gram of hydrogen, for example, contains approximately 602,000,000,000,000,000,000,000 hydrogenatoms. The mass of an atom of gold is 0.000 000 000 000 000 000 000 327 gram. Writing and using such large and small numbers is very cumbersome. You can work more easily with these numbers by writing them in scientific, or exponential, notation.

In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. For example, the number 602,000,000,000,000,000,000,000written in scientific notation is 6.02 × 1023. The coefficient in this number is 6.02. In scientific notation, the coefficient is always a number equal to or greater than one and less than ten. The power of 10, or exponent, in this example is 23. Figure 3.1 illustrate show to express the number of stars in a galaxy by using scientific notation. For more practice on writing numbers in scientific notation, refer to page R56 of Appendix C.

Accuracy, Precision, and Error

Your success in the chemistry lab and in many of your daily activities depends on your ability to make reliable measurements. Ideally, measurements should be both correct and reproducible.
Accuracy and Precision

Correctness and reproducibility relate to the concepts of accuracy and precision, two words that mean the same thing to many people. In chemistry, however, their meanings are quite different. Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another.To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Darts on a dartboard illustrate accuracy and precision in measurement. Let the bull’s-eye of the dartboard represent the true, or correct, value of what you are measuring. The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closer it comes to the bull’s-eye,the more accurately the dart was thrown. The closeness of several darts to one another corresponds to the degree of precision. The closer together the darts are, the greater the precision and the reproducibility.

Look at Figure 3.2 as you consider the following outcomes.

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All of the darts land close to the bull’s-eye and to one another. Closeness to the bull’s-eye means that the degree of accuracy is great. Each dart in the bull’s-eye corresponds to an accurate measurement of a value. Closeness of the darts to one another indicates high precision.
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All of the darts land close to one another but far from the bull’s-eye. The precision is high because of the closeness of grouping and thus the high level of reproducibility. The results are inaccurate, however, because of the distance of the darts from the bull’s-eye.
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The darts land far from one another and from the bull’s-eye.The results are both inaccurate and imprecise.

Figure 3.2 The distribution of darts illustrates the difference between accuracy and precision. a Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. b Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. c Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another.

Reading Checkpoint
Determining Error

Note that an individual measurement may be accurate or inaccurate. Suppose you use a thermometer to measure the boiling point of pure water at standard pressure. The thermometer reads 99.1°C. You probably know that the true or accepted value of the boiling point of pure water under these conditions is actually 100.0°C. There is a difference between the accepted value, which is the correct value based on reliable references, and the experimental value, the value measured in the lab. The difference between the experimental value and the accepted value is called the error.

Error = experimental value − accepted value

Error can be positive or negative depending on whether the experimental value is greater than or less than the accepted value.

For the boiling-point measurement, the error is 99.1°C − 100.0°C,or − 0.9° C. The magnitude of the error shows the amount by which the experimental value differs from the accepted value. Often, it is useful to calculate the relative error, or percent error. The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.

Word Origins

Percent comes from the Latin words per, meaning “by” or “through,” and centum, meaning “100.” What do you think the phrase per annum means?

Using the absolute value of the error means that the percent error will always be a positive value. For the boiling-point measurement, the percent error is calculated as follows.

= 0.009 × 100%

= 0.9%

Just because a measuring device works doesn’t necessarily mean that it is accurate. As Figure 3.3 shows, a weighing scale that does not read zero when nothing is on it is bound to yield error. In order to weigh yourself accurately, you must first make sure that the scale is zeroed.

Figure 3.3 The scale below has not been properly zeroed so the reading obtained for the person's weight is inaccurate. There is a difference between the person's correct weight and the measured value. Calculating What is the percent error of a measured value of 114 lb if the person's actual weight is 107 lb?

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Significant Figures in Measurements

Supermarkets often provide scales like the one in Figure 3.4. Customers use these scales to measure the weight of produce that is priced per pound. If you use a scale that is calibrated in 0.1-lb intervals, you can easily read the scale to the nearest tenth of a pound. With such a scale, however, you can also estimate the weight to the nearest hundredth of a pound by noting the position of the pointer between calibration marks.

Figure 3.4 The precision of a weighing scale depends on how finely it is calibrated.

Suppose you estimate a weight that lies between 2.4 lb and 2.5 lb to be 2.46 lb. The number in this estimated measurement has three digits. The first two digits in the measurement (2 and 4) are known with certainty. But the rightmost digit (6) has been estimated and involves some uncertainty. These three reported digits all convey useful information, however, and are called significant figures. The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated. Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

Significant Figures in Calculations

Suppose you use a calculator to find the area of a floor that measures 7.7 meters by 5.4 meters. The calculator would give an answer of 41.58 square meters. The calculated area is expressed to four significant figures. However, each of the measurements used in the calculation is expressed to only two significant figures. So the answer must also be reported to two significant figures (42 m2). In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.
Rounding

To round a number, you must first decide how many significant figures the answer should have. This decision depends on the given measurements and on the mathematical process used to arrive at the answer. Once you know the number of significant figures your answer should have, round to that many digits, counting from the left. If the digit immediately to the right of the last significant digit is less than 5, it is simply dropped and the value of the last significant digit stays the same. If the digit in question is 5 or greater, the value of the digit in the last significant place is increased by 1.

Reading Checkpoint

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3.1 Rounding Measurements

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Problem-Solving 3.3 Solve Problem 3 with the help of an interactive guided tutorial.
Addition and Subtraction

The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. Work through Sample Problem 3.2 below which provides an example of rounding in an addition calculation.

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3.2 Significant Figures in Addition

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Problem Solving 3.6 Solve Problem 6 with the help of an interactive guided tutorial.
Multiplication and Division

In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements. The position of the decimal point is important only in rounding the answers of addition or subtraction problems.