Problem 19 - Chemistry: Principles and Reactions

Question

Round off the following quantities to the indicated number of significant figures. (a) \(7.4855 \mathrm{~g}\) (three significant figures) (b) \(298.693 \mathrm{~cm}\) (five significant figures) (c) \(11.698 \mathrm{lb}\) (one significant figure) (d) \(12.05 \mathrm{oz}\) (three significant figures)

Step-by-Step Solution

Answer

Question: Round off each of the given quantities to the indicated number of significant figures. (a) 7.4855 g (three significant figures) (b) 298.693 cm (five significant figures) (c) 11.698 lb (one significant figure) (d) 12.05 oz (three significant figures) Answer: (a) 7.49 g (b) 298.69 cm (c) 12 lb (d) 12.0 oz

Step 1: Significant Figures Rules

To round off the given quantities to the specified number of significant figures, keep these rules in mind: 1. Non-zero digits (1-9) are always significant. 2. Any zeros between significant digits are also significant. 3. Leading zeros, or zeros at the beginning of a number, are never significant. 4. Trailing zeros, or zeros at the end of a number, are only significant if the number has a decimal point.

Step 2: Round off the given quantities

Now we will round off each quantity to the indicated number of significant figures using the rules discussed above. (a) \(7.4855 \mathrm{~g}\) (three significant figures) To round off to three significant figures, look at the fourth digit. Since it's greater than 5, the third digit will be increased by 1. Thus, \(7.4855\) is rounded to \(7.49 \mathrm{~g}\). (b) \(298.693 \mathrm{~cm}\) (five significant figures) To round off to five significant figures, look at the sixth digit. Since it's less than 5, the fifth digit remains unchanged. Thus, \(298.693\) is rounded to \(298.69 \mathrm{~cm}\). (c) \(11.698 \mathrm{lb}\) (one significant figure) To round off to one significant figure, look at the second digit. Since it's greater than 5, the first digit will be increased by 1. Thus, \(11.698\) is rounded to \(12 \mathrm{~lb}\). (d) \(12.05 \mathrm{oz}\) (three significant figures) To round off to three significant figures, look at the fourth digit. Since it's less than 5, the third digit remains unchanged. Thus, \(12.05\) is rounded to \(12.0 \mathrm{~oz}\).

Key Concepts

Rounding Rules

Rounding numbers is crucial for simplifying complex data and making it more comprehensible. It's a skill used in everyday life, as well as in scientific and mathematical calculations. To round numbers accurately, you must first understand the rules behind it.

• Look at the digit immediately after your last significant figure to decide how to round. If this digit is 5 or greater, increase the last significant figure by one.
• If it's less than 5, just keep the last significant figure as it is.
• In situations involving a sequence of digits 5 followed by zeros, it's a rule of thumb to round up to the next digit, ensuring precision is maintained.

For example, when rounding a number like 7.4855 to three significant figures, you look at the fourth. Since this digit, "5", is equal to 5, the third digit is increased by one, resulting in 7.49. Round numbers thoughtfully to maintain integrity and precision in your work.

Measurement Precision

Measurement precision refers to how consistently a measurement can be repeated with the same results. The more precise an instrument, the closer its repeated measurements are to each other. Precision is essential in scientific settings where accuracy in data plays a significant role.

• High precision doesn't necessarily mean the measurement is accurate, it just means it's repeatable.
• Precision can be affected by factors like the instrument used and the method applied.
• It's often dictated by the instrument's limitations, meaning more decimal places typically indicate higher precision.

When performing calculations or using measurements, maintaining precision ensures results are reliable. As in the example of 298.693 cm rounded to five significant figures, the resulting precision is 298.69 cm.

Significant Digits

Significant digits or figures are the digits in a number that carry meaningful contributions to its measurement. They help in determining the accuracy and precision of a measurement. In any given number, significant figures are used to express the precision of a value based on the context in which it's presented.

• All non-zero numbers are significant because they directly contribute to a number's meaning.
• Zeros can be tricky; for example, leading zeros (zeros before non-zero digits) don't count as they're not holding any value.
• Zeros between non-zero digits or after the decimal point are significant as they influence the precision of your measurement.
• Trailing zeros in whole numbers are only significant if stated with a decimal point.

In rounding to one significant figure, such as with 11.698 lb, only the leading non-zero digit is focused on, leading to 12 lb.