Question

Indicate the number of significant figures in each of the following: a. This book contains more than 1000 pages. b. A mile is about \(5300 \mathrm{ft}\). c. A liter is equivalent to \(1.059\) qt. d. The population of the United States is approaching \(3.0 \times 10^{2}\) million. e. A kilogram is \(1000 \mathrm{~g}\). f. The Boeing 747 cruises at around \(600 \mathrm{mi} / \mathrm{h}\).

Short answer

a. 1 significant figure. b. 2 significant figures. c. 4 significant figures. d. 2 significant figures. e. 3 significant figures. f. 1 significant figure.

Step-by-step solution

Identifying the significant figures in 1000 pages

Since the number is more than 1000 pages, it implies that the exact number of pages is uncertain. In this case, the significant figures are not explicitly stated so we consider just the significant figure at the unit place, the 1. There is 1 significant figure in this case. #b. A mile is about \(5300 \mathrm{ft}\).#

Identifying the significant figures in \(5300\)

The given value is an approximation; hence all non-zero digits are significant. So, there are 2 significant figures in \(5300\). #c. A liter is equivalent to \(1.059\) qt.#

Identifying the significant figures in 1.059

All non-zero digits and zeros between non-zero digits are significant. In this case, all 4 digits (1, 0, 5, and 9) contribute to the value of the number, making it 4 significant figures. #d. The population of the United States is approaching \(3.0 \times 10^{2}\) million.#

Identifying the significant figures in \(3.0 \times 10^{2}\)

In a number written in scientific notation, all digits in the decimal part are significant. So, the significant figures are 3 and 0, making it 2 significant figures. #e. A kilogram is \(1000 \mathrm{~g}\).#

Identifying the significant figures in 1000

The given value is an exact conversion, not an approximation or a measurement. Therefore, all digits are considered significant, making it 3 significant figures. #f. The Boeing 747 cruises at around \(600 \mathrm{mi}/\mathrm{h}\).#

Identifying the significant figures in 600

The given value is an approximation; hence all non-zero digits are significant. In this case, there is 1 significant figure which is 6.

Key concepts

Measurement Accuracy

Measurement accuracy is vital in chemistry and is closely tied to the concept of significant figures. It reflects the closeness of a measurement to the true or accepted value. Accurate measurement is crucial as it ensures correctness in scientific research and experiments. Significant figures play a crucial role in expressing this accuracy. They comprise all the certain digits in a number plus the first uncertain digit. For example, in the measure of about 5300 feet for a mile, the '53' are the certain digits, reflecting that the mile is definitely not less than 5300 feet, but could be a bit more. Hence, it has 2 significant figures.

However, when we say a book has 'more than 1000 pages', it is not clear how many pages over 1000 there are, and thus we only consider the '1' as a significant figure. The way in which we report the significant figures directly influences the perceived accuracy of the measurement. For instance, stating a population as '3.0 x 10^2 million' provides a different level of accuracy compared to simply saying '300 million', even though numerically they may appear equivalent.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. In chemistry, it assists in maintaining measurement accuracy by allowing scientists to clearly indicate the precision of a number through significant figures. For example, the population of the United States is given as '3.0 x 10^2 million'. The scientific notation here, specifically the '3.0', tells us that there are 2 significant figures, highlighting the precision of the measurement. In scientific notation, each non-zero digit after the decimal point is also considered a significant figure.

Why use scientific notation?

Scientific notation is not only a concise way to represent numbers but it also provides clarity about the certainty of measurements. When dealing with very large or small quantities, it allows chemists to focus on the meaningful digits—those that contribute to the measure's precision—while readily conveying the scale of the measurement.

Unit Conversion

Unit conversion is the process of changing one unit of measure to another without changing the quantity itself. It is critical in chemistry for comparing, communicating, and interpreting measurements. For example, converting a kilogram into 1000 grams is a direct conversion where no approximate figures are involved, and all the digits (1,0,0,0) are significant figures because they represent an exact count.

Navigating through different units

Unit conversions often use conversion factors that are defined with an exact number of significant figures. These conversion factors are based on definitions and agreements in the scientific community, as opposed to measurements, which might contain a degree of uncertainty. For instance, '1 liter is equivalent to 1.059 quarts' shows a conversion from one volume unit to another and requires the consideration of the significant figures (all four digits in 1.059 are significant) to maintain the accuracy of the converted measurement. By understanding how significant figures translate during unit conversions, students can ensure that the integrity of the data is preserved from one unit to another.