Question

How many significant figures do the following measured quantities have? (a) \(37.53 \mathrm{~cm}\) (d) \(0.00024 \mathrm{~kg}\) (b) \(37.240 \mathrm{~cm}\) (e) \(2400 \mathrm{~mL}\) (c) \(202.0 \mathrm{~g}\)

Short answer

(a) has 4 significant figures, (b) has 2 significant figures, (c) has 5 significant figures, (d) potentially has 4 significant figures (context-dependent), and (e) has 4 significant figures.

Step-by-step solution

Understanding Significant Figures

Significant figures are the digits in a number that are known with some degree of reliability. The rules to determine significant figures are: (1) All non-zero digits are significant. (2) Any zeros between non-zero digits are significant. (3) Leading zeros are not significant. (4) Trailing zeros are significant if there is a decimal point in the number.

Identifying Significant Figures in (a) 37.53 cm

For the measurement 37.53 cm, all the digits are non-zero. Therefore, according to rule 1, each digit is significant. There are four significant figures in this measurement.

Identifying Significant Figures in (b) 0.00024 kg

For the measurement 0.00024 kg, the first three zeros are leading zeros and not significant (rule 3). The two non-zero digits, 2 and 4, are significant (rule 1). Hence, there are two significant figures in this measurement.

Identifying Significant Figures in (c) 37.240 cm

For the measurement 37.240 cm, the zeroes between 7 and 2 and after 4 are significant (rules 2 and 4) because they're between non-zero digits and at the end of the number which has a decimal point. Therefore, there are five significant figures in this measurement.

Identifying Significant Figures in (d) 2400 mL

For the measurement 2400 mL, the trailing zeros may or may not be significant as there's no decimal point given which makes this ambiguous without additional context such as a bar over the zero or underlining to indicate they are measured (rule 4). If all digits are measured, then they're all significant, giving four significant figures. However, if the context does not specify that the zeros are measured, we assume that there are only two significant figures.

Identifying Significant Figures in (e) 202.0 g

For the measurement 202.0 g, the zero between 2 and 2 is significant because it is between two non-zero digits (rule 2), and the trailing zero is also significant because there is a decimal point present (rule 4). This gives us four significant figures.

Key concepts

Measuring Accuracy

Measuring accuracy is critical when it comes to scientific data collection and interpretation. The precision of a measurement refers to the extent to which repeated observations under unchanged conditions show the same results. Accuracy, meanwhile, is about how close a measurement comes to the actual or true value.

For instance, if you were to use a ruler to measure a table's length, accuracy would be reflected in how close your measurement is to the table's true length. Precision would be gauged by measuring the table multiple times and noting the consistency of the results.

Digits in Measurement

The digits in any measurement are significant because they convey the precision of a measurement. For instance, if a scientist records a length as 3.00 meters, this implies a higher precision than just stating 3 meters. The additional zeros indicate that the measurement was precise up to the hundredths of a meter.

Importance of Non-Zero Digits

Every non-zero digit expressed in a number is there because the measurer was able to quantify something specific at that level of precision. For example, in the measurement of 37.53 cm from the exercise, scientists assert that they've measured up to a hundredth of a centimeter accurately.

Significant Figures Rules

Understanding the rules of significant figures is important for students and professionals dealing with precise measurements. These rules help in identifying the amount of precision in a given number.

Key Rules of Significant Figures

• All non-zero digits are significant, indicating that they were intentionally measured and are meaningful.
• Zeros situated between non-zero digits are significant, reflecting additional accuracy in the measurement.
• Leading zeros are placeholders and not significant, as they do not represent measured quantities.
• Trailing zeros in a number containing a decimal point are significant. This signifies that the measurement was precise to that decimal place.

These rules need to be applied thoughtfully, considering the context in which the measurement takes place, and are especially relevant when recording, reporting, or calculating with these measurements.

Chemistry Education

In chemistry education, significant figures play an integral role. Grasping how to correctly determine the number of significant figures in a measurement equips students to handle data responsibly and understand the implications of precision in chemical experiments.

Exercises that involve significant figures, such as the textbook examples listed, are not only about learning to count the correct digits; they're about appreciating the scientific method's attention to detail. Chemistry educators emphasize these concepts because accurate and precise measurements underpin the reliability of experimental results and subsequent theories derived from them.