Problem 67 - Chemistry The Science in Context

Question

Which of these uncertain values has the smallest number of significant figures? (a) \(1 / 545 ;\) (b) \(1 / 6.4 \times 10^{-3} ;\) (c) \(1 / 6.50\) (d) \(1 / 1.346 \times 10^{2}\)

Step-by-Step Solution

Answer

a) \(\displaystyle \frac{1}{545}\) b) \(\displaystyle \frac{1}{6.4 \times 10^{-3}}\) c) \(\displaystyle \frac{1}{6.50}\) d) \(\displaystyle \frac{1}{1.346 \times 10^{2}}\) Answer: b) \(\displaystyle \frac{1}{6.4 \times 10^{-3}}\)

Step 1: Recognizing significant figures

In each value mentioned, realize that the number of significant figures corresponds to the number of digits in the denominator of the given fractions. (a) \(\displaystyle \frac{1}{545}\) has three significant figures (545), (b) \(\displaystyle \frac{1}{6.4 \times 10^{-3}}\) has two significant figures (6.4), (c) \(\displaystyle \frac{1}{6.50}\) has three significant figures (6.50), (d) \(\displaystyle \frac{1}{1.346 \times 10^{2}}\) has four significant figures (1.346). Keep in mind that the significant figures are only considered for the denominators since that's where the uncertainty lies.

Step 2: Comparing the significant figures

Now compare the significant figures in each option: (a) 3 significant figures, (b) 2 significant figures, (c) 3 significant figures, (d) 4 significant figures.

Step 3: Finding the smallest number of significant figures

From the comparison, option (b) with \(\displaystyle \frac{1}{6.4 \times 10^{-3}}\) has the smallest number of significant figures, which is 2.

Key Concepts

Understanding Uncertainty in Measurements

When dealing with measurements, uncertainty is an inherent aspect to consider. Uncertainty arises because every measurement has limitations due to the tools, conditions, and methods used.
Understanding this uncertainty helps us convey how precise or reliable a measurement is. In our exercise, significant figures play a crucial role in expressing uncertainty.

• Significant figures indicate how precise a measurement is. The more significant figures, the less uncertainty it implies.
• It's about the number of meaningful digits in a measurement, not just any digit.
• For example, the measurement of \(1 / 545\) has three significant figures, suggesting a certain level of precision, whereas \(1 / 6.4 \times 10^{-3}\) has two, indicating higher uncertainty.

Understanding which measurement is more uncertain helps in making more informed decisions in scientific calculations.

The Role of Scientific Notation

Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It's especially useful in conveying significant figures accurately.
This notation expresses numbers as a product of a coefficient and a power of 10.

• For example, \(6.4 \times 10^{-3}\) indicates that the number is 6.4 multiplied by 0.001.
• Scientific notation clarifies the number of significant figures present. Here, "6.4" shows two significant digits.
• By maintaining the precision of a value, scientific notation aids in identifying uncertainty in a measurement.

By using this method, scientists and students can compare and analyze values effectively, maintaining clarity in accuracy.

Comparing Numerical Values Using Significant Figures

Numerical comparison using significant figures allows us to understand which measurements are more precise or uncertain. By examining the number of significant figures, we can prioritize the importance of measurements in calculations.

• In the given exercise, each option is assessed based on the denominator's significant figures.
• For instance, \(1 / 6.50\) with three significant figures is more precise than \(1 / 6.4 \times 10^{-3}\) with two significant figures.
• The exercise revealed that the fewest significant figures were in option (b), making it the most uncertain variable.

By comparing the number of significant figures, one can quickly determine which values to trust more in precise scientific or mathematical operations.