Problem 36 - Chemistry: The Central Science

Question

Indicate the number of significant figures in each of the following measured quantities: (a) \(3.774 \mathrm{~km}\), (b) \(205 \mathrm{~m}^{2}\), (c) \(1.700 \mathrm{~cm}\), (d) \(350.00 \mathrm{~K}_{1}\) (e) \(307.080 \mathrm{~g}\).

Step-by-Step Solution

Answer

The number of significant figures in the given measured quantities are as follows: (a) 3.774 km has \(4\) significant figures, (b) 205 m² has \(3\) significant figures, (c) 1.700 cm has \(4\) significant figures, (d) 350.00 K has \(5\) significant figures, and (e) 307.080 g has \(6\) significant figures.

Step 1: (a) Determine significant figures in 3.774 km

Considering the rules, there are no leading or trailing zeros, and all digits are nonzero. Therefore, all digits are significant. The number has 4 significant figures.

Step 2: (b) Determine significant figures in 205 m^2

All nonzero digits are significant, which are 2 and 5. The zero in between is also significant as it is between nonzero digits. The number has 3 significant figures.

Step 3: (c) Determine significant figures in 1.700 cm

In this case, all nonzero digits are significant, and trailing zeros after the decimal point are also significant. Therefore, the number of significant figures is 4.

Step 4: (d) Determine significant figures in 350.00 K

All nonzero digits, in this case, 3 and 5, are significant. Also, the zeros after the decimal point and trailing zeros are significant. Hence, there are 5 significant figures.

Step 5: (e) Determine significant figures in 307.080 g

All nonzero digits (3, 0, 7) are significant. Since the zeros are between nonzero digits and also trailing zeros after a decimal point, they are significant as well. The number of significant figures is 6.

Key Concepts

Measurement Precision

Precision in measurements refers to the consistency and reproducibility of a series of readings or values. In the context of significant figures in chemistry, precise measurements are crucial because they reflect the reliability of the data collected during experiments and quantitative analysis.

When looking at a value like our example, 1.700 cm, with four significant figures, we see a high level of precision. This suggests that the measurement was taken to the nearest tenth of a millimeter, indicating not only the tool used for measurement had the ability to measure such fine detail, but also that the measurement was intended to be that precise. In this case, the trailing zeros after the decimal point signify that the measurement was exact up to that point, confirming the intention to convey precision.

Improving precision may involve using more sensitive equipment or clarifying measurement procedures to ensure consistency. This aspect is vital across all scientific disciplines but particularly in chemistry where reactions and properties can be highly sensitive to variations in quantities.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is also used to easily communicate the precision of a measurement with significant figures. This method uses powers of ten to simplify figures and make them easier to work with, especially in calculations.

For instance, 205 m² might appear simple enough, but when dealing with measurements like Avogadro's number (approximately 6.022 x 10²³), scientific notation becomes essential for practical use and comprehension. To express this number in scientific notation, we identify the significant digits (6.022) and count the places to move the decimal point to get from 6.022 to the actual number. Here, we move it 23 places to the right, hence 10 to the power of 23.

Quantitative Analysis

Quantitative analysis in chemistry involves the determination of the exact amount or concentration of a substance or chemical in a sample. It relies heavily on the correct use of significant figures since the accuracy of results depends on correct measurement and data interpretation.

When we observe a mass of 307.080 grams, there is an implication that the quantity was measured to the nearest microgram. This level of detail is crucial in quantitative analysis, as a small mistake or misinterpretation could lead to inaccurate conclusions about the chemical composition or reaction stoichiometry.

To improve quantitative analysis, chemists take multiple measurements to ensure reproducibility and employ calibration methods for their instruments. Moreover, they closely adhere to significant figure rules to represent the certainty of their findings accurately, making sure that no false precision is implied.