Problem 45 - Chemistry: The Central Science in SI Units, Global Edition

Question

What is the number of significant figures in each of the following measured quantities? (a) \(902.5 \mathrm{~kg}\), (b) \(3 \times 10^{-6} \mathrm{~m}\), (c) \(0.0096 \mathrm{~L}\), (d) \(2.94 \times 10^{3} \mathrm{~m}^{2}\) (e) \(92.03 \mathrm{~km}\) (f) \(782.234 \mathrm{~g}\).

Step-by-Step Solution

Answer

The number of significant figures for the given measured quantities is as follows: (a) 902.5 kg: 4 significant figures, (b) \(3 \times 10^{-6}\) m: 1 significant figure, (c) 0.0096 L: 2 significant figures, (d) \(2.94 \times 10^{3}\) m²: 3 significant figures, (e) 92.03 km: 4 significant figures, (f) 782.234 g: 6 significant figures.

Step 1: (1. Significant figures for 902.5 kg)

The number 902.5 contains no leading or trailing zeros. All nonzero digits are significant and the zero between 9 and 2 is also significant. Therefore, there are 4 significant figures in this quantity.

Step 2: (2. Significant figures for \(3 \times 10^{-6}\) m)

In scientific notation, all digits in the base (3 in this case) are significant. Therefore, the quantity has 1 significant figure.

Step 3: (3. Significant figures for 0.0096 L)

According to the rules, leading zeros are not significant. Therefore, the quantity has 2 significant figures—9 and 6.

Step 4: (4. Significant figures for \(2.94 \times 10^{3}\) m²)

In scientific notation, all digits in the base (2.94 in this case) are significant. Therefore, the quantity has 3 significant figures.

Step 5: (5. Significant figures for 92.03 km)

There are no leading or trailing zeros and the zero between 9 and 2 is significant. Therefore, there are 4 significant figures in this quantity.

Step 6: (6. Significant figures for 782.234 g)

There are no leading or trailing zeros, and all nonzero digits are significant. Therefore, there are 6 significant figures in this quantity.

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a more manageable form. It is especially useful in scientific and engineering calculations. In scientific notation, a number is represented as the product of two factors: a base and an exponent of ten.

• The base is generally a number greater than or equal to 1 and less than 10.
• The exponent signifies how many times the base is multiplied by ten.

Performing calculations with numbers in scientific notation can be easier and maintains measurement precision. For example, expressing a number like 0.000003 as \(3 \times 10^{-6}\) highlights its small magnitude succinctly. When dealing with scientific notation, remember that all digits in the base are considered significant figures.

Measurement Precision

Measurement precision refers to the exactness and consistency of a measurement. It indicates how close multiple measurements are to one another. Achieving high precision in measurements typically requires the use of fine instruments and meticulous techniques. Precision is important in scientific experiments because it shows the reliability of results. While measurement accuracy concerns how close a measurement is to the "true" or accepted value, precision focuses on the repeatability of measurements. It's essential to be aware of instrument calibration and usage:

• Check calibration of measuring devices to ensure consistent results.
• Use instruments according to the manufacturer's guidelines to avoid bias in measurements.

Additionally, keeping a record of measurements allows you to assess measurement precision over time.

Counting Significant Figures

Counting significant figures is crucial when recording measurements accurately. Significant figures are the digits in a measurement that contribute to its precision. This includes all nonzero digits, any zeros between significant digits, and trailing zeros when they are to the right of a decimal point.

• Leading zeros, such as in 0.0096, do not count as significant figures.
• All non-zero numbers are always significant like in 782.234.

To count significant figures, it is important to understand the specific positioning and context of zeros in numbers: 1. **Non-zero digits**: Always significant. 2. **Zeros between non-zero digits**: Also significant, for example, 902.5 has four significant figures. 3. **Leading zeros**: Not significant, they merely indicate the position of the decimal point. 4. **Trailing zeros to the right of a decimal point**: These are significant as they indicate measured precision. Counting significant figures helps in assessing how exact a measurement is and ensures consistency in rounding numbers during calculations.