Problem 46 - 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) \(1.700 \mathrm{cm},(\mathbf{d}) 350.00 \mathrm{K},(\mathbf{e}) 307.080 \mathrm{g},(\mathbf{f}) 1.3 \times 10^{3} \mathrm{m} / \mathrm{s}\) .

Step-by-Step Solution

Answer

The number of significant figures in the given measured quantities are: (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, (e) 307.080 g has 6 significant figures, and (f) \(1.3 \times 10^{3}\) m/s has 2 significant figures.

Step 1: (a) 3.774 km

In the number 3.774, all four digits are non-zero and therefore significant. So, there are 4 significant figures in 3.774 km.

Step 2: (b) 205 m²

In the number 205, the digit 2 and 5 are non-zero and significant. The middle digit, 0, is also significant because it is between non-zero numbers. So, there are 3 significant figures in 205 m².

Step 3: (c) 1.700 cm

In the number 1.700, the digits 1 and 7 are non-zero and significant. The following two zeroes are also significant because they occur after the decimal point. Therefore, there are 4 significant figures in 1.700 cm.

Step 4: (d) 350.00 K

In the number 350.00, the digits 3 and 5 are non-zero and significant. The two zeroes before the decimal point are significant because they are between non-zero numbers. The two zeroes after the decimal point are also significant. So, there are 5 significant figures in 350.00 K.

Step 5: (e) 307.080 g

In the number 307.080, the digits 3, 0, and 7 are significant because they are non-zero. The zeroes occurring between them are also significant. The last two zeroes after the decimal point carry information about the precision and are significant as well. Therefore, there are 6 significant figures in 307.080 g.

Step 6: (f) 1.3 x 10³ m/s

In the number 1.3 x 10³, there are two significant figures: 1 and 3. There is no need to count the power of 10 (3), as it only indicates the order of magnitude and doesn't impact significant figures directly. So, there are 2 significant figures in 1.3 x 10³ m/s.

Key Concepts

Scientific Notation

Scientific notation is a method of writing numbers to clearly express very large or very small quantities in an efficient manner. This is particularly useful when dealing with measurements in scientific calculations, as it helps to quickly convey scale and precision without unnecessary zeros. For example, a number like 3000 can be written as \(3 \times 10^{3}\) in scientific notation.

• The digit before the multiplication symbol is the coefficient, which contains the significant figures of the number.
• The exponent of 10 indicates how many places the decimal point has been moved.
• A positive exponent shows a large number, while a negative exponent indicates a small number.

Scientific notation ensures clarity in communication of magnitude and precision, aiding quick and accurate understanding among both scientists and students.

Measurement Precision

Precision in measurement reflects how exact or refined an assessment is. It indicates the level of detail in the measurement, often dictated by the instrument used and the method of measurement. Higher precision reduces the uncertainty but doesn't automatically imply accuracy.
Precision is often communicated through the number of significant figures a measurement includes. Each digit plays a role in expressing the precision.

• If more digits are used to express a number, the measurement is considered to be more precise. For instance, 3.774 km is more precise than 3.8 km.
• Significant zeros can also indicate higher precision, such as in 1.700 cm, where the additional zeros demonstrate refined measurement.

Consistency in measurement precision across a dataset adds credibility and enables proper comparisons.

Decimal Places

Decimal places are the number of digits to the right of the decimal point in a value. They play a crucial role in conveying the precision and exactness of measurements. The more decimal places, the greater the potential precision of a measurement.

• Each digit in the decimal places represents a fraction of a unit and carries precision in interpreting a measurement.
• In numbers like 1.700, the decimals not only extend precision but also highlight intentional accuracy, as opposed to simply rounding to a lesser decimal.
• When dealing with calculations, maintaining consistent decimal places helps preserve precision throughout the process, avoiding errors due to rounding too early.

Understanding decimal places and their effect is pivotal in scientific and engineering fields, where precise calculations are non-negotiable.