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) \(350.00 \mathrm{~K}\), (e) \(307.080 \mathrm{~g}\), (f) \(1.3 \times 10^{3} \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The number of significant figures for each measured quantity are: (a) 3.774 km: \(4\) significant figures, (b) 205 m²: \(3\) significant figures, (c) 1.700 cm: \(4\) significant figures, (d) 350.00 K: \(5\) significant figures, (e) 307.080 g: \(6\) significant figures, (f) 1.3 × 10³ m/s: \(2\) significant figures.

Step 1: a) 3.774 km

The measured quantity consists of all non-zero digits, which are all significant. There are 4 significant figures in this measurement: 3, 7, 7, and 4.

Step 2: b) 205 m²

In this quantity, there are no decimal points, but we have a zero between two non-zero digits. The zero is significant in this case. Therefore, there are 3 significant figures: 2, 0, and 5.

Step 3: c) 1.700 cm

This quantity has non-zero digits and zeros to the right of the decimal point and after a non-zero digit. These zeros are significant. There are 4 significant figures: 1, 7, 0, and 0.

Step 4: d) 350.00 K

Here, we have non-zero digits and zeros both to the right and left of the decimal point. Since these zeros follow non-zero digits and there is a decimal point, they are considered significant. There are 5 significant figures: 3, 5, 0, 0, and 0.

Step 5: e) 307.080 g

In this measured quantity, we have non-zero digits and zeros between them, as well as zeros to the right of the decimal point. All these zeros are significant. This results in 6 significant figures: 3, 0, 7, 0, 8, and 0.

Step 6: f) 1.3 × 10³ m/s

In scientific notation, all the digits in the coefficient are considered significant. In this case, there are 2 significant figures in the coefficient: 1 and 3.

Key Concepts

Scientific Measurement

Understanding scientific measurement is fundamental to the precision and accuracy of scientific inquiry. Measurement is the process of quantifying physical quantities, such as length, mass, temperature, and many others. Scientific measurements are used not only for experimental findings but also for theoretical predictions and engineering applications.

These measurements are based on established units and scales like meters for distance, kilograms for mass, and seconds for time. Every measurement is accompanied by a level of uncertainty; this uncertainty can depend on the tool used for the measurement, such as rulers, scales, and thermometers, or the intrinsic variability of the quantity being measured.

Significant figures play a pivotal role in communicating the precision of these measurements. They are the digits in a measurement that carry meaningful contributions to its precision. This includes all non-zero digits, any zeros between them, and any trailing zeros when they follow a decimal point. However, leading zeros, or zeros that merely serve as placeholders are not considered significant. Therefore, the significant figures in a measurement reflect the precision of the measurement and the limitations of the measuring instrument used.

Measurement Precision

Precision refers to the closeness of two or more measurements to each other. It is an indication of the 'repeatability' or 'consistency' of a measurement. The concept is different from accuracy, which is the closeness of a measurement to the true or accepted value.

For example, if you were to use a ruler to measure the length of a table several times, and each time the measurement reads 1.540 meters, the consistency of this result suggests high precision. However, if the actual length of the table is 1.550 meters, then your measurements although precise, are not accurate.

An understanding of precision is crucial when making scientific measurements since it affects the reliability of the data. Whether you are recording the mass of a chemical or the time taken for a reaction, being consistent in using equipment and method is key to precision. While tools with finer scales (i.e., more decimal places) provide potential for greater precision, the significant figures in a reported measurement will ultimately convey the true level of precision achieved.

Significant Digits

Significant digits, also known as significant figures, are a representation of the precision of a measurement. Identifying significant digits in a measurement is key to maintaining the integrity of scientific data.

Here are the basic rules:

• Non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Leading zeros, to the left of the first non-zero digits, are not significant as they only indicate the position of the decimal point.
• Trailing zeros in a decimal portion are significant, as they indicate precision.
• In scientific notation, all digits in the coefficient are significant, and the exponential part simply specifies the decimal's placement.

For example, in the measurement of 1.700 cm, the zeroes are significant because they are trailing a non-zero number after the decimal point, indicating that the measurement was precise to the thousandth of a centimeter. Similarly, in the scientific notation 1.3 × 10³ m/s, we understand that the measurement was made to a precision of tenths; hence, there are two significant figures. Mastering the use of significant digits is an essential skill in reporting scientific measurements accurately, preserving the meaning and precision of the original measurement.