Problem 77 - Chemistry A Molecular Approach

Question

How many significant figures are in each number? MISSED THIS? Read Section 1.7; Watch KCV \(1.6,\) IWE 1.5 a. \(0.000312 \mathrm{~m}\) b. \(312,000 \mathrm{~s}\) c. \(3.12 \times 10^{5} \mathrm{~km}\) d. 13,127 s e. 2000

Step-by-Step Solution

Answer

a. 3 significant figures, b. 3 significant figures (assuming the zeros are not measured), c. 3 significant figures, d. 5 significant figures, e. 1 significant figure (assuming the zeros are not measured).

Step 1: Identifying Significant Figures in a. 0.000312 m

To determine the number of significant figures, we count all the nonzero digits and any zero that are between nonzero digits or are at the end of the number after a decimal point. Here, we have '312', which are all nonzero digits. The leading zeros do not count, so we have 3 significant figures.

Step 2: Identifying Significant Figures in b. 312,000 s

The number 312,000 can be tricky because it depends on whether the trailing zeros are considered significant. If the value was measured and the zeros are significant, there would be 6 significant figures. However, without a decimal point or additional context, we generally assume these are not significant, so it would be 3 significant figures.

Step 3: Identifying Significant Figures in c. 3.12 x 10^5 km

Scientific notation makes it easier to identify significant figures since all the numbers before the 'x 10' are significant. Here, '3.12' has two non-zero digits and one digit after the decimal, all of which are significant. Therefore, there are 3 significant figures in this number.

Step 4: Identifying Significant Figures in d. 13,127 s

All non-zero digits are significant. Since this number has no leading or trailing zeros, all of its digits count. Therefore, there are 5 significant figures in this number.

Step 5: Identifying Significant Figures in e. 2000

Trailing zeros in a whole number without a decimal point are not usually considered significant unless specified. Therefore, unless additional context is given, this number has 1 significant figure.

Key Concepts

Scientific Notation

Scientific notation is a system used to express very large or very small numbers in a compact form. It allows scientists and students to easily handle and perform calculations with numbers that would otherwise be cumbersome to write out in full. For example, the mass of the Earth is about 5,972,000,000,000,000,000,000,000 kilograms, which can be neatly written as 5.972 x 10²⁴ kg in scientific notation.

In scientific notation, a number is usually expressed as a product of two parts: a coefficient (a number greater than or equal to 1 and less than 10) and a power of 10. This is particularly useful in chemistry and physics, where quantities can span an enormous range of magnitudes.

Chemical Measurement Precision

Measurement precision in chemistry refers to the reliability and consistency of measurements and how closely they agree with each other. Precision is vital in chemical measurements to ensure that experiments and results are repeatable and accurate. Precision is often confused with accuracy, which is how close a measurement is to the true value, but the two are distinct concepts.

For example, if you weigh a chemical substance three times and get 2.04 g, 2.05 g, and 2.03 g, your measurements are precise because they are very close to one another. However, if the true weight is 3.00 g, your measurements are not accurate. Understanding the precision of measuring tools, along with significant figures, helps convey the reliability of the reported measurements.

Identifying Significant Figures

Identifying significant figures is essential in science because it allows for uniformity in reporting measurements and ensures that the communicated values reflect the precision of those measurements. In our exercise solutions, we encounter different scenarios for identifying significant figures.

• For a number such as 0.000312 m, only the non-zero digits (312) are considered significant.
• In a number like 312,000 s, the significance of trailing zeros depends on measurement precision or context provided.
• Scientific notation, as in 3.12 x 10⁵ km, simplifies significant figures identification because only the coefficient's digits are significant.
• All non-zero digits are significant, which is straightforward in a number like 13,127 s.
• For whole numbers such as 2000 without a decimal point, trailing zeros are not considered significant unless otherwise indicated.

It is always important to default to the guideline that significant figures include all non-zero digits, any zeros between them, and trailing zeros that are to the right of the decimal point.