Problem 76 - Introductory Chemistry Atoms First

Question

How many significant figures are there in the following measured values, and what is the uncertainty in each measurement? (a) \(0.001 \mathrm{~kg}\) (b) \(0.00010 \mathrm{~m}\) (c) \(102 \mathrm{~L}\) (d) \(2.600 \times 10^{-3} \mathrm{~m}\) (e) \(1.1 \times 10^{6} \mathrm{~km}\)

Step-by-Step Solution

Answer

(a) There is one significant figure in \(0.001 \mathrm{~kg}\), and the uncertainty is \(\pm 0.001 \mathrm{~kg}\). (b) There are two significant figures in \(0.00010 \mathrm{~m}\), and the uncertainty is \(\pm 0.00001 \mathrm{~m}\). (c) There are two significant figures in \(102 \mathrm{~L}\), and the uncertainty is \(\pm 1 \mathrm{~L}\). (d) There are three significant figures in \(2.600 \times 10^{-3} \mathrm{~m}\), and the uncertainty is \(\pm 0.001 \times 10^{-3} \mathrm{~m}\). (e) There are two significant figures in \(1.1 \times 10^{6} \mathrm{~km}\), and the uncertainty is \(\pm 0.1 \times 10^{6} \mathrm{~km}\).

Step 1: (a) Number of significant figures for \(0.001 \mathrm{~kg}\)

To find the significant figures, let's identify the non-zero digits and zeros according to the rules mentioned in the analysis. In this measurement, there is only one non-zero digit (1), so there is only one significant figure in \(0.001 \mathrm{~kg}\).

Step 2: (a) Uncertainty for \(0.001 \mathrm{~kg}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.001 \mathrm{~kg}\). So the uncertainty for this measurement is \(\pm 0.001 \mathrm{~kg}\).

Step 3: (b) Number of significant figures for \(0.00010 \mathrm{~m}\)

In this measurement, the significant figures include the 1 and the trailing zero after the decimal point (0). So there are two significant figures for \(0.00010 \mathrm{~m}\).

Step 4: (b) Uncertainty for \(0.00010 \mathrm{~m}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.00001 \mathrm{~m}\). So the uncertainty for this measurement is \(\pm 0.00001 \mathrm{~m}\).

Step 5: (c) Number of significant figures for \(102 \mathrm{~L}\)

In this measurement, both non-zero digits (1 and 2) are significant figures. There are no trailing zeros with a decimal point. So there are two significant figures for \(102 \mathrm{~L}\).

Step 6: (c) Uncertainty for \(102 \mathrm{~L}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(1 \mathrm{~L}\). So the uncertainty for this measurement is \(\pm 1 \mathrm{~L}\).

Step 7: (d) Number of significant figures for \(2.600 \times 10^{-3} \mathrm{~m}\)

In this measurement, all non-zero digits (2, 6) and the trailing zero after the decimal point (0) are significant figures. There are three significant figures for \(2.600 \times 10^{-3} \mathrm{~m}\).

Step 8: (d) Uncertainty for \(2.600 \times 10^{-3} \mathrm{~m}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.001 \times 10^{-3} \mathrm{~m}\). So the uncertainty for this measurement is \(\pm 0.001 \times 10^{-3} \mathrm{~m}\).

Step 9: (e) Number of significant figures for \(1.1 \times 10^{6} \mathrm{~km}\)

In this measurement, both non-zero digits (1 and 1) are significant figures. There are no zeros with a decimal point. So there are two significant figures for \(1.1 \times 10^{6} \mathrm{~km}\).

Step 10: (e) Uncertainty for \(1.1 \times 10^{6} \mathrm{~km}\)

The uncertainty in this case is associated with the least significant figure's value, which is \(0.1 \times 10^{6} \mathrm{~km}\). So the uncertainty for this measurement is \(\pm 0.1 \times 10^{6} \mathrm{~km}\).

Key Concepts

Measurement Uncertainty

Measurement uncertainty refers to the doubt about how accurate a measured value is. It indicates the margin of error, or how close the measured value might be to the true value. Any measurement always has some level of uncertainty since no instrument can measure with perfect precision. This uncertainty is often described using significant figures.

When stating a measurement, the uncertainty typically corresponds to the value of the last significant figure. For example, if you measure something as \(102 \pm 1 \mathrm{~L}\), this suggests that the true volume could be between 101 and 103 L. This ± value represents an estimate of how much the measurement could differ from the actual value, considering factors like the limitations of the measuring tool and the method of measurement.

Understanding measurement uncertainty is crucial as it helps scientists and engineers determine the reliability of their data, guiding better decision-making in practical applications.

Non-Zero Digits

Non-zero digits are all numbers from 1 to 9. They are always considered significant in a measured value because they provide information about the size or magnitude of the measurement. For instance, in the number 102, the digits 1 and 2 are non-zero, and they both count towards the significant figures.

These digits are straightforward in terms of their significance, unlike zeros, which can have different roles. Whether these non-zero digits appear at the beginning, middle, or end, they are always counted when determining the number of significant figures. Thus, the number 2.600 has four significant figures since it includes three non-zero digits (2 and 6) and a zero that follows a decimal point.

Trailing Zeros

Trailing zeros are zeros that appear after the last non-zero digit in a number. Their significance depends on the presence of a decimal point, which can sometimes make their role a bit tricky to understand.

• If a number has trailing zeros and a decimal point, those zeros are significant. For example, in \(2.600\), the zeros count as significant figures, indicating precision in the measurement.
• However, in numbers without a decimal point, trailing zeros are often not considered significant. For instance, \(1020\) has three significant figures: 1, 0, and 2, as the zero is simply a placeholder.

Trailing zeros are essential in expressing measurements to the correct precision, especially when using tools with a high degree of accuracy.

Decimal Point

The decimal point is a dot used to separate the whole part of a number from the fractional part. In terms of significant figures, a decimal point plays a critical role in determining which zeros in a number are significant.

Including a decimal point means that all trailing zeros in the number are significant. This can be seen in numbers like \(2.600\), where the decimal indicates that all four digits are significant. The decimal point emphasizes precision in the measurement, often highlighting how exact the measurement can be.

It's also the reason why \(0.00010\) has two significant figures: the 1 and the trailing zero after the decimal. The decimal ensures that the zeros leading up to the 1 are understood as non-significant, merely acting as placeholders to show value center around the decimal point.