Question

What is the number of significant figures in each of the following measured quantities? (a) \(601 \mathrm{~kg}\), (b) \(0.054 \mathrm{~s}\), (c) \(6.3050\) \(\mathrm{cm}\), (d) \(0.0105 \mathrm{~L}\), (e) \(7.0500 \times 10^{-3} \mathrm{~m}^{3}\), (f) \(400 \mathrm{~g}\).

Short answer

The number of significant figures in each given quantity are: (a) 3, (b) 2, (c) 5, (d) 2, (e) 4, and (f) 1.

Step-by-step solution

(a) 601 kg

Initially, all the non-zero numbers are significant, i.e., \(6\) and \(1\). Further, the zero number lies between two significant figures, so it is also significant. Therefore, there are \(3\) significant figures in \(601 \mathrm{~kg}\).

(b) 0.054 s

Leading zeroes are not significant, so we do not count the two zeroes before \(54\). The remaining digits, \(5\) and \(4\), are both significant. So, there are \(2\) significant figures in \(0.054 \mathrm{~s}\).

(c) 6.3050 cm

All non-zero numbers are significant, so \(6, 3\) and \(5\) are all significant. The trailing zeroes after the decimal point are also significant in this case. So, there are \(5\) significant figures in \(6.3050 \mathrm{~cm}\).

(d) 0.0105 L

The three leading zeroes are not significant. The remaining digits, \(1\) and \(5\), are significant. Therefore, there are \(2\) significant figures in \(0.0105 \mathrm{~L}\).

(e) 7.0500 × 10^{-3} m^3

In scientific notation, all the non-zero numbers in the coefficient are significant. Here, \(7\) is significant. Additionally, all trailing zeroes after a decimal are significant, so both the zeroes after \(7\) are also significant. Therefore, there are \(4\) significant figures in \(7.0500 \times 10^{-3} \mathrm{~m}^{3}\).

(f) 400 g

The non-zero number \(4\) is significant. However, the trailing zeroes do not have a clear indication of uncertainty in this case. Hence, we cannot assume that they are significant. So, there is only \(1\) significant figure in \(400 \mathrm{~g}\).

Key concepts

Scientific Notation

Scientific notation is a method used to handle very large or very small numbers by simplifying them into two components: a coefficient and a power of ten. This is particularly useful in chemistry as it aids in the clear communication of measurements where precision matters.

For example, the number 7.0500 x 10^-3 is in scientific notation. It's broken down into a coefficient (7.0500) and an exponent (-3), which corresponds to the power of ten. Importantly, when we consider significant figures, only the digits in the coefficient are examined. Leading and trailing zeros within the coefficient matter here because they indicate precision. Therefore, every non-zero digit and zeros that are between non-zero digits or at the end of the decimal part, are significant. This can be seen in our given exercise, as the number 7.0500 has four significant figures given its specific form in scientific notation.

Measurement in Chemistry

Measurement in chemistry is foundational for experimental science and involves quantifying substances or phenomena in a reliable way. It uses units like kilograms for mass, seconds for time, liters for volume, etc. Precision in measurement is vital to ensure repeatability and accuracy of scientific work.

When we express these measurements, we use significant figures as a way to indicate the precision of the measured quantities. For instance, in the exercise 601 kg indicates three significant figures, implying that the mass has been measured to the nearest kilogram. Each digit conveys a certain level of confidence in its accuracy, with zeros sometimes playing a critical role depending on their placement within the number. Overall, careful consideration of significant figures can avoid overstatements of precision and potential miscommunications.

Accuracy and Precision

In the context of chemistry, accuracy refers to how close a measurement is to the true value, while precision speaks to the repeatability or consistency of the measurements. Accuracy and precision are both critical, yet distinct aspects of data quality and they are not interchangeable.

For example, if you're weighing a substance and the scale reports 400 g, with only the '4' being treated as significant, this is indicative of a low-precision measurement — it tells us the weight is roughly 400 grams but could be significantly more or less since the zeros do not confirm the measurement beyond the hundreds place. On the other hand, if significant figures were reported as 400.0 g, the implication would be a much more precise measurement indicating confidence up to the ones place. In sum, significant figures are a crucial expression of both precision and potentially, indirectly, of accuracy in a given measurement in chemistry.