Question

Round off the following quantities to the indicated number of significant figures. (a) \(7.4855 \mathrm{~g}\) (three significant figures) (b) \(298.693 \mathrm{~cm}\) (five significant figures) (c) \(11.698 \mathrm{lb}\) (one significant figure) (d) 12.05 oz (three significant figures)

Short answer

Question: Round off the following quantities to the specified number of significant figures: (a) 7.4855 g to three significant figures (b) 298.693 cm to five significant figures (c) 11.698 lb to one significant figure (d) 12.05 oz to three significant figures Answer: (a) 7.49 g (b) 298.69 cm (c) 10 lb (d) 12.1 oz

Step-by-step solution

1. Identify the Significant Figures

For each quantity, determine the significant figures and which digit should be rounded. The following numbers are underlined to indicate the digit that should be rounded: (a) \(7.4855\mathrm{~g}\) (three significant figures) - \(7.48\underline{5}\mathrm{~g}\) (b) \(298.693\mathrm{~cm}\) (five significant figures) - \(298.69\underline{3}\mathrm{~cm}\) (c) \(11.698 \mathrm{lb}\) (one significant figure) - \(\underline{1}1.698 \mathrm{lb}\) (d) \(12.05\mathrm{~oz}\) (three significant figures) - \(12.0\underline{5}\mathrm{~oz}\)

2. Round off Each Quantity

Apply the rounding rules to the identified digits. If the chosen digit is 5 or above, round up; otherwise, round down. (a) \(7.48\underline{5}\mathrm{~g}\) → \(7.49\mathrm{~g}\) for three significant figures. (b) \(298.69\underline{3}\mathrm{~cm}\) → \(298.69\mathrm{~cm}\) for five significant figures. (c) \(\underline{1}1.698\mathrm{~lb}\) → \(10\mathrm{~lb}\) for one significant figure. (d) \(12.0\underline{5}\mathrm{~oz}\) → \(12.1\mathrm{~oz}\) for three significant figures. In summary, the rounded off quantities are: (a) \(7.49\mathrm{~g}\) (three significant figures) (b) \(298.69\mathrm{~cm}\) (five significant figures) (c) \(10\mathrm{~lb}\) (one significant figure) (d) \(12.1\mathrm{~oz}\) (three significant figures)

Key concepts

Rounding Off Significant Figures

When it comes to rounding off significant figures, the objective is to convey precision relative to the accuracy of the measurement. Imagine you're baking a cake. If the recipe asks for approximately 3.2 cups of flour, it's more precise than just saying 3 cups, because that extra 0.2 can make a difference in the outcome.

Here's how you go about it: begin by identifying how many significant figures are in the number, then locate the figure that needs to be rounded. If this digit is followed by a number that is 5 or higher, round up. If it's 4 or lower, you keep it as is or round down. For instance, in the example given, to round 7.4855 g to three significant figures, focus on the third significant figure (the 8), and look at the next number (5). Since it's 5 or higher, bump the 8 up to 9, yielding 7.49 g.

Remember, the goal is precision, but it's also about knowing when additional digits don't add to the value of the measurement—you don't measure the weight of an elephant with a bathroom scale!

Measurement in Chemistry

How Do We Measure?

In measurement in chemistry, the units and precision matter. Like getting the right temperature on a thermometer to gauge a fever accurately, chemists must use the correct units and significant figures to precisely represent their findings. The choice of unit depends on what's being measured—grams for mass, liters for volume, and meters for distance, for example.

Chemists also use lab equipment calibrated to provide measurements that reflect the amount of significant figures needed. A balance might report mass to the nearest thousandth of a gram, while a graduated cylinder might only be accurate to the nearest milliliter. Working with these measurements involves understanding the limitations of the tools and the data they provide. For the case of 298.693 cm rounded to five significant figures, as in example (b), the reported value of 298.69 acknowledges the precision of the tool used for measuring.

Scientific Notation

Think of scientific notation as a shortcut for writing very large or very small numbers. It's like when you abbreviate words to save time. In science, a lengthy number can be cumbersome, so to streamline things, it's written as a product of a number between 1 and 10 and a power of 10. This makes comparison and calculations with these big or small numbers way easier.

Here's a simple example: the distance from Earth to the Sun is about 149,600,000 km. In scientific notation, this distance becomes approximately 1.496 x 10⁸ km. Couldn't be shorter! Similarly, for tiny measurements like the mass of a hydrogen atom (about 0.00000000000000000000000167 grams), instead of counting all those zeroes, represent it as 1.67 x 10^-24 grams. Whether you're talking astronomical distances or atomic scales, scientific notation keeps your numbers neat and manageable.