Chapter 2

What is the volume of a sample that has a mass of 20 \(\mathrm{g}\) and a density of 4 \(\mathrm{g} / \mathrm{mL}\) ?

Challenge \(A 147-\) giece of metal has a density of \(7.00 \mathrm{g} / \mathrm{mL} . \mathrm{A} 50-\mathrm{mL}\) graduated cylinder contains 20.0 \(\mathrm{mL}\) of water. What is the final volume after the metal is added to the graduated cylinder?

Define the Sl units for length, mass, time, and temperature.

Describe how adding the prefix mega- to a unit affects the quantity being described.

Compare a base unit and a derived unit, and list the derived units used for density and volume.

Define the relationships among the mass, volume, and density of a material.

Apply Why does oil float on water?

Calculate Samples \(\mathrm{A}, \mathrm{B}\) , and \(C\) have masses of \(80 \mathrm{g}, 12 \mathrm{g},\) and \(33 \mathrm{g},\) and volumes of \(20 \mathrm{mL}, 4 \mathrm{cm}^{3},\) and 11 \(\mathrm{mL}\) , respectively. Which of the samples have the same density?

Design a concept map that shows the relationships among the following terms: volume, derived unit, mass, base unit, time, and length.

Express each number in scientific notation. a. 700\(\quad\) c. 4,500,000 \(\quad\) e. 0.0054\(\quad\) g. 0.000000076 b. 38,000\(\quad\) d. 685,000,000,000 \(\quad\) f. 0.00000687\(\quad\) h. 0.0000000008

Challenge Express each quantity in regular notation along with its appropriate unit. a. \(3.60 \times 10^{5} \mathrm{s} \quad\) b. \(5.4 \times 10^{-5} \mathrm{g} / \mathrm{cm}^{3}\) \(\quad\) c. \(5.060 \times 10^{3} \mathrm{km} \quad\) d. \(8.9 \times 10^{10} \mathrm{Hz}\)

Solve each problem, and express the answer in scientific notation. a. \(\left(5 \times 10^{-5}\right)+\left(2 \times 10^{-5}\right) \quad\) c. \(\left(9 \times 10^{2}\right)-\left(7 \times 10^{2}\right)\) b. \(\left(7 \times 10^{8}\right)-\left(4 \times 10^{8}\right) \quad\) d. \(\left(4 \times 10^{-12}\right)+\left(1 \times 10^{-12}\right)\)

Challenge Express each answer in scientific notation in the units indicated. a. \(\left(1.26 \times 10^{4} \mathrm{kg}\right)+\left(2.5 \times 10^{6} \mathrm{g}\right)\) in kg b. \((7.06 \mathrm{g})+\left(1.2 \times 10^{-4} \mathrm{kg}\right)\) in \(\mathrm{kg}\) c. \(\left(4.39 \times 10^{5} \mathrm{kg}\right)-\left(2.8 \times 10^{7} \mathrm{g}\right)\) in \(\mathrm{kg}\) d. \(\left(5.36 \times 10^{-1} \mathrm{kg}\right)-\left(7.40 \times 10^{-2} \mathrm{kg}\right)\) in \(\mathrm{g}\)

PRACTICE Problems Solve each problem, and express the answer in scientific notation. a. \(\left(4 \times 10^{2}\right) \times\left(1 \times 10^{8}\right) \quad\) c. \(\left(6 \times 10^{2}\right) \div\left(2 \times 10^{1}\right)\) b. \(\left(2 \times 10^{-4}\right) \times\left(3 \times 10^{2}\right) \quad\) d. \(\left(8 \times 10^{4}\right) \div\left(4 \times 10^{1}\right)\)

Challenge Calculate the areas and densities. Report the answers in the correct units. a. the area of a rectangle with sides measuring \(3 \times 10^{1} \mathrm{cm}\) and \(3 \times 10^{-2} \mathrm{cm}\) b. the area of a rectangle with sides measuring \(1 \times 10^{3} \mathrm{cm}\) and \(5 \times 10^{-1} \mathrm{cm}\) c. the density of a substance having a mass of \(9 \times 10^{5} \mathrm{g}\) and a volume of \(3 \times 10^{-1} \mathrm{cm}^{3}\) d. the density of a substance having a mass of \(4 \times 10^{-3} \mathrm{g}\) and a volume of \(2 \times 10^{-2} \mathrm{cm}^{3}\)

Write two conversion factors for each of the following. a. a 16\(\%\) (by mass) salt solution b. a density of 1.25 \(\mathrm{g} / \mathrm{mL}\) c. a speed of 25 \(\mathrm{m} / \mathrm{s}\)

Challenge What conversion factors are needed to convert: a. nanometers to meters? b. density given in \(\mathrm{g} / \mathrm{cm}^{3}\) to a value in \(\mathrm{kg} / \mathrm{m}^{3}\) ?

Challenge Write the conversion factors needed to determine the number of seconds in one year.

How many seconds are in 24 h?

Challenge Vinegar is 5\(\%\) acetic acid by mass and has a density of 1.02 \(\mathrm{g} / \mathrm{mL}\) . What mass of acetic acid, in grams, is present in 185 \(\mathrm{mL}\) of vinegar?

Describe how scientific notation makes it easier to work with very large or very small numbers.

Express the numbers 0.00087 and 54,200,000 in scientific notation.

Write the measured distance quantities \(3 \times 10^{-4} \mathrm{cm}\) and \(3 \times 10^{4} \mathrm{km}\) in regular notation.

Write a conversion factor relating cubic centimeters and milliliters.

Solve How many millimeters are there in \(2.5 \times 10^{2} \mathrm{km} ?\)

Explain how dimensional analysis is used to solve problems.

Apply Concepts A classmate converts 68 \(\mathrm{km}\) to meters and gets 0.068 \(\mathrm{m}\) as the answer. Explain why this answer is incorrect, and identify the likely source of the error.

Organize Create a flowchart that outlines when to use dimensional analysis and when to use scientific notation.

Determine the number of significant figures in each measurement. a. 508.0 \(\mathrm{L}\) \(\quad\) c. \(1.0200 \times 10^{5} \mathrm{kg}\) b. \(820,400.0 \mathrm{L}\) \(\quad\) d. \(807,000 \mathrm{kg}\)

Determine the number of significant figures in each measurement. a. 0.049450 \(\mathrm{s}\) \(\quad\) c. \(3.1587 \times 10^{-4} \mathrm{g}\) b. 0.000482 \(\mathrm{mL}\) \(\quad\) d. 0.0084 \(\mathrm{mL}\)

Challenge Write the numbers 10,100, and 1000 in scientific notation with two, three, and four significant figures, respectively.

Round each number to four significant figures. a. \(84,791 \mathrm{kg} \quad\) c. 256.75 \(\mathrm{cm}\) b. 38.5432 \(\mathrm{g} \quad\) d. 4.9356 \(\mathrm{m}\)

Challenge Round each number to four significant figures, and write the answer in scientific notation. a. 0.00054818 \(\mathrm{g} \quad\) c. 308,659,000 \(\mathrm{mm}\) b. 136,758 \(\mathrm{kg} \quad\) d. 2.0145 \(\mathrm{mL}\)

Add and subtract as indicated. Round off when necessary. a. \(43.2 \mathrm{cm}+51.0 \mathrm{cm}+48.7 \mathrm{cm}\) \(\quad\) b. \(258.3 \mathrm{kg}+257.11 \mathrm{kg}+253 \mathrm{kg}\)

Challenge Add and subtract as indicated. Round off when necessary. a. \(\left(4.32 \times 10^{3} \mathrm{cm}\right)-\left(1.6 \times 10^{6} \mathrm{mm}\right)\) \(\quad\) b. \(\left(2.12 \times 10^{7} \mathrm{mm}\right)+\left(1.8 \times 10^{3} \mathrm{cm}\right)\)

Perform the following calculations. Round the answers. a. 24 \(\mathrm{m} \times 3.26 \mathrm{m} \quad\) b. 120 \(\mathrm{m} \times 0.10 \mathrm{m}\) \(\quad\) c. 1.23 \(\mathrm{m} \times 2.0 \mathrm{m} \quad\) d. 53.0 \(\mathrm{m} \times 1.53 \mathrm{m}\)

Perform the following calculations. Round the answers. a. 4.84 \(\mathrm{m} \div 2.4 \mathrm{s} \quad\) b. 60.2 \(\mathrm{m} \div 20.1 \mathrm{s}\) \(\quad\) c. 102.4 \(\mathrm{m} \div 51.2 \mathrm{s} \quad\) d. 168 \(\mathrm{m} \div 58 \mathrm{s}\)

Perform the following calculations. Round the answers. Challenge \(\left(1.32 \times 10^{3} \mathrm{g}\right) \div\left(2.5 \times 10^{2} \mathrm{cm}^{3}\right)\)

State how a measured value is reported in terms of known and estimated digits.

Define accuracy and precision.

Identify the number of significant figures in each of these measurements of an object's length: \(76.48 \mathrm{cm}, 76.47 \mathrm{cm},\) and 76.59 \(\mathrm{cm} .\)

Apply Write an expression for the quantity \(506,000 \mathrm{cm}\) in which it is clear that all the zeros are significant.

Explain why graphing can be an important tool for analyzing data.

Infer What type of data must be plotted on a graph for the slope of the line to represent density?

Relate If a linear graph has a negative slope, what can you say about the dependent variable?

Why must a measurement include both a number and a unit?

Explain why standard units of measurement are particularly important to scientists.

What role do prefixes play in the metric system?

What role do prefixes play in the metric system?

SI Units What is the relationship between the SI unit for volume and the SI unit for length?