Chapter 1

The following densities are given at \(20^{\circ} \mathrm{C}\) : water, \(0.998 \mathrm{g} / \mathrm{cm}^{3} ;\) iron, \(7.86 \mathrm{g} / \mathrm{cm}^{3} ;\) aluminum, \(2.70 \mathrm{g} / \mathrm{cm}^{3}\). Arrange the following items in terms of increasing mass. (a) a rectangular bar of iron,$$81.5 \mathrm{cm} \times 2.1 \mathrm{cm} \times 1.6 \mathrm{cm}$$ (b) a sheet of aluminum foil,$$12.12 \mathrm{m} \times 3.62 \mathrm{m} \times 0.003 \mathrm{cm}$$ (c) 4.051 L of water

To determine the approximate mass of a small spherical shot of copper, the following experiment is performed. When 125 pieces of the shot are counted out and added to \(8.4 \mathrm{mL}\) of water in a graduated cylinder, the total volume becomes \(8.9 \mathrm{mL}\). The density of copper is \(8.92 \mathrm{g} / \mathrm{cm}^{3} .\) Determine the approximate mass of a single piece of shot, assuming that all of the pieces are of the same dimensions.

The density of aluminum is \(2.70 \mathrm{g} / \mathrm{cm}^{3} .\) A square piece of aluminum foil, \(22.86 \mathrm{cm}\) on a side is found to weigh 2.568 g. What is the thickness of the foil, in millimeters?

In normal blood, there are about \(5.4 \times 10^{9}\) red blood cells per milliliter. The volume of a red blood cell is about \(90.0 \times 10^{-12} \mathrm{cm}^{3},\) and its density is \(1.096 \mathrm{g} / \mathrm{mL}\) How many liters of whole blood would be needed to collect \(0.5 \mathrm{kg}\) of red blood cells?

A technique once used by geologists to measure the density of a mineral is to mix two dense liquids in such proportions that the mineral grains just float. When a sample of the mixture in which the mineral calcite just floats is put in a special density bottle, the weight is 15.4448 g. When empty, the bottle weighs 12.4631 g, and when filled with water, it weighs 13.5441 g. What is the density of the calcite sample? (All measurements were carried out at \(25^{\circ} \mathrm{C}\), and the density of water at \(25^{\circ} \mathrm{C}\) is \(0.9970 \mathrm{g} / \mathrm{mL}\) ). At the left, grains of the mineral calcite float on the surface of the liquid bromoform \((d=2.890 \mathrm{g} / \mathrm{mL})\) At the right, the grains sink to the bottom of liquid chloroform \((d=1.444 \mathrm{g} / \mathrm{mL}) .\) By mixing bromoform and chloroform in just the proportions required so that the grains barely float, the density of the calcite can be determined (Exercise 62).

In a class of 76 students, the results of a particular examination were \(7 \mathrm{A}^{\prime} \mathrm{s}, 22 \mathrm{B}^{\prime} \mathrm{s}, 37 \mathrm{C}^{\prime} \mathrm{s}, 8 \mathrm{D}^{\prime} \mathrm{s}, 2 \mathrm{F}^{\prime} \mathrm{s}\). What was the percent distribution of grades, that is, \(\%\) A's, \(\%\) B's, and so on?

A class of 84 students had a final grade distribution of 18\(\%\) A's, 25\(\%\) B's, 32\(\%\) C's, 13\(\%\) D's, 12\(\%\) F's. How many students received each grade?

A solution of sucrose in water is \(28.0 \%\) sucrose by mass and has a density of \(1.118 \mathrm{g} / \mathrm{mL} .\) What mass of sucrose, in grams, is contained in 3.50 L of this solution?

A solution containing \(12.0 \%\) sodium hydroxide by mass in water has a density of \(1.131 \mathrm{g} / \mathrm{mL}\). What volume of this solution, in liters, must be used in an application requiring \(2.25 \mathrm{kg}\) of sodium hydroxide?

According to the rules on significant figures, the product of the measured quantities \(99.9 \mathrm{m}\) and \(1.008 \mathrm{m}\) should be expressed to three significant figures-\(101 \mathrm{m}^{2} .\) Yet, in this case, it would be more appropriate to express the result to four significant figures-\(100.7 \mathrm{m}^{2} .\) Explain why.

For a solution containing \(6.38 \%\) para-diclorobenzene by mass in benzene, the density of the solution as a function of temperature ( \(t\) ) in the temperature range 15 to \(65^{\circ} \mathrm{C}\) is given by the equation \(d(\mathrm{g} / \mathrm{mL})=1.5794-1.836 \times 10^{-3}(t-15)\) At what temperature will the solution have a density of \(1.543 \mathrm{g} / \mathrm{mL} ?\)

A standard \(1.000 \mathrm{kg}\) mass is to be cut from a bar of steel having an equilateral triangular cross section with sides equal to 2.50 in. The density of the steel is \(7.70 \mathrm{g} / \mathrm{cm}^{3} .\) How many inches long must the section of bar be?

The volume of seawater on Earth is about \(330,000,000 \mathrm{mi}^{3} .\) If seawater is \(3.5 \%\) sodium chloride by mass and has a density of \(1.03 \mathrm{g} / \mathrm{mL}\), what is the approximate mass of sodium chloride, in tons, dissolved in the seawater on Earth ( 1 ton \(=\) 2000 lb)?

The diameter of metal wire is often referred to by its American wire-gauge number. A 16-gauge wire has a diameter of 0.05082 in. What length of wire, in meters, is found in a 1.00 lb spool of 16 -gauge copper wire? The density of copper is \(8.92 \mathrm{g} / \mathrm{cm}^{3}\).

Magnesium occurs in seawater to the extent of \(1.4 \mathrm{g}\) magnesium per kilogram of seawater. What volume of seawater, in cubic meters, would have to be processed to produce \(1.00 \times 10^{5}\) tons of magnesium \((1 \text { ton }=2000 \mathrm{lb}) ?\) Assume a density of \(1.025 \mathrm{g} / \mathrm{mL}\) for seawater.

A typical rate of deposit of dust ("dustfall") from unpolluted air was reported as 10 tons per square mile per month. (a) Express this dustfall in milligrams per square meter per hour. (b) If the dust has an average density of \(2 \mathrm{g} / \mathrm{cm}^{3}\), how long would it take to accumulate a layer of dust \(1 \mathrm{mm}\) thick?

In the United States, volume of irrigation water is usually expressed in acre- feet. One acre-foot is a volume of water sufficient to cover 1 acre of land to a depth of 1 ft \(\left(640 \text { acres }=1 \mathrm{mi}^{2} ; 1 \mathrm{mi}=5280 \mathrm{ft}\right)\) The principal lake in the California Water Project is Lake Oroville, whose water storage capacity is listed as \(3.54 \times 10^{6}\) acre-feet. Express the volume of Lake Oroville in (a) cubic feet; (b) cubic meters; (c) U.S. gallons.

A Fahrenheit and a Celsius thermometer are immersed in the same medium. At what Celsius temperature will the numerical reading on the Fahrenheit thermometer be (a) \(49^{\circ}\) less than that on the Celsius thermometer; (b) twice that on the Celsius thermometer; (c) one-eighth that on the Celsius thermometer; (d) \(300^{\circ}\) more than that on the Celsius thermometer?

A pycnometer (see Exercise 78 ) weighs 25.60 g empty and \(35.55 \mathrm{g}\) when filled with water at \(20^{\circ} \mathrm{C}\) The density of water at \(20^{\circ} \mathrm{C}\) is \(0.9982 \mathrm{g} / \mathrm{mL}\). When \(10.20 \mathrm{g}\) lead is placed in the pycnometer and the pycnometer is again filled with water at \(20^{\circ} \mathrm{C}\), the total mass is \(44.83 \mathrm{g}\). What is the density of the lead in grams per cubic centimeter?

The Greater Vancouver Regional District (GVRD) chlorinates the water supply of the region at the rate of 1 ppm, that is, 1 kilogram of chlorine per million kilograms of water. The chlorine is introduced in the form of sodium hypochlorite, which is \(47.62 \%\) chlorine. The population of the GVRD is 1.8 million persons. If each person uses 750 L of water per day, how many kilograms of sodium hypochlorite must be added to the water supply each week to produce the required chlorine level of 1 ppm?

A Boeing 767 due to fly from Montreal to Edmonton required refueling. Because the fuel gauge on the aircraft was not working, a mechanic used a dipstick to determine that 7682 L of fuel were left on the plane. The plane required \(22,300 \mathrm{kg}\) of fuel to make the trip. In order to determine the volume of fuel required, the pilot asked for the conversion factor needed to convert a volume of fuel to a mass of fuel. The mechanic gave the factor as \(1.77 .\) Assuming that this factor was in metric units (kg/L), the pilot calculated the volume to be added as 4916 L. This volume of fuel was added and the 767 subsequently ran out the fuel, but landed safely by gliding into Gimli Airport near Winnipeg. The error arose because the factor 1.77 was in units of pounds per liter. What volume of fuel should have been added?

The following equation can be used to relate the density of liquid water to Celsius temperature in the range from \(0^{\circ} \mathrm{C}\) to about \(20^{\circ} \mathrm{C}:\) $$d\left(\mathrm{g} / \mathrm{cm}^{3}\right)=\frac{0.99984+\left(1.6945 \times 10^{-2} t\right)-\left(7.987 \times 10^{-6} t^{2}\right)}{1+\left(1.6880 \times 10^{-2} t\right)}$$ (a) To four significant figures, determine the density of water at \(10^{\circ} \mathrm{C}\). (b) At what temperature does water have a density of \(0.99860 \mathrm{g} / \mathrm{cm}^{3} ?\) (c) In the following ways, show that the density passes through a maximum somewhere in the temperature range to which the equation applies. (i) by estimation (ii) by a graphical method (iii) by a method based on differential calculus

A tabulation of data lists the following equation for calculating the densities \((d)\) of solutions of naphthalene in benzene at \(30^{\circ} \mathrm{C}\) as a function of the mass percent of naphthalene. $$d\left(\mathrm{g} / \mathrm{cm}^{3}\right)=\frac{1}{1.153-1.82 \times 10^{-3}(\% \mathrm{N})+1.08 \times 10^{-6}(\% \mathrm{N})^{2}}$$ Use the equation above to calculate (a) the density of pure benzene at \(30^{\circ} \mathrm{C} ;\) (b) the density of pure naphthalene at \(30^{\circ} \mathrm{C} ;\) (c) the density of solution at \(30^{\circ} \mathrm{C}\) that is 1.15\% naphthalene; (d) the mass percent of naphthalene in a solution that has a density of \(0.952 \mathrm{g} / \mathrm{cm}^{3}\) at \(30^{\circ} \mathrm{C} .[\text { Hint: For }(\mathrm{d}),\) you need to use the quadratic formula. See Section A-3 of Appendix A.]

The total volume of ice in the Antarctic is about \(3.01 \times 10^{7} \mathrm{km}^{3} .\) If all the ice in the Antarctic were to melt completely, estimate the rise, \(h,\) in sea level that would result from the additional liquid water entering the oceans. The densities of ice and fresh water are \(0.92 \mathrm{g} / \mathrm{cm}^{3}\) and \(1.0 \mathrm{g} / \mathrm{cm}^{3},\) respectively. Assume that the oceans of the world cover an area, \(A,\) of about \(3.62 \times 10^{8} \mathrm{km}^{2}\) and that the increase in volume of the oceans can be calculated as \(A \times h\).

The filament in an incandescent light bulb is made from tungsten metal \(\left(d=19.3 \mathrm{g} / \mathrm{cm}^{3}\right)\) that has been drawn into a very thin wire. The diameter of the wire is difficult to measure directly, so it is sometimes estimated by measuring the mass of a fixed length of wire. If a \(0.200 \mathrm{m}\) length of tungsten wire weighs \(42.9 \mathrm{mg}\), then what is the diameter of the wire? Express your answer in millimeters.

Blood alcohol content (BAC) is sometimes reported in weight-volume percent and, when it is, a BAC of \(0.10 \%\) corresponds to \(0.10 \mathrm{g}\) ethyl alcohol per \(100 \mathrm{mL}\) of blood. In many jurisdictions, a person is considered legally intoxicated if his or her BAC is 0.10\%. Suppose that a 68 kg person has a total blood volume of 5.4 L and breaks down ethyl alcohol at a rate of 10.0 grams per hour. \(^{*}\) How many 145 mL glasses of wine, consumed over three hours, will produce a BAC of \(0.10 \%\) in this 68 kg person? Assume the wine has a density of \(1.01 \mathrm{g} / \mathrm{mL}\) and is \(11.5 \%\) ethyl alcohol by mass. (* The rate at which ethyl alcohol is broken down varies dramatically from person to person. The value given here for the rate is a realistic, but not necessarily accurate, value.)

In an attempt to determine any possible relationship between the year in which a U.S. penny was minted and its current mass (in grams), students weighed an assortment of pennies and obtained the following data. $$\begin{array}{lllllll}\hline 1968 & 1973 & 1977 & 1980 & 1982 & 1983 & 1985 \\\\\hline 3.11 & 3.14 & 3.13 & 3.12 & 3.12 & 2.51 & 2.54 \\\3.08 & 3.06 & 3.10 & 3.11 & 2.53 & 2.49 & 2.53 \\\3.09 & 3.07 & 3.06 & 3.08 & 2.54 & 2.47 & 2.53 \\\\\hline\end{array}$$ What valid conclusion(s) might they have drawn about the relationship between the masses of the pennies within a given year and from year to year?

In the third century \(\mathrm{BC}\), the Greek mathematician Archimedes is said to have discovered an important principle that is useful in density determinations. The story told is that King Hiero of Syracuse (in Sicily) asked Archimedes to verify that an ornate crown made for him by a goldsmith consisted of pure gold and not a gold-silver alloy. Archimedes had to do this, of course, without damaging the crown in any way. Describe how Archimedes did this, or if you don't know the rest of the story, rediscover Archimedes's principle and explain how it can be used to settle the question.

The canoe gliding gracefully along the water in the photograph is made of concrete, which has a density of about \(2.4 \mathrm{g} / \mathrm{cm}^{3}\). Explain why the canoe does not sink.

As mentioned on page \(13,\) the MCO was lost because of a mix-up in the units used to calculate the force needed to correct its trajectory. Ground-based computers generated the force correction file. On September \(29,1999,\) it was discovered that the forces reported by the ground-based computer for use in MCO navigation software were low by a factor of \(4.45 .\) The erroneous trajectory brought the MCO \(56 \mathrm{km}\) above the surface of Mars; the correct trajectory would have brought the MCO approximately \(250 \mathrm{km}\) above the surface. At \(250 \mathrm{km},\) the MCO would have successfully entered the desired elliptic orbit. The data contained in the force correction file were delivered in lb-sec instead of the required SI units of newton-sec for the MCO navigation software. The newton is the SI unit of force and is described in Appendix B. The British Engineering (gravitational) system uses a pound (lb) as a unit of force and \(\mathrm{ft} / \mathrm{s}^{2}\) as a unit of acceleration. In turn, the pound is defined as the pull of Earth on a unit of mass at a location where the acceleration due to gravity is \(32.174 \mathrm{ft} / \mathrm{s}^{2} .\) The unit of mass in this case is the slug, which is \(14.59 \mathrm{kg}\). Thus, BE unit of force \(=1\) pound \(=(\text { slug })\left(\mathrm{ft} / \mathrm{s}^{2}\right)\) Use this information to confirm that BE unit of force \(=4.45 \times\) SI unit of force 1 pound \(=4.45\) newton

In your own words, define or explain the following terms or symbols: (a) \(\mathrm{mL}\) (b) \(\%\) by mass (c) \(^{\circ}$$\text{C}\) (d) density (e) element.

Briefly describe each of the following ideas: (a) SI base units; (b) significant figures; (c) natural law; (d) exponential notation.

Explain the important distinctions between each pair of terms: (a) mass and weight; (b) intensive and extensive properties; (c) substance and mixture; (d) systematic and random errors; (e) hypothesis and theory.

The fact that the volume of a fixed amount of gas at a fixed temperature is inversely proportional to the gas pressure is an example of (a) a hypothesis; (b) a theory; (c) a paradigm; (d) the absolute truth; (e) a natural law.

A good example of a homogeneous mixture is (a) a cola drink in a tightly capped bottle (b) distilled water leaving a distillation apparatus (c) oxygen gas in a cylinder used in welding (d) the material produced in a kitchen blender

Compared with its mass on Earth, the mass of the same object on the moon should be (a) less; (b) more; (c) the same; (d) nearly the same, but somewhat less.

Of the following masses, two are expressed to the nearest milligram. The two are (a) \(32.7 \mathrm{g}\); (b) \(0.03271 \mathrm{kg} ;(\mathrm{c}) 32.7068 \mathrm{g} ;(\mathrm{d}) 32.707 \mathrm{g} ;(\mathrm{e}) 30.7 \mathrm{mg};\) (f) \(3 \times 10^{3} \mu g.\)

The highest temperature of the following group is (a) \(217 \mathrm{K} ;\) (b) \(273 \mathrm{K} ;\) (c) \(217^{\circ} \mathrm{F} ;\) (d) \(105^{\circ} \mathrm{C} ;\) (e) \(373 \mathrm{K}\).

Which of the following quantities has the greatest mass? (a) \(752 \mathrm{mL}\) of water at \(20^{\circ} \mathrm{C}\) (b) 1.05 L of ethanol at \(20^{\circ} \mathrm{C}(d=0.789 \mathrm{g} / \mathrm{mL})\) (c) \(750 \mathrm{g}\) of chloroform at \(20^{\circ} \mathrm{C}(d=1.483 \mathrm{g} / \mathrm{mL})\) (d) a cube of balsa wood \(\left(d=0.11 \mathrm{g} / \mathrm{cm}^{3}\right)\) that is \(19.20 \mathrm{cm}\) on edge

The density of water is \(0.9982 \mathrm{g} / \mathrm{cm}^{3}\) at \(20^{\circ} \mathrm{C}\). Express the density of water at \(20^{\circ} \mathrm{C}\) in the following units: (a) \(\mathrm{g} / \mathrm{L} ;\) (b) \(\mathrm{kg} / \mathrm{m}^{3} ;\) (c) \(\mathrm{kg} / \mathrm{km}^{3}\).

The reported value for the volume of a rectangular piece of cardboard with the dimensions \(36 \mathrm{cm} \times\) \(20.2 \mathrm{cm} \times 9 \mathrm{mm}\) should be \((\mathrm{a}) \quad 6.5 \times 10^{3} \mathrm{cm}^{3};\) (b) \(7 \times 10^{2} \mathrm{cm}^{3} ;\) (c) \(655 \mathrm{cm}^{3} ;\) (d) \(6.5 \times 10^{2} \mathrm{cm}^{3}\).

List the following in the order of increasing precision, indicating any quantities about which the precision is uncertain: (a) \(1400 \mathrm{km} ;\) (b) \(1516 \mathrm{kg} ;\) (c) \(0.00304 \mathrm{g};\) (d) \(125.34 \mathrm{cm} ;\) (e) \(2000 \mathrm{mg}\).

Without doing detailed calculations, explain which of the following objects contains the greatest mass of the element iron. (a) \(\mathrm{A} 1.00 \mathrm{kg}\) pile of pure iron filings. (b) A cube of wrought iron, \(5.0 \mathrm{cm}\) on edge. Wrought iron contains \(98.5 \%\) iron by mass and has a density of \(7.7 \mathrm{g} / \mathrm{cm}^{3}\). (c) A square sheet of stainless steel \(0.30 \mathrm{m}\) on edge and \(1.0 \mathrm{mm}\) thick. The stainless steel is an alloy (mixture) containing iron, together with \(18 \%\) chromium, \(8\%\) nickel, and 0.18\% carbon by mass. Its density is \(7.7 \mathrm{g} / \mathrm{cm}^{3}\). (d) \(10.0 \mathrm{L}\) of a solution characterized as follows: \(d=1.295 \mathrm{g} / \mathrm{mL} .\) This solution is \(70.0 \%\) water and \(30.0 \%\) of a compound of iron, by mass. The iron compound consists of \(34.4 \%\) iron by mass.

A lump of pure copper weighs \(25.305 \mathrm{g}\) in air and 22.486 g when submerged in water \((d=0.9982 \mathrm{g} / \mathrm{mL})\) at \(20.0^{\circ} \mathrm{C} .\) Suppose the copper is then rolled into a \(248 \mathrm{cm}^{2}\) foil of uniform thickness. What will this thickness be, in millimeters?

Water, a compound, is a substance. Is there any circumstance under which a sample of pure water can exist as a heterogeneous mixture? Explain.

Appendix E describes a useful study aid known as concept mapping. Using the method presented in Appendix \(E,\) construct a concept map illustrating the different concepts presented in Sections \(1-2,\) \(1-3,\) and \(1-4\).