One further physical quantity that we will take up in this chapter is density, which relates the mass of an object to its volume. Density is usually expressed in units of grams per cubic centimeter (g/cm3) for solids and grams per milliliter (g/mL) for liq- uids. Thus, if we know the density of a substance, we know both the mass of a given volume and the volume of a given mass. The densities of some common materials are listed in Table 1.11.
Density = Mass 1g2 Volume 1mL or cm32
Although most substances contract when cooled and expand when heated, wa- ter behaves differently. Water contracts when cooled from 100 °C to 3.98 °C, but below this temperature it begins to expand again. The density of liquid water is at its maximum of 1.0000 g/mL at 3.98 °C but decreases to 0.999 87 g/mL at 0 °C.
When freezing occurs, the density drops still further to a value of 0.917 g/cm3 for ice at 0 °C. Since a less dense substance will f loat on top of a more dense f luid, ice and any other substance with a density less than that of water will f loat in water.
Conversely, any substance with a density greater than that of water will sink in water.
Knowing the density of a liquid is useful because it is often easier to measure a liquid’s volume rather than its mass. Suppose, for example, that you need 1.50 g of
Density The physical property that relates the mass of an object to its volume; mass per unit volume.
▲ The Galileo thermometer contains several weighted bulbs which rise or fall as the density of the liquid changes with temperature.
TABLE 1.11 Densities of Some Common Materials at 25 °C
Substance Density* Substance Density*
Gases Solids
Helium 0.000 194 Ice (0 °C)
Gold Human fat Cork Table sugar Balsa wood Earth
0.917 19.3 0.94 0.22–0.26 1.59 0.12 5.54
Air 0.001 185
Liquids
Water (3.98 °C) 1.0000
Urine 1.003–1.030
Blood plasma 1.027
*Densities are in g/cm3 for solids and g/mL for liquids and gases.
ethanol. Rather than use a dropper to weigh out exactly the right amount, it would be much easier to look up the density of ethanol (0.7893 g/mL at 20 °C) and measure the correct volume (1.90 mL) with a syringe or graduated cylinder. Thus, density acts as a conversion factor between mass (g) and volume (mL).
1.50 g ethanol * 1 mL ethanol
0.7893 g ethanol = 1.90 mL ethanol
For many purposes, ranging from winemaking to medicine, it is more con- venient to use specific gravity than density. The specific gravity (sp gr) of a sub- stance (usually a liquid) is simply the density of the substance divided by the density of water at the same temperature. Because all units cancel, specific grav- ity is unitless:
Specific gravity = Density of substance 1g>mL2
Density of water at the same temperature 1g>mL2
At typical temperatures, the density of water is very close to 1 g/mL. Thus, the specific gravity of a substance is numerically equal to its density and is used in the same way.
The specific gravity of a liquid can be measured using an instrument called a hydrometer, which consists of a weighted bulb on the end of a calibrated glass tube, as shown in Figure 1.11. The depth to which the hydrometer sinks when placed in a fluid indicates the fluid’s specific gravity: the lower the bulb sinks, the lower the specific gravity of the fluid.
In medicine, a hydrometer called a urinometer is used to indicate the amount of solids dissolved in urine. Although the specific gravity of normal urine is about 1.003–1.030, conditions such as diabetes mellitus or a high fever cause an abnor- mally high urine specific gravity, indicating either excessive elimination of solids or decreased elimination of water. Abnormally low specific gravity is found in individuals using diuretics—drugs that increase water elimination.
Specific gravity The density of a substance divided by the density of water at the same temperature.
1.020 1.030
▲ Figure 1.11
A hydrometer for measuring specific gravity.
The instrument has a weighted bulb at the end of a calibrated glass tube.
The depth to which the hydrometer sinks in a liquid indicates the liquid’s specific gravity.
Worked Example 1.16 Density: Mass-to-Volume Conversion
What volume of isopropyl alcohol (rubbing alcohol) would you use if you needed 25.0 g? The density of isopropyl alcohol is 0.7855 g/mL at 20 °C.
ANALYSIS The known information is the mass of isopropyl alcohol needed (25.0 g). The density (0.7855 g/mL) acts as a conversion factor between mass and the unknown volume of isopropyl alcohol.
BALLPARK ESTIMATE Because 1 mL of isopropyl alcohol contains only 0.7885 g of the alcohol, obtaining 1 g of alcohol would require almost 20% more than 1 mL, or about 1.2 mL. Therefore, a volume of about 25 * 1.2 mL = 30 mL is needed to obtain 25 g of alcohol.
SOLUTION
STEP 1: Identify known information.
STEP 2: Identify answer and units.
STEP 3: Identify conversion factors. Starting with the mass of isopropyl alcohol (in g), the corresponding vol- ume (in mL) can be calculated using density (g/mL) as the conversion factor.
STEP 4: Solve. Starting with the known information, set up the equation with conversion factors so that unwanted units cancel.
Mass of rubbing alcohol = 25.0 g
Density of rubbing alcohol = 0.7855 g/mL Volume of rubbing alcohol = ?? mL Density = g/mLS1/density = mL/g
25.0 g alcohol * 1 mL alcohol
0.7855 g alcohol = 31.8 mL alcohol BALLPARK CHECK Our estimate was 30 mL.
S E C T I O N 1 . 1 4 Density and Specific Gravity 35
A Measurement Example:
Obesity and Body Fat
According to the U.S. Centers for Disease Control and Prevention, the U.S. population is suffering from a fat epi- demic. Over the last 25 years, the percentage of adults 20 years or older identified as obese increased from 15%
in the late 1970s to nearly 33% in 2008. Even children and adolescents are gaining too much weight: The number of overweight children in all age groups increased by nearly a factor of 3, with the biggest increase seen among teenagers (from 5% to 18.1%). Of particular concern is the fact that 80% of children who were overweight as teenagers were identified as obese at age 25. Obesity increases the risk for many adverse health conditions, including type 2 diabetes and heart disease.
How do we define obesity, however, and how is it mea- sured? Obesity is defined by reference to body mass index (BMI), which is equal to a person’s mass in kilograms divided by the square of his or her height in meters. BMI can also be calculated by dividing a person’s weight in pounds by the square of her or his height in inches multiplied by 703. For instance, someone 5 ft 7 in. (67 inches; 1.70 m) tall weighing 147 lb (66.7 kg) has a BMI of 23:
BMI = weight 1kg2
3height 1m242, or weight 1lb2
3height 1in.242 * 703 A BMI of 25 or above is considered overweight, and a BMI of 30 or above is obese. By these standards, approximately 61% of the U.S. population is overweight. Health professionals are concerned by the rapid rise in obesity in the United States because of the link between BMI and health problems. Many reports have documented the correlation between health and BMI, including a recent study on more than 1 million adults. The
lowest death risk from any cause, including cancer and heart disease, is associated with a BMI between 22 and 24. Risk in- creases steadily as BMI increases, more than doubling for a BMI above 29.
An individual’s percentage of body fat is most easily measured by the skinfold-thickness method. The skin at sev- eral locations on the arm, shoulder, and waist is pinched, and the thickness of the fat layer beneath the skin is mea- sured with calipers. Comparing the measured results to those in a standard table gives an estimation of percent- age body fat. As an alternative to skinfold measurement, a more accurate assessment of body fat can be made by un- derwater immersion. The person’s underwater body weight is less than her or his weight on land because water gives the body buoyancy. The higher the percentage of body fat, the more buoyant the person and the greater the difference between land weight and underwater body weight. Check- ing the observed buoyancy on a standard table then gives an estimation of body fat.
See Chemistry in Action Problems 1.100 and 1.101 at the end of the chapter.
CHEMISTRY IN ACTION
▲A person’s percentage body fat can be estimated by measur- ing the thickness of the fat layer under the skin.
21 20 19 18 17 16 15 14 13
110
Weight (lb)
Body Mass Index (numbers in boxes)
Height
5’0”
5’2”
5’4”
5’6”
5’8”
5’10”
6’0”
6’2”
6’4”
22 21 20 19 17 17 16 15 14
115
23 22 21 19 18 17 16 15 15
120
24 23 21 20 19 18 17 16 15
125
25 24 22 21 20 19 18 17 16
130
26 25 23 22 21 19 18 17 16
135
27 26 24 23 21 20 19 18 17
140
28 27 25 23 22 21 20 19 18
145
29 27 26 24 23 22 20 19 18
150
30 28 27 25 24 22 21 20 19
155
31 29 27 26 24 23 22 21 19
160
32 30 28 27 25 24 22 21 20
165
33 31 29 27 26 24 23 22 21
170
34 32 30 28 27 25 24 22 21
175
35 33 31 29 27 26 24 23 22
180
36 34 32 30 28 27 25 24 23
185
37 35 33 31 29 27 26 24 23
190
38 36 33 31 30 28 26 25 24
195
39 37 34 32 30 29 27 26 24
200
▲The specific gravity of urine, measured by a urinometer, is used to diagnose conditions such as diabetes.
PROBLEM 1.26
A sample of pumice, a porous volcanic rock, weighs 17.4 grams and has a volume of 27.3 cm3. If this sample is placed in a container of water, will it sink or will it float?
Explain.
PROBLEM 1.27
Chloroform, once used as an anesthetic agent, has a density of 1.474 g/mL. What volume would you use if you needed 12.37 g?
PROBLEM 1.28
The sulfuric acid solution in an automobile battery typically has a specific gravity of about 1.27. Is battery acid more dense or less dense than pure water?