## 2.5 Density and Temperature

### Learning Objectives

By the end of this section, you will be able to:

• Define density.
• Calculate density from experimental results.
• Use density as a conversion factor.
• Learn about the various temperature scales that are commonly used in chemistry.
• Convert units of temperature

### Density

We use the mass and volume of a substance to determine its density. Thus, the units of density are defined by the base units of mass and length.

$\large Density = \frac{mass}{volume}$

The density of a substance is the ratio of the mass of a sample of the substance to its volume. The SI unit for density is the kilogram per cubic meter (kg/m3). For many situations, however, this as an inconvenient unit, and we often use grams per cubic centimeter (g/cm3) for the densities of solids and liquids, and grams per liter (g/L) for gases. Common units for density include g/mL, g/cm3, g/L, or kg/L. Although there are exceptions, most liquids and solids have densities that range from about 0.7 g/mL (the density of gasoline) to 19 g/mL (the density of gold). The density of air is about 1.2 g/L. Table 1 shows the densities of some common substances.

Table 1. Densities of Common Substances
Solids Liquids Gases (at 25 °C and 1 atm)
ice (at 0 °C) 0.92 g/cm3 water 1.0 g/mL dry air 1.20 g/L
oak (wood) 0.60–0.90 g/cm3 ethanol 0.79 g/mL oxygen 1.31 g/L
iron 7.9 g/cm3 acetone 0.79 g/mL nitrogen 1.14 g/L
copper 9.0 g/cm3 glycerin 1.26 g/mL carbon dioxide 1.80 g/L
lead 11.3 g/cm3 olive oil 0.92 g/mL helium 0.16 g/L
silver 10.5 g/cm3 gasoline 0.70–0.77 g/mL neon 0.83 g/L
gold 19.3 g/cm3 mercury 13.6 g/mL radon 9.1 g/L

While there are many ways to determine the density of an object, perhaps the most straightforward method involves separately finding the mass and volume of the object, and then dividing the mass of the sample by its volume. In the following example, the mass is found directly by weighing, but the volume is found indirectly through length measurements.

### Example 1: Calculation of Density

Gold—in bricks, bars, and coins—has been a form of currency for centuries. In order to swindle people into paying for a brick of gold without actually investing in a brick of gold, people have considered filling the centers of hollow gold bricks with lead to fool buyers into thinking that the entire brick is gold. It does not work: Lead is a dense substance, but its density is not as great as that of gold, 19.3 g/cm3. What is the density of lead if a cube of lead has an edge length of 2.00 cm and a mass of 90.7 g?

1. To three decimal places, what is the volume of a cube (cm3) with an edge length of 0.843 cm?
2. If the cube in part 1 is copper and has a mass of 5.34 g, what is the density of copper to two decimal places?

To learn more about the relationship between mass, volume, and density, use this PhET Density Simulator to explore the density of different materials, like wood, ice, brick, and aluminum.

### Example 2: Using Displacement of Water to Determine Density

This PhET simulation illustrates another way to determine density, using displacement of water. Determine the density of the red and yellow blocks.

Remove all of the blocks from the water and add the green block to the tank of water, placing it approximately in the middle of the tank. Determine the density of the green block.

Because of how it is defined, density can act as a conversion factor for switching between units of mass and volume. For example, suppose you have a sample of aluminum that has a volume of 7.88 cm3. How can you determine what mass of aluminum you have without measuring it? You can use the volume to calculate it. If you multiply the given volume by the known density (2.7 g/cm3), the volume units will cancel and leave you with mass units, telling you the mass of the sample:

$\large 7.88\cancel{{ cm^{3}}}\times\frac{2.7{\text{ g}}}{1\cancel{cm^{3}}}=21\text{ g of Al}$

where we have limited our answer to two significant figures.

### Example 3:  Using Density as a Conversion factor

What is the mass of 44.6 mL of mercury? Mercury has a density of 13.6 g/mL.

What is the mass of 25.0 cm3 of iron? Density of iron can be found in Table 1.

Density can also be used as a conversion factor to convert mass to volume—but care must be taken. We have already demonstrated that the number that goes with density normally goes in the numerator when density is written as a fraction. Take the density of gold, for example:

$\large \text{density of Au} =19.3\text{ g/mL}=\frac{19.3{\text{ g}}}{1\text{ mL}}$

That is, the density value tells us that we have 19.3 grams for every 1 milliliter of volume, and the 1 is an exact number. When we want to use density to convert from mass to volume, the numerator and denominator of density need to be switched. For example, if we want to know the volume of 45.9 g of gold, we would set up the conversion as follows:

$\large 45.9\cancel{ g}\times\frac{1{\text{ mL}}}{19.3\cancel{ g}}=2.38\text{ mL of Au}$

Note how the mass units cancel, leaving the volume unit, which is what we’re looking for.

### Example 4: Using Density as a Conversion factor

A cork stopper from a bottle of wine has a mass of 3.78 g. If the density of cork is 0.22 g/mL, what is the volume (in mL) of the cork?

What is the volume (in mL) of 3.78 g of glycerin? Density of glycerin can be found in Table 1.

Care must be used with density as a conversion factor. Make sure the mass units are the same, or the volume units are the same, before using density to convert to a different unit. Often, the unit of the given quantity must be first converted to the appropriate unit before applying density as a conversion factor (see Example 7).

### Example 5: Using Density as a Conversion factor

Acetone has a density of 0.79 g/mL, what is the volume (in L) of the 25.0 g of acetone?  Hint: 1 L = 103 mL

What is the mass (in grams) of 0.050 L of olive oil? Density of olive oil can be found in Table 1. (1 L = 103 mL)

### Measurement of Temperature (temperature scales)

There are other units in chemistry that are important, and we will cover others in the course of the entire book. One of the fundamental quantities in science is temperature. Temperature is a measure of the average amount of energy of motion, or kinetic energy, a system contains. Temperatures are expressed using scales that use units called degrees, and there are several temperature scales in use. In the United States, the commonly used temperature scale is the Fahrenheit scale (symbolized by °F and spoken as “degrees Fahrenheit”). On this scale, the freezing point of liquid water (the temperature at which liquid water turns to solid ice) is 32 °F, and the boiling point of water (the temperature at which liquid water turns to steam) is 212 °F.

Science also uses other scales to express temperature. The Celsius scale (symbolized by °C and spoken as “degrees Celsius”) is a temperature scale where 0 °C is the freezing point of water and 100 °C is the boiling point of water; the scale is divided into 100 divisions between these two landmarks and extended higher and lower. By comparing the Fahrenheit and Celsius scales, a conversion between the two scales can be determined:

$\large ^{o}C =(^{o}F-32)\times\frac{5}{9}$                    $\large ^{o}F =(^{o}C\times\frac{9}{5})+ 32$

Using these formulas, we can convert from one temperature scale to another. The number 32 in the formulas is exact and does not count in significant figure determination.

### Example 6: Celsius/Fahrenheit temperature Conversions

1. The average internal temperature of a human is 98.6 °F. What is this temperature in °C?

2. Room temperature is typically considered to be 25.0 °C. What is this temperature in °F?

1. The average temperature during the month of January at the summit of Mount Everest is -36 °C. What is this temperature in °F?

2. The warmest month for Rochester, New York is July with an average high temperature of 83.2 °F. What is this temperature in °C?

The fundamental unit of temperature (another fundamental unit of science, bringing us to four) in SI is the Kelvin (K). The Kelvin temperature scale (note that the name of the scale capitalizes the word Kelvin, but the unit itself is lowercase) uses degrees that are the same size as the Celsius degree, but the numerical scale is shifted up by 273.15 units. That is, the conversion between the Kelvin and Celsius scales is as follows:

$\large K =\ ^{o}C + 273.15$                    $\large \ ^{o}C =K - 273.15$

For most purposes, it is acceptable to use 273 instead of 273.15. Note that the Kelvin scale does not use the word degrees; a temperature of 295 K is spoken of as “two hundred ninety-five kelvins” and not “two hundred ninety-five degrees Kelvin.”

The reason that the Kelvin scale is defined this way is because there exists a minimum possible temperature called absolute zero. The Kelvin temperature scale is set so that 0 K is absolute zero, and temperature is counted upward from there. Normal room temperature is about 295 K, as seen in the following example.

### Example 7: Celsius/Kelvin Temperature conversions

In Denver, Colorado which has an elevation of approximately 1 mile, water boils at about 95 °C.  What is this temperature in Kelvin?

Figure 1. compares the three temperature scales. Note that science uses the Celsius and Kelvin scales almost exclusively; virtually no practicing chemist expresses laboratory-measured temperatures with the Fahrenheit scale. (In fact, the United States is one of the few countries in the world that still uses the Fahrenheit scale on a daily basis. The other two countries are Liberia and Myanmar [formerly Burma].

Figure 1. Fahrenheit, Celsius, and Kelvin Temperatures

### Food and Drink App: Cooking Temperatures

Because degrees Fahrenheit is the common temperature scale in the United States, kitchen appliances, such as ovens, are calibrated in that scale. A cool oven may be only 150°F, while a cake may be baked at 350°F and a chicken roasted at 400°F. The broil setting on many ovens is 500°F, which is typically the highest temperature setting on a household oven.

People who live at high altitudes, typically 2,000 ft above sea level or higher, are sometimes urged to use slightly different cooking instructions on some products, such as cakes and bread, because water boils at a lower temperature the higher in altitude you go, meaning that foods cook slower. For example, in Cleveland water typically boils at 212°F (100°C), but in Denver, the Mile-High City, water boils at about 200°F (93.3°C), which can significantly lengthen cooking times. Good cooks need to be aware of this.

At the other end is pressure cooking. A pressure cooker is a closed vessel that allows steam to build up additional pressure, which increases the temperature at which water boils. A good pressure cooker can get to temperatures as high as 252°F (122°C); at these temperatures, food cooks much faster than it normally would. Great care must be used with pressure cookers because of the high pressure and high temperature. (When a pressure cooker is used to sterilize medical instruments, it is called an autoclave.)

Other countries use the Celsius scale for everyday purposes. Therefore, oven dials in their kitchens are marked in degrees Celsius. It can be confusing for US cooks to use ovens abroad—a 425°F oven in the United States is equivalent to a 220°C oven in other countries. These days, many oven thermometers are marked with both temperature scales.

### Key Takeaways

• Density relates a substance’s mass and volume.
• Density is a derived unit and can be used as a conversion factor to calculate volume from a given mass or mass from a given volume.
• Chemistry uses the Celsius and Kelvin scales to express temperatures.
• A temperature on the Kelvin scale is the Celsius temperature plus 273.15.
• The minimum possible temperature is absolute zero and is assigned 0 K on the Kelvin scale.

### Exercise

Calculating Density from Experimental Results

1. The length, height, and width of an unknown block of metal was found to be 4.53 cm by 13.62 cm by 6.00 cm.  The mass of the block was determined to be 3.330 × 103 g.  Determine the density of the block and identify the metal using Table 1.
2. An unknown metal is weighed in a 150 mL beaker, giving a total mass of 125.476 g.  The mass of the empty beaker was 61.439 g. The metal was then placed into a graduated cylinder with an initial volume of 26.5 mL.  The volume was displaced to give a final volume of 29.8 mL.  Determine the density of the unknown metal and identify the metal using Table 1.

Using Density as a Conversion Factor

1. A sample of iron has a volume of 48.2 cm3. What is its mass (in grams)?
2. The volume of an Olympic-sized swimming pool is 2.50 × 109 cm3. If the pool is filled with alcohol (d = 0.789 g/cm3), what mass (in grams) of alcohol is in the pool?
3. Neon, a Noble gas, is commonly used in signs. What is the volume in liters of 222 g of neon if neon has a density of 0.900 g/L?
4. What is the volume in cubic centimeters of 100.0 g of lead if lead has a density of 11.34 g/cm3?
5. A sample of air has a volume of 1,015 mL. What is its mass (in grams)?
6. The volume of hydrogen used by the Hindenburg, the German airship that exploded in New Jersey in 1937, was 2.000 × 108 L. If hydrogen gas has a density of 0.0899 g/L, what mass in kilograms of hydrogen was used by the airship?
7. The gas tank of a Reliant Robin automobile holds 7.1 gallons according to the owner’s manual. If the density of gasoline is 0.8206 g/mL, determine the mass of the fuel in a full tank in pounds.
8. A typical engagement ring has 0.77 cm3 of gold. What mass (in ounces) of gold is present?
9. The density of orange juice is 1.05 g/mL, what is the volume (in pints) of 9.91 × 105 mg of juice?

Converting Units of Temperature

1. Convert 255°F to degrees Celsius
2. Convert−255°F to degrees Celsius
3. Convert 50.0°C to degrees Fahrenheit
4. Convert −50.0°C to degrees Fahrenheit
5. Convert 0 K to degrees Celsius. What is the significance of the temperature in degrees Celsius?
6. The hottest temperature ever recorded on the surface of the earth was 136°F in Libya in 1922. What is the temperature in degrees Celsius and in kelvins?
7. The coldest temperature ever recorded on the surface of the earth was −128.6°F in Vostok, Antarctica, in 1983. What is the temperature in degrees Celsius and in kelvins?