Chapter 2 The Metric System Lecture Presentation Chapter 2 The Metric System John Singer Jackson College

Basic Units and Symbols The English system was used primarily in the British Empire. The French organized a committee to devise a universal measuring system. After about 10 years, the committee designed and agreed on the metric system. The metric system offers simplicity with a single base unit for each measurement.

Metric System Basic Units

Original Metric Unit Definitions A meter was defined as 1/10,000,000 of the distance from the North Pole to the equator. A kilogram (1000 grams) was equal to the mass of a cube of water measuring 0.1 m on each side. A liter was set equal to the volume of one kilogram of water at 4 °C.

Metric Prefixes The following table lists the common prefixes used in the metric system:

Metric Symbols The names of metric units are abbreviated using symbols. Use the prefix symbol followed by the symbol for the base unit: Kilometer is abbreviated km. Milligram is abbreviated mg. Microliter is abbreviated mL. Nanosecond is abbreviated ns.

Critical Thinking: The International System of Units (SI) An advantage of the metric system (i.e., International System of Units, SI) is that it is a decimal system. It uses prefixes to enlarge or reduce the basic units. For example: A kilometer is 1000 meters. A millimeter is 1/1000 of a meter.

Metric Conversion Factors A unit equation relates two quantities that are equal. For example: 1 kilometer = 1000 meters 1 km = 1000 m Also, we can write: 1 centimeter = 1/100 of a meter or 100 cm = 1 m

For the unit equation 1 m = 100 cm, we can write two unit factors: A unit conversion factor, or unit factor, is a ratio of two equivalent quantities. For the unit equation 1 m = 100 cm, we can write two unit factors: 1 m or 100 cm 100 cm 1 m

Metric–Metric Conversions An effective method for solving problems in science is the unit analysis method. It is also often called dimensional analysis or the factor-label method.

Applying the Unit Analysis Method Step 1: Write down the unit asked for in the answer. Step 2: Write down the given value related to the answer. Step 3: Apply unit factor(s) to convert the unit in the given value to the unit in the answer.

Metric–Metric Conversion Problem What is the mass in grams of a 325-mg aspirin tablet? Step 1: We want grams. Step 2: We write down the given: 325 mg. Step 3: We apply a unit factor (1000 mg = 1 g) and round to three significant figures. 325 mg × = 0.325 g 1000 mg 1 g

Two Metric–Metric Conversions A hospital has 125 deciliters of blood plasma. What is the volume in milliliters? Step 1: We want the answer in mL. Step 2: We have 125 dL. Step 3: We need to first convert dL to L and then convert L to mL: 1 L and 1000 mL 10 dL 1 L

Two Metric–Metric Conversions In working a problem where there is not a direct conversion from the given unit to the desired unit, first convert to the metric unit with no prefix. Apply both unit factors, and round the answer to three significant digits. Notice that both dL and L units cancel, leaving us with units of mL. 125 dL × = 12,500 mL × 10 dL 1 L 1000 mL

Another Example The mass of the Earth’s moon is 7.35 × 1022 kg. What is the mass expressed in nanograms, ng? We want ng; we have 7.35 × 1022 kg. Convert kilograms to grams, and then grams to nanograms. 7.35 × 1022 kg × = 7.35 × 1031 ng × 1 kg 1000 g 1 × 106 ng 1 g

Metric–English Conversions The English system is still very common in the United States. We often have to convert between English and metric units.

Metric–English Conversions

Metric–English Conversions The length of an American football field, including the end zones, is 120 yards. What is the length in meters? Convert 120 yd to meters (given that 1 yd = 0.914 m). 120 yd × = 110 m 1 yd 0.914 m

Metric–English Conversions A half-gallon carton contains 64.0 fl oz of milk. How many milliliters of milk are in a carton? We want mL; we have 64.0 fl oz. Use 1 qt = 32 fl oz, and 1 qt = 946 mL. 64.0 fl oz × = 1,890 mL × 32 fl oz 1 qt 946 mL

Another English–Metric Problem A marathon is 26.2 miles. What is the distance in kilometers (1 km = 0.62 mi)? Step 1: We want km. Step 2: We write down the given: 26.2 mi. Step 3: We apply a unit factor (1 km = 0.62 mi) and round to three significant figures.

Some measurements have a ratio of units. Compound Units Some measurements have a ratio of units. For example, the speed limit on many highways is 55 miles per hour. How would you convert this to meters per second? Convert one unit at a time using unit factors. First, miles meters Next, hours seconds

We have km/h; we want m/s. Use 1 km = 1000 m and 1 h = 3600 s. Compound Unit Problem A motorcycle is traveling at 105 km/h. What is the speed in meters per second? We have km/h; we want m/s. Use 1 km = 1000 m and 1 h = 3600 s. = 29.2 m/s × 1 km 1000 m 1 h 3600 s 105 km h

Critical Thinking: World Trade Center When discussing measurements, it is critical that we use the proper units. The World Trade Center footprint was 150 feet square, not 150 square feet. NASA engineers mixed metric and English units when designing the Mars Climate Orbiter. The engineers used kilometers rather than miles. The spacecraft approached too close to the Martian surface and burned up in the atmosphere.

The Percent Concept A percent (%) expresses the amount of a single quantity compared to an entire sample. A percent is a ratio of parts per 100 parts. The formula for calculating percent is shown below:

Calculating Percentages Bronze is an alloy of copper and tin. If a sample of bronze contains 79.2 g of copper and 10.8 g of tin, what is the percent of copper in bronze? = 88.0% × (79.2 + 10.8) g 79.2 g 100%

Percent Unit Factors A percent can be expressed as parts per 100 parts. 25% can be expressed as 25/100, and 10% can be expressed as 10/100. We can use a percent expressed as a ratio as a unit factor. A rock is 4.70% iron, so

Percent Unit Factor Calculation The Earth and Moon have a similar composition; each contains 4.70% iron. What is the mass of iron in a lunar sample that weighs 92 g? Step 1: We want g iron. Step 2: We write down the given: 92 g sample. Step 3: We apply a unit factor (4.70 g iron = 100 g sample) and round to three significant figures.

Volume by Calculation The volume of an object is calculated by multiplying the length (l) times the width (w) times the thickness (t). volume = l × w × t All three measurements must be in the same units. If an object measures 3 cm by 2 cm by 1 cm, the volume is 6 cm3 (cm3 is cubic centimeters).

Volumes of Solids, Liquids, and Gases The liter (L) is the basic unit of volume in the metric system. One liter is defined as the volume occupied by a cube that is 10 cm on each side.

Volumes of Solids, Liquids, and Gases 1 liter is equal to 1000 cubic centimeters. 10 cm × 10 cm × 10 cm = 1000 cm3 1000 cm3 = 1 L = 1000 mL Therefore, 1 cm3 = 1 mL

Cubic-Liquid Volume Conversion An automobile engine displaces a volume of 498 cm3 in each cylinder. What is the displacement of a cylinder in cubic inches (in3)? We want in3; we have 498 cm3. Use 1 in = 2.54 cm three times. = 30.4 in3 × 1 in 2.54 cm 498 cm3 ×

Volume by Displacement If a solid has an irregular shape, its volume cannot be determined by measuring its dimensions. You can determine its volume indirectly by measuring the amount of water it displaces. This technique is called volume by displacement. Volume by displacement can also be used to determine the volume of a gas, as long as the gas is not soluble in water.

Solid Volume by Displacement You want to measure the volume of an irregularly shaped piece of jade. Partially fill a volumetric flask with water and measure the volume of the water. Add the jade, and measure the difference in volume. The volume of the jade is 10.5 mL.

Gas Volume by Displacement You want to measure the volume of gas given off in a chemical reaction. The gas produced displaces the water in the flask into the beaker. The volume of water displaced is equal to the volume of gas.

The Density Concept mass = density volume The density of an object is a measure of its concentration of mass. Density is defined as the mass of an object divided by the volume of the object. = density volume mass

Density Density is expressed in different units. It is usually grams per milliliter (g/mL) for liquids, grams per cubic centimeter (g/cm3) for solids, and grams per liter (g/L) for gases.

Densities of Common Substances

Estimating Density We can estimate the density of a substance by comparing it to another object. A solid object will float on top of a liquid with a higher density. Object S1 has a density less than that of water, but larger than that of L1. Object S2 has a density less than that of L2, but larger than that of water.

Calculating Density What is the density of a platinum nugget that has a mass of 224.50 g and a volume of 10.0 cm3 ? Recall, density is mass/volume. = 22.5 g/cm3 10.0 cm3 224.50 g

Density as a Unit Factor We can use density as a unit factor for conversions between mass and volume. An automobile battery contains 1275 mL of acid. If the density of battery acid is 1.84 g/mL, how many grams of acid are in an automobile battery? – We have 1275 mL; we want grams: 1275 mL × = 2350 g mL 1.84 g

There are three temperature scales: Temperature is a measure of the average kinetic energy of individual particles in motion. There are three temperature scales: Fahrenheit Celsius Kelvin Kelvin is the absolute temperature scale.

Temperature Scales On the Fahrenheit scale, water freezes at 32 °F and boils at 212 °F. On the Celsius scale, water freezes at 0 °C and boils at 100 °C. These are the reference points for the Celsius scale. Water freezes at 273 K and boils at 373 K on the Kelvin scale.

Temperature Conversions The relationship between the Fahrenheit and Celsius temperature scales is linear. Therefore this relationship can be represented in the standard format of an equation for a straight line. This is the equation for converting °C to °F.

Temperature Scales Solving the previous equation for °C leads to the following: This is the equation for converting °F to °C.

Temperature Scales To convert from °C to K, add 273. °C + 273 = K

Fahrenheit–Celsius Conversions Body temperature is 98.6 °F. What is body temperature in degrees Celsius? In Kelvin? K = °C + 273 = 37.0 °C + 273 = 310 K

The Heat Concept Heat is a measure of total energy. Temperature measures the average energy of particles in a system. Heat is often expressed in terms of joules (J) or calories (cal).

Heat Versus Temperature Although both beakers below have the same temperature (100 °C), the beaker on the right has twice the amount of heat because it has twice the amount of water.

Specific Heat The specific heat of a substance is the amount of heat required to raise the temperature of one gram of substance one degree Celsius. It is expressed with units of calories per gram per degree Celsius. The larger the specific heat, the more heat is required to raise the temperature of the substance.

Chapter Summary The basic units in the metric system are grams for mass, liters for volume, and meters for distance. The base units are modified using prefixes to reduce or enlarge the base units by factors of 10. We can use unit factors to convert between metric units. We can convert between metric and English units using unit factors.

Chapter Summary A unit equation is a statement of two equivalent quantities. A unit factor is a ratio of two equivalent quantities. Unit factors can be used to convert measurements between different units. A percent is the ratio of parts per 100 parts.

Chapter Summary Volume is defined as length × width × thickness. Volume can also be determined by displacement of water. Density is mass divided by volume.

Chapter Summary, Continued Temperature is a measure of the average energy of the particles in a sample. Heat is a measure of the total energy of a substance. Specific heat is a measure of how much heat is required to raise the temperature of a substance.