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Chapter 3

Chapter 3. The Metric System. Vanessa N. Prasad- Permaul CHM 1025 Valencia Community College. The Metric System. The English system was used primarily in the British Empire and wasn’t very standardized. The French organized a committee to devise a universal measuring system.

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Chapter 3

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  1. Chapter 3 The Metric System Vanessa N. Prasad-Permaul CHM 1025 Valencia Community College

  2. The Metric System • The English system was used primarily in the British Empire and wasn’t very standardized. • 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.

  3. 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.

  4. Metric System Basic Units

  5. Metric System Advantage • Another advantage of the metric system 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.

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

  7. Metric Prefixes, Continued • For example, the prefix kilo- increases a base unit by 1000: • 1 kilogram is 1000 grams. • The prefix milli- decreases a base unit by a factor of 1000: • There are 1000 millimeters in 1 meter.

  8. EXAMPLE 3.1Metric Basic Units and Prefixes Give the symbol for each of the following metric units and state the quantity measured by each unit: (a) gigameter (b) kilogram (c) centiliter (d) microsecond Solution We compose the symbol for each unit by combining the prefix symbol and the basic unit symbol. If we refer to Tables 3.1 and 3.2, we have the following: (a) Gm, length(b) kg, mass(c) cL, volume (d) s, time

  9. Practice Exercise Give the symbol for each of the following metric units and state the quantity measured by each unit: (a) nanosecond (b) microiliter (c) kilogram (d) millimeter Concept Exercise What is the basic unit of length, mass, and volume in the metric system? EXERCISE 3.1 Metric Basic Units and Prefixes

  10. Metric Symbols • The names of metric units are abbreviated using symbols. Use the prefix symbol followed by the symbol for the base unit, so: • Nanometer is abbreviated nm. • Microgram is abbreviated mg. • Deciliter is abbreviated dL. • Gigasecond is abbreviated Gs.

  11. Nanotechnology • Nanotechnology refers to devices and processes on the 1–100 nm scale. • For reference, a human hair is about 100,000 nm thick! • A DNA helix is a nanoscale substance, with a diameter of about 1 nm. • Nanoscale hollow tubes, called carbon nanotubes, have slippery inner surfaces that allow for the easy flow of fluids.

  12. Metric Equivalents • We can write unit equations for the conversion between different metric units. • The prefix kilo- means 1000 basic units, so 1 kilometer is 1000 meters. • The unit equation is 1 km = 1000 m. • Similarly, a millimeter is 1/1000 of a meter, so the unit equation is 1000 mm = 1 m.

  13. Metric Unit Factors • Since 1000 m = 1 km, we can write the following unit factors for converting between meters and kilometers: 1 km or 1000 m 1000 m 1 km • Since 1000 mm = 1 m, we can write the following unit factors: 1000 mm or 1 m 1 m 1000 mm

  14. Solution • We can refer to Table 3.2 as necessary. • The prefix mega- means 1,000,000 basic units; thus, 1 Mm = 1,000,000 m. • 1 Mm = 1  106 m • The prefix kilo- means 1000 basic units; thus, 1 kg = 1000 g. • 1 kg = 1  103 g • The prefix deci- means 0.1 of a basic unit, thus, 1 L = 10 dL. • 1 L = 1  101dL • The prefix nano- means 0.000 000 001 of a basic unit, thus; 1 s = 1,000,000,000 ns. • 1 s = 1  109 ns EXAMPLE 3.2Metric Unit Equations Complete the unit equation for each of the following exact metric equivalents: (a) 1 Mm = ? m (b) 1 kg = ? g (c) 1 L = ? dL (d) 1 s = ? ns

  15. Practice Exercise Complete the unit equation for each of the following exact metric equivalents: (a) 1 nm = ? m (b) 1 g = ? mg (c) 1 L = ? L (d) 1 s = ? Ms EXERCISE 3.2 Metric Unit Equations

  16. Solution • We start by writing the unit equation to generate the • two unit factors. • The prefix kilo- means 1000 basic units; thus, 1 km = 1000 m. The two unit factors are • (b) The prefix deci- means 0.1 basic unit; thus, 1 g = 10 dg. The two unit factors are EXAMPLE 3.3Metric Unit Factors Write two unit factors for each of the following metric relationships: (a) kilometers and meters (b)grams and decigrams

  17. Practice Exercise Write two unit factors for each of the following metric relationships: (a) liters and microliters (b)milliseconds and seconds EXERCISE 3.3 Metric Unit Factors

  18. Metric–Metric Conversions • We will use the unit analysis method we learned in Chapter 2 to do metric–metric conversion problems. • Remember, there are three steps: • Write down the unit asked for in the answer. • Write down the given value related to the answer. • Apply unit factor(s) to convert the given unit to the units desired in the answer.

  19. 1 g 325 mg x = 0.325 g 1000 mg 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.

  20. EXAMPLE 3.4Metric Unit Factors 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 and1000 mL 10 dL 1 L

  21. 1 L 1000 mL 125 dLx x = 12,500 mL 10 dL 1 L EXAMPLE 3.4Metric Unit Factors Continued… • 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.

  22. Practice Exercise Concept Exercise A dermatology patient is treated with ultraviolet light having a wavelength of 305 nm. What is the wavelength expressed in meters? In micrometers? Express the volume of a cube 1 cm on a side in milliliters. EXERCISE 3.4 Metric-Metric Conversions

  23. 1 Mg 1000 g 7.35 x 1022 kg × x = 5.98 x 1019 Mg 1000000 g 1 kg EXAMPLE EXERCISE 3.5Metric–Metric Conversion Another Example The mass of the Earth’s moon is 7.35 × 1022 kg. What is the mass expressed in megagrams, Mg? • We want Mg; we have 7.35 x 1022 kg. • Convert kilograms to grams, and then grams to megagrams.

  24. Practice Exercise Concept Exercise Light travels through the universe at a velocity of 3.00  1010 cm/s. How many megameters does light travel in one second? How many significant digits are in the following unit factor? 1 g/1000 mg EXERCISE 3.5 Metric–Metric Conversion

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

  26. 0.914 m 120 yd x = 110 m 1 yd Metric–English Conversion 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).

  27. 1 qt 946 mL 64.0 fl oz x x = 1,890 mL 32 fl oz 1 qt EXAMPLE EXERCISE 3.6Metric–English Conversion English–Metric Conversion 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 = 32.0 fl oz, and 1 qt = 946 mL.

  28. EXERCISE 3.6 Metric–English Conversion Practice Exercise A plastic bottle contains 4.00 gallons of distilled water. How many liters of distilled water are in the bottle (given that 1 gal = 4 qt)? Concept Exercise How many significant digits are in the following unit factor? 1 qt/946 mL

  29. EXAMPLE 3.7Metric–English Conversion We apply the unit factor 1 lb/16 oz to cancel ounces , and 454 g/1 lb to cancel pounds . The given value, 2.0 oz, limits the answer to two significant digits. Unit factor 1 has no effect as it is derived from an exact equivalent, and unit factor 2 has three significant digits.

  30. EXAMPLE 3.7Metric–English Conversion If a tennis ball weighs 2.0 oz, what is the mass of the tennis ball in grams? Unit Analysis Map

  31. Concept Exercise How many significant digits are in the following unit factor? 1 kg/2.20 lb EXERCISE 3.7 Metric–English Conversion Practice Exercise If a tennis ball has a diameter of 2.5 inches, what is the diameter in millimeters?

  32. 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

  33. 75 km 1 hr 1000 m x x = 21 m/s 1 km hr 3600 s Compound Unit Problem A motorcycle is traveling at 75 km/hour. 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.

  34. EXAMPLE 3.8Conversion of a Unit Ratio If a Mazda Miata is traveling at 95 km/h, what is the speed in meters per second (given that 1 km = 1000 m, and 1 h = 3600 s)? Unit Analysis Map

  35. EXAMPLE 3.8Conversion of a Unit Ratio Solution We apply the unit factor 1000 m/1 km to cancel kilometers , and 1 h/3600 s to cancel hours . The given value has two significant digits, so the answer is limited to two digits. Since each unit factor is derived from an exact equivalent, neither affects the number of significant digits in the answer.

  36. Practice Exercise Concept Exercise If a runner completes a 10K race in 32.50 minutes (min), what is the 10.0 km pace in miles per hour (given that 1 mi = 1.61 km)? Which speed is faster: 65 mi/h or 65 km/h? EXERCISE 3.8 Conversion of a Unit Ratio

  37. Volume by Calculation • The volume of an object is calculated by multiplying the length (l) by the width (w) by the thickness (t). volume = lxwxt • 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).

  38. Solution We can calculate the volume of the rectangular solid by multiplying length times width times thickness: lw t. 5.55 cm  3.75 cm  2.25 cm = 46.8 cm3 The answer is rounded off to three significant digits because each given value has three significant digits. EXAMPLE 3.9Volume Calculation for a Rectangular Solid If a stainless steel rectangular solid measures 5.55 cm long, 3.75 cm wide, and 2.25 cm thick, what is the volume in cubic centimeters?

  39. Practice Exercise If a rectangular brass solid measures 52.0 mm by 25.0 mm by 15.0 mm, what is the volume in cubic millimeters? Concept Exercise Express the volume of a cube 10 cm on a side in liters. EXERCISE 3.9 Volume Calculation for a Rectangular Solid

  40. Solution We can calculate the thickness of the foil by dividing the volume by length and width. Since the unit of volume is mm3, we obtain the thickness in mm by unit cancellation. The answer is rounded off to three significant digits because each given value has three significant digits. EXAMPLE 3.10:Thickness Calculation for a Rectangular Solid A sheet of aluminum foil measures 25.0 mm by 10.0 mm, and the volume is 3.75 mm3. What is the thickness of the foil in millimeters?

  41. EXERCISE 3.10: Thickness Calculation for a Rectangular Solid Practice Exercise A sheet of aluminum foil measures 35.0 cm by 25.0 cm, and the volume is 1.36 cm3. What is the thickness of the foil in centimeters? Aluminum foil :A thin sheet of aluminum foil. Concept Exercise Which of the following is the greatest thickness? 1 mm, 0.1 cm, or 0.001 m

  42. Cubic Volume and Liquid Volume • 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.

  43. Cubic and Liquid Volume Units • 1 liter is equal to 1000 cubic centimeters. • 10 cm x 10 cm x 10 cm = 1000 cm3 • 1000 cm3 = 1 L = 1000 mL. • Therefore, 1 cm3 = 1 mL.

  44. 1 in 1 in 1 in 498 cm3x x x = 30.4 in3 2.54 cm 2.54 cm 2.54 cm EXAMPLE 3.11Metric–English Volume Conversion 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.

  45. Concept Exercise Which of the following is the greater volume? 500 mL or 500 cm3 EXERCISE 3.11: Metric–English Volume Conversion Practice Exercise Given that an SUV has a 304 in.3 engine, express the engine volume in liters.

  46. 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.

  47. 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.

  48. 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.

  49. Solution We can calculate the displaced volume in milliliters by subtracting the initial volume from the final volume. 36.5 mL– 25.0 mL = 11.5 mL EXAMPLE 3.12:Volume by Displacement A quartz stone weighing 30.475 g is dropped into a graduated cylinder. If the water level increases from 25.0 mL to 36.5 mL, what is the volume of the quartz stone?

  50. Practice Exercise Concept Exercise Hydrogen peroxide decomposes to give oxygen gas, which displaces a volume of water into a beaker. If the water level in the beaker increases from 50.0 mL to 105.5 mL, what is the volume of oxygen gas? Which of the following has the greater volume? 1 mL or 1 cm3 EXERCISE 3.12: Volume by Displacement

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