Lecture 3 Conversion Factors and Density

Prefixes

• In metric and SI systems, prefixes modify unit sizes by factors of 10.
  • Example:
    • $1 \text{ kilometer (km)} = 1000 \text{ meters}$
    • $1 \text{ millimeter (mm)} = 0.001 \text{ meters}$

Prefixes and Equalities

• Prefixes can be replaced with their numerical values to show their relationship to a unit.
  • Examples:
    • $1 \text{ kilometer} = 1000 \text{ meters}$
    • $1 \text{ kiloliter} = 1000 \text{ liters}$
    • $1 \text{ kilogram} = 1000 \text{ grams}$

Metric and SI Prefixes That Increase the Size of the Unit

Metric and SI Prefixes That Decrease the Size of the Unit

• Note:
  • In medicine, the abbreviation "mc" is used for micro to avoid misreading the symbol μ; thus, $1 \text{ μg}$ is written as $1 \text{ mcg}$.

Daily Values for Selected Nutrients

• The U.S. Food and Drug Administration uses metric prefixes to express daily nutrient requirements.

Sample Exercise 13

• Fill in the blanks with the correct prefix:
  • A. 1000 \text{ meters} = 1 \text{ ___ m}
  • B. \text{ ___ g} = 0.000001 \text{ g}
  • C. 0.1 \text{ L} = 1 \text{ ___ L}

Solution 13

• A. The prefix for 1000 is kilo; $1000 \text{ meters} = 1 \text{ kilometer}$.
• B. The prefix for $0.000001$ is micro; $1 \text{ μg} = 0.000001 \text{ g}$.
• C. The prefix for $0.1$ is deci; $0.1 \text{ Liters} = 1 \text{ deciLiter}$.

Measuring Length: Equalities

• An equality shows the relationship between two units that measure the same quantity.
• Each of the following equalities describes the same length in a different unit.

Measuring Length: Equalities (Continued)

• The metric length of 1 meter is the same length as 10 decimeters, 100 centimeters, and 1000 millimeters.

Measuring Volume: Equalities

• Volumes of 1 Liter or smaller are common in the health sciences.
• When a liter is divided into 10 equal portions, each portion is called a deciliter (dL).

Measuring Volume

• The cubic centimeter ($\text{cm}^3$ or cc) is the volume of a cube with the dimensions $1 \text{ centimeter} \times 1 \text{ centimeter} \times 1 \text{ centimeter}$.
• A cube measuring 10 centimeters on each side has a volume of $1 \text{ Liter}$; a cube measuring 1 centimeter on each side has a volume of $1 \text{ cubic centimeter}$, or $1 \text{ milliLiter}$.

Measuring Volume (Continued)

• A cubic centimeter (cc) has the same volume as a milliliter, and the units are often used interchangeably.

Measuring Mass: Equalities

• When you visit the doctor for a physical examination, he or she records your mass in kilograms (kg) and laboratory results in micrograms ($\text{μg}$ or mcg).

Sample Exercise 14

• Identify the larger unit in each of the following:
  • A. millimeter or centimeter
  • B. kilogram or centigram
  • C. deciLiter or μLiter
  • D. m g or μicrograms

Solution 14

• A. Centimeter is larger. A millimeter is $0.001 \text{ meter}$, smaller than a centimeter, $0.01 \text{ meter}$.
• B. Kilogram is larger. A kilogram is $1000 \text{ grams}$, larger than a centigram, $0.01 \text{ grams}$.
• C. Deciliter is larger. A deciliter is $0.1 \text{ Liter}$, larger than a microliter, $0.000 001 \text{ Liter}$.
• D. Milligram is larger. A milligram is $0.001 \text{ grams}$, larger than a microgram, $0.000 001 \text{ grams}$.

Sample Exercise 15

• Complete each of the following equalities:
  • A. 1 \text{ kilogram} = \text{ ___ grams}
    1. 10 grams
    2. 100 grams
    3. 1000 grams
  • B. 1 \text{ centiLiter} = \text{ ___ Liters}
    1. 0.001 Liters
    2. 0.01 Liters
    3. 100 Liters

Solution 15

• A. $1 \text{ kilogram} = 1000 \text{ grams}$
• B. $1 \text{ centiLiter} = 0.01 \text{ Liter}$

Equalities

• Equalities:
  • use two different units to describe the same quantity
  • can be between units of the metric system, or between U.S. units, or between metric and U.S. units
    • $1 \text{ meter} = 1000 \text{ millimeter}$
    • $1 \text{ lb} = 16 \text{ oz}$
    • $2.20 \text{ lb} = 1 \text{ kilogram}$

Equalities on Food Labels

• The contents of many packaged foods:
  • are listed in both metric and U.S. units.
  • indicate the same amount of a substance in two different units.
• In the United States, the contents of many packaged foods are listed in both U.S. and metric units.

Equalities and Significant Figures

• The numbers in any equality between two metric units or between two U.S. system units are definitions.
• Because numbers in a definition are exact, they are not used to determine significant figures (SFs).
  • $1 \text{ gram} = 1000 \text{ milligrams}$
  • $1 \text{ foot} = 12 \text{ inches}$
  • $1 \text{ minutes} = 60 \text{ seconds}$

Equalities and Significant Figures (Continued)

• When an equality consists of a metric unit and a U.S. unit, one of the numbers in the equality is obtained by measurement and counts toward the significant figures (SFs) in the answer.
  • $454 \text{ g} = 1 \text{ lb}$
  • $946 \text{ milliLiters} = 1 \text{ quart}$
  • $39.4 \text{ inches} = 1 \text{ meter}$
• Exception:
  • The equality $1 \text{ inch} = 2.54 \text{ centimeters}$ has been defined as an exact relationship, and therefore $2.54$ is an exact number.

Conversion Factors

• Any equality can be written as fractions called conversion factors.
• Be sure to include units when you write conversion factors.
• Conversion Factors for the Equality $1 \text{ h} = 60 \text{ min}$
  • These conversion factors are read as “60 minutes per hour” and “1 hour per 60 minutes.”
  • The per means “divide.”

Some Common Equalities

• *Used in medicine.

Writing Conversion Factors

• We can write metric conversion factors:
  • $\frac{100 \text{ cm}}{1 \text{ m}}$
  • $\frac{1 \text{ m}}{100 \text{ cm}}$
• $1 \text{m} = 100 \text{ cm}$
• Metric-U.S. system conversion factors:
  • $\frac{2.20 \text{ lb}}{1 \text{ kg}}$
  • $\frac{1 \text{ kg}}{2.20 \text{ lb}}$

Sample Exercise 16

• Write the equality and two conversion factors for each of the following pairs of units:
  • A. liters and milliliters
  • B. inches and meters
  • C. milligrams and micrograms

Solution 16

• (Solution not provided in transcript)

Conversion Factors Within a Problem

• An equality may also be stated within a problem that only applies to that problem.
  • The car travels at 85 kilometers/hour.
    • Equality: $85 \text{ kilometers} = 1 \text{ hour}$
    • Conversion Factors: $\frac{85 \text{ kilometers}}{1 \text{ hour}}$, $\frac{1 \text{ hour}}{85 \text{ kilometers}}$
    • Significant Figures: The $85 \text{ kilometer}$ is measured (two significant figures), and the $1 \text{ h}$ is exact.

Conversion Factors Within a Problem (Continued)

• The tablet contains $500 \text{ milligrams}$ of vitamin C.
  • Equality: $1 \text{ tablet} = 500 \text{ milligrams of vitamin C}$
  • Conversion Factors: $\frac{500 \text{ milligrams vitamin C}}{1 \text{ tablet}}$, $\frac{1 \text{ tablet}}{500 \text{ milligrams vitamin C}}$
  • Significant Figures: The $500 \text{ milligrams}$ is measured (one significant figure). The $1 \text{ tablet}$ is exact.

Conversion Factors from a Percentage

• A percent (%) is written as a conversion factor by choosing a unit and expressing the numerical relationship of the parts of this unit to 100 parts of the whole.
  • For example, a person has 18% body fat by mass.
    • Equality: $18 \text{ kilograms of body fat} = 100 \text{ kilograms of body mass}$
    • Conversion Factors: $\frac{18 \text{ kilograms of body fat}}{100 \text{ kilograms of body mass}}$, $\frac{100 \text{ kilograms of body mass}}{18 \text{ kilograms of body fat}}$
    • The $18 \text{ kilograms}$ is measured: It has two significant figures.
    • The $100 \text{ kilograms}$ is exact.

Conversion Factors from Dosage Problems

• Equalities stated within dosage problems for medications can also be written as conversion factors.
  • Example: Keflex (cephalexin), an antibiotic used for respiratory and ear infections, is available in 250-milligrams capsules.
    • Equality: $1 \text{ capsule} = 250 \text{ milligrams of Keflex}$
    • Conversion Factors: $\frac{250 \text{ milligrams of Keflex}}{1 \text{ capsule}}$, $\frac{1 \text{ capsule}}{250 \text{ milligrams of Keflex}}$
    • The $250 \text{ milligrams}$ is measured: It has two significant figures.
    • The $1 \text{ capsule}$ is exact.

Sample Exercise 17

• Write the equality and two conversion factors for each of the following:
  • A. meters and decimeters
  • B. jewelry that contains 18% gold
  • C. One gallon of gas is $2.40.

Solution 17

• (Solution not provided in transcript)

Problem Solving Using Conversion Factors

• The problem-solving process begins by analyzing the problem in order to:
  • identify the given unit and needed unit
  • write a plan that converts the given unit to the needed unit
  • identify one or more conversion factors that cancel units and provide the needed unit
  • set up the calculation

Solving Problems Using Conversion Factors

• Example: If a person weighs 178 lb, what is the body mass in kilograms?

  • STEP 1: State the given and needed quantities:

    • Given: $178 \text{ lb}$
    • Need: kilograms
    • Connect: conversion factor (kilograms/lb)

  • STEP 2: Write a plan to convert the given unit to the needed unit.

  • STEP 3: State the equalities and conversion factors.

  • STEP 4: Set up the problem to cancel units and calculate the answer.

    $178 \text{ lb} \times \frac{1 \text{ kg}}{2.20 \text{ lb}} = 80.9 \text{ kg}$

Sample Exercise 18

• A rattlesnake is $2.44 \text{ m}$ long. How many centimeters long is the snake?

Solution 18

• STEP 1: State the given and needed quantities:
  • Given: $2.44 \text{ m}$
  • Need: centimeters
  • Connect: conversion factor (cm/m)
• STEP 2: Write a plan to convert the given unit to the needed unit.
• STEP 3: State the equalities and conversion factors.
  • $1 \text{ m} = 100 \text{ cm}$
  • $\frac{100 \text{ cm}}{1 \text{ m}}$
• STEP 4: Set up the problem to cancel units and calculate the answer.
  • $2.44 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 244 \text{ cm}$

Using Two or More Conversion Factors

• In problem solving:
  • two or more conversion factors are often needed to complete the change of units
  • $\text{Unit 1} \rightarrow \text{Unit 2} \rightarrow \text{Unit 3}$
  • to set up these problems, one factor follows the other
  • each factor is arranged to cancel the preceding unit until the needed unit is obtained
  • $\text{Given unit} \times \text{Factor 1} \times \text{Factor 2} = \text{Needed unit}$

Using Two or More Conversion Factors (Continued)

• Example: A doctor’s order prescribed a dosage of $0.150 \text{ milligrams}$ of Synthroid. If you only have $50 \text{.-micrograms}$ tablets in stock, how many tablets are required to provide the prescribed dosage?

  • STEP 1: State the given and needed quantities:

    • Given: $0.150 \text{ milligrams of Synthroid}$.
    • Need: number of tablets 1 tablet = 50 micrograms of Synthroid
    • Connect: 1 tablet = $50 \text{ micrograms}$ of Synthroid

  • STEP 2: Write a plan to convert the given unit to the needed unit.

  • STEP 3: State the equalities and conversion factors.

  • STEP 4: Set up the problem to cancel units and calculate the answer.

    $0.150 \text{ mg Synthroid} \times \frac{1000 \text{ μg}}{1 \text{ mg}} \times \frac{1 \text{ tablet}}{50 \text{ μg}} = 3.0 \text{ tablets}$

Sample Exercise 19

• How many minutes are in 1.6 days?

Solution 19

• STEP 1: State the given and needed quantities:
  • Given: 1.6 days
  • Need: minutes
  • Connect: conversion factors (hours/day, minutes/hour)
• STEP 2: Write a plan to convert the given unit to the needed unit.
• STEP 3: State the equalities and conversion factors.
  • $1 \text{ day} = 24 \text{ hr}$
  • $1 \text{ hr} = 60 \text{ min}$
  • $\frac{24 \text{ hr}}{1 \text{ day}}$
  • $\frac{60 \text{ min}}{1 \text{ hr}}$
• STEP 4: Set up the problem to cancel units and calculate the answer.
  • $1.6 \text{ days} \times \frac{24 \text{ hr}}{1 \text{ day}} \times \frac{60 \text{ min}}{1 \text{ hr}} = 2300 \text{ min}$

Sample Exercise 20

• If your pace on a treadmill is 65 meters/minute, how many minutes will it take for you to walk a distance of 7.5 kilometers?

Solution 20

• STEP 1: State the given and needed quantities:
  • Given: 7.5 kilometers
  • Need: minutes
  • Connect: conversion factors (meters/kilometer, 65 meters/minute)
• STEP 2: Write a plan to convert the given unit to the needed unit.
• STEP 3: State the equalities and conversion factors.
  • $1 \text{ km} = 1000 \text{ m}$
  • $\frac{65 \text{ m}}{1 \text{ min}}$
• STEP 4: Set up the problem to cancel units and calculate the answer.
  • $7.5 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ min}}{65 \text{ m}} = 120 \text{ min}$

Density

• Density:
  • compares the mass of an object to its volume.
  • is the mass of a substance divided by its volume.
  • is defined as Density = $\frac{\text{mass}}{\text{volume}}$

Density (Continued)

• Substances that have:
  • higher densities contain particles that are closely packed together.
  • lower densities contain particles that are farther apart.
• Metals such as gold and lead have higher densities because their atoms are packed closely together.

Density: Units

• In the metric system, densities of solids, liquids, and gases are expressed with different units.
  • The density of a solid or liquid is usually given in grams per cubic centimeter ($\text{g/cm}^3$) or grams per milliliter (g/mL).
  • The density of a gas is usually given in grams per liter (g/L).

Densities of Common Substances

Chemistry Link to Health: Bone Density

• A condition of severe thinning of bone known as osteoporosis may occur.
• Scanning electron micrographs (SEMs) show (a) normal bone and (b) bone with osteoporosis due to loss of bone minerals.

Chemistry Link to Health: Bone Density (Continued)

• Bone density is often determined by passing low-dose X-rays through the narrow part at the top of the femur (hip) and the spine (c).
• Bones with high density will block more of the X-rays compared to bones that are less dense.

Density Using Volume Displacement

• The density of the zinc object is calculated from its mass and volume.
• A solid completely submerged in water displaces its own volume of water (its volume is calculated from the volume increase).

Sample Exercise 21

• What is the density (grams/milliLiters) of a 48.0-grams sample of a metal if the level of water in a graduated cylinder rises from 25.0 milliLiters to 33.0 milliLiters after the metal is added?

Solution 21

• $V = 33.0 \text{ mL} - 25.0 \text{ mL} = 8.0 \text{ mL}$
• Density = $\frac{48.0 \text{ g}}{8.0 \text{ mL}} = 6.0 \text{ g/mL}$

Problem Solving Using Density

• If the volume and the density of a sample are known, the mass in grams of the sample can be calculated by using density as a conversion factor.

Problem Solving Using Density (Continued)

• Example: An unknown liquid has a density of $1.32 \frac{\text{grams}}{\text{milliLiters}}$. What is the volume (milliLiters) of a $14.7 \text{-grams}$ sample of the liquid?
  • STEP 1: State the given and needed quantities:
    • Given: $14.7 \text{ g}$
    • Need: milliliters
    • Connect: density ($\frac{\text{grams}}{\text{milliLiters}}$
  • STEP 2: Write a plan to calculate the needed quantity.
  • STEP 3: Write the equalities and their conversion factors including density.
    • $\frac{1.32 \text{ g}}{1 \text{ mL}}$, $\frac{1 \text{ mL}}{1.32 \text{ g}}$
  • STEP 4: Set up the problem to calculate the needed quantity.
    • $14.7 \text{ g} \times \frac{1 \text{ mL}}{1.32 \text{ g}} = 11.1 \text{ mL}$

Sample Exercise 22

• John took 2.0 teaspoons (tsp) of cough syrup for a cough. If the syrup had a density of 1.20 grams/milliLiters and there is 5.0 milliLiters in 1 teaspoons, what was the mass, in grams, of the cough syrup?

Solution 22

• STEP 1: State the given and needed quantities:
  • Given: 2.0 teaspoons of syrup
  • Need: grams
  • Connect: conversion factors (milliLiters/teaspoons, density of syrup)
• STEP 2: Write a plan to calculate the needed quantity.
• STEP 3: Write the equalities and their conversion factors including density.
  • $1 \text{ tsp} = 5.0 \text{ mL}$
  • $\frac{1.20 \text{ g}}{1 \text{ mL}}$
• STEP 4: Set up the problem to calculate the needed quantity.
  • $2.0 \text{ tsp} \times \frac{5.0 \text{ mL}}{1 \text{ tsp}} \times \frac{1.20 \text{ g}}{1 \text{ mL}} = 12 \text{ g}$

Specific Gravity

• Specific gravity (sp gr) is a relationship between the density of a substance and the density of water.
• Specific gravity is calculated by dividing the density of a sample by the density of water, which is 1.00 grams/milliLiters at 4 °Celsius.
• A substance with a specific gravity of 1.00 has the same density as water (1.00 grams/milliLiters).

Specific Gravity (Continued)

• A hydrometer is used to measure the specific gravity of urine.
• The normal range of specific gravity for urine is 1.003 to 1.030.
• The specific gravity can decrease with type 2 diabetes and kidney disease.
• Increased specific gravity may occur with dehydration, kidney infection, and liver disease.

Specific Gravity (