Complete Lesson: Unit Conversions in Chemistry

Table of Contents

PART 1: Understanding Reciprocals and Dimensional Analysis

What Are Reciprocals?

A reciprocal is just flipping a fraction upside down:

The Key Concept: Conversion Factors Work Both Ways!

If \(1 \text{ inch} = 2.54 \text{ cm}\), then we can write TWO conversion factors:

Both are valid! You choose which one makes your units cancel correctly.

The Dimensional Analysis Method (Factor-Label Method)

  1. STEP 1: Write what you're starting with
  2. STEP 2: Multiply by a conversion factor
  3. STEP 3: Choose the conversion factor so unwanted units cancel
  4. STEP 4: Calculate and check your units

Example: Convert 5 inches to centimeters

Given: \(1 \text{ inch} = 2.54 \text{ cm}\)

Setup:

\[5 \text{ inches} \times \frac{2.54 \text{ cm}}{1 \text{ inch}} = ?\]

Notice: "inches" cancels out!

\[5 \cancel{\text{inches}} \times \frac{2.54 \text{ cm}}{1 \cancel{\text{inch}}} = 12.7 \text{ cm}\]

Why this conversion factor? We needed inches on the bottom to cancel with inches on top.

Counter-Example: Converting cm to inches

Convert 50 cm to inches

Setup:

\[50 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} = ?\]

Notice: Now we use the RECIPROCAL! "cm" cancels out!

\[50 \cancel{\text{cm}} \times \frac{1 \text{ inch}}{2.54 \cancel{\text{cm}}} = 19.7 \text{ inches}\]

PART 2: One-Step Conversions (Same Measure)

Your Conversion Factors Reference Sheet

LENGTH:

  • 1 inch = 2.54 cm
  • 1 foot = 12 inches
  • 1 meter = 3.2808 feet
  • 1 yard = 3 feet
  • 1 mile = 5280 feet
  • 1 mile = 1.609 km

VOLUME:

  • 1 gallon = 3.785 liters
  • 1 gallon = 4 quarts
  • 1 pint = 2 cups
  • 1 cup = 8 fluid ounces
  • 1 fluid ounce = 29.575 mL

WEIGHT/MASS:

  • 1 ounce = 28.35 grams
  • 1 pound = 16 ounces
  • 1 kilogram = 2.2046 pounds

PRESSURE:

  • 1 atmosphere = 1013.25 millibars
  • 1 atmosphere = 101.325 kilopascals

ENERGY/POWER:

  • 1 calorie = 4.184 J
  • 1 horsepower = 745.7 watts

Study Guide - Part 2 (One-Step Conversions)

Set up each conversion showing the conversion factor. Circle the units that cancel.

1. Convert 8 feet to inches
2. Convert 144 inches to feet
3. Convert 5 pounds to ounces
4. Convert 80 ounces to pounds
5. Convert 2.5 gallons to quarts
6. Convert 12 quarts to gallons
7. Convert 100 meters to feet
8. Convert 500 feet to meters
9. Convert 3.5 kilograms to pounds
10. Convert 50 pounds to kilograms

Answers

PART 3: Multi-Step Conversions (Same Measure)

Sometimes you need to use multiple conversion factors in a chain to get from one unit to another.

Example: Convert 3 miles to centimeters

Available conversions:

  • \(1 \text{ mile} = 5280 \text{ feet}\)
  • \(1 \text{ foot} = 12 \text{ inches}\)
  • \(1 \text{ inch} = 2.54 \text{ cm}\)

Setup (chain them together!):

\[3 \text{ miles} \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{2.54 \text{ cm}}{1 \text{ in}}\]

Watch the units cancel:

\[3 \cancel{\text{miles}} \times \frac{5280 \cancel{\text{ft}}}{1 \cancel{\text{mile}}} \times \frac{12 \cancel{\text{in}}}{1 \cancel{\text{ft}}} \times \frac{2.54 \text{ cm}}{1 \cancel{\text{in}}} = 482{,}803.2 \text{ cm}\]

Pro Tip: Always Check Your Units!

Before you calculate:

  1. Cross out units that appear on both top and bottom
  2. Make sure only your target unit remains
  3. If units don't cancel correctly, flip one of your conversion factors!

Study Guide - Part 3 (Multi-Step Conversions)

Set up each conversion showing ALL conversion factors needed.

1. Convert 2 miles to inches (use: mile→feet→inches)
2. Convert 50,000 centimeters to miles (use: cm→inches→feet→miles)
3. Convert 5 gallons to cups (use: gallons→quarts→pints→cups)
Note: You'll need to figure out quarts→pints: 1 quart = 2 pints
4. Convert 128 fluid ounces to gallons (reverse the path from #3)
5. Convert 5 pounds to grams (use: pounds→ounces→grams)

Answers

PART 4: Squared and Cubed Unit Conversions (AREA AND VOLUME)

The Critical Rule: Use the Conversion Factor Multiple Times!

When converting area (square units) or volume (cubic units), you need to apply the conversion factor ONCE FOR EACH DIMENSION.

Why? Let's Think About It

If \(1 \text{ foot} = 12 \text{ inches}\), then:

Think about it: A square that is \(1 \text{ ft} \times 1 \text{ ft}\) is actually \(12 \text{ in} \times 12 \text{ in} = 144 \text{ in}^2\)

We need to convert BOTH dimensions!

Area Conversions - The Long Form Method

Key Idea: Square units mean TWO dimensions (length × width). So we need to use our conversion factor TWICE - once for each dimension!

Example: Convert 5 square feet to square inches

Given: \(1 \text{ foot} = 12 \text{ inches}\)

Step 1: Write out "square feet" as ft × ft

\[5 \text{ ft}^2 = 5 \text{ ft} \times \text{ft}\]

Step 2: Apply the conversion factor TWICE (once for each ft)

\[5 \text{ ft} \times \text{ft} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{12 \text{ in}}{1 \text{ ft}}\]

Step 3: Cancel the units

\[5 \cancel{\text{ft}} \times \cancel{\text{ft}} \times \frac{12 \text{ in}}{1 \cancel{\text{ft}}} \times \frac{12 \text{ in}}{1 \cancel{\text{ft}}}\]

Step 4: What's left?

\[5 \times 12 \text{ in} \times 12 \text{ in} = 5 \times 144 \text{ in}^2 = 720 \text{ in}^2\]

Notice: Each "ft" in the original cancels with one "ft" in the denominator of a conversion factor. That's why we needed the conversion factor TWICE!

Another Area Example: Convert 2 square yards to square feet

Given: \(1 \text{ yard} = 3 \text{ feet}\)

Step 1: Write it out long form

\[2 \text{ yd}^2 = 2 \text{ yd} \times \text{yd}\]

Step 2: Apply conversion factor twice

\[2 \text{ yd} \times \text{yd} \times \frac{3 \text{ ft}}{1 \text{ yd}} \times \frac{3 \text{ ft}}{1 \text{ yd}}\]

Step 3: Cancel

\[2 \cancel{\text{yd}} \times \cancel{\text{yd}} \times \frac{3 \text{ ft}}{1 \cancel{\text{yd}}} \times \frac{3 \text{ ft}}{1 \cancel{\text{yd}}}\]

Step 4: Calculate

\[2 \times 3 \text{ ft} \times 3 \text{ ft} = 2 \times 9 \text{ ft}^2 = 18 \text{ ft}^2\]

Volume Conversions - The Long Form Method

Key Idea: Cubic units mean THREE dimensions (length × width × height). So we need to use our conversion factor THREE TIMES - once for each dimension!

Example: Convert 2 cubic feet to cubic inches

Given: \(1 \text{ foot} = 12 \text{ inches}\)

Step 1: Write out "cubic feet" as ft × ft × ft

\[2 \text{ ft}^3 = 2 \text{ ft} \times \text{ft} \times \text{ft}\]

Step 2: Apply the conversion factor THREE TIMES (once for each ft)

\[2 \text{ ft} \times \text{ft} \times \text{ft} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{12 \text{ in}}{1 \text{ ft}}\]

Step 3: Cancel the units

\[2 \cancel{\text{ft}} \times \cancel{\text{ft}} \times \cancel{\text{ft}} \times \frac{12 \text{ in}}{1 \cancel{\text{ft}}} \times \frac{12 \text{ in}}{1 \cancel{\text{ft}}} \times \frac{12 \text{ in}}{1 \cancel{\text{ft}}}\]

Step 4: What's left?

\[2 \times 12 \text{ in} \times 12 \text{ in} \times 12 \text{ in} = 2 \times 1728 \text{ in}^3 = 3456 \text{ in}^3\]

Notice: Each "ft" cancels individually! We had three ft's, so we needed three conversion factors.

Another Volume Example: Convert 0.5 cubic yards to cubic feet

Given: \(1 \text{ yard} = 3 \text{ feet}\)

Step 1: Write it out long form

\[0.5 \text{ yd}^3 = 0.5 \text{ yd} \times \text{yd} \times \text{yd}\]

Step 2: Apply conversion factor three times

\[0.5 \text{ yd} \times \text{yd} \times \text{yd} \times \frac{3 \text{ ft}}{1 \text{ yd}} \times \frac{3 \text{ ft}}{1 \text{ yd}} \times \frac{3 \text{ ft}}{1 \text{ yd}}\]

Step 3: Cancel

\[0.5 \cancel{\text{yd}} \times \cancel{\text{yd}} \times \cancel{\text{yd}} \times \frac{3 \text{ ft}}{1 \cancel{\text{yd}}} \times \frac{3 \text{ ft}}{1 \cancel{\text{yd}}} \times \frac{3 \text{ ft}}{1 \cancel{\text{yd}}}\]

Step 4: Calculate

\[0.5 \times 3 \text{ ft} \times 3 \text{ ft} \times 3 \text{ ft} = 0.5 \times 27 \text{ ft}^3 = 13.5 \text{ ft}^3\]

The Pattern You Should See:

  • Linear units (ft): Use conversion factor ONCE
  • Square units (ft × ft): Use conversion factor TWICE
  • Cubic units (ft × ft × ft): Use conversion factor THREE TIMES

This is why people say "square the conversion factor" or "cube the conversion factor" - it's shorthand for applying it multiple times!

Understanding the Numbers

When you apply conversion factors multiple times, you end up multiplying the numbers together. Here's what you get:

AREA (Apply conversion factor TWICE):

  • \(1 \text{ ft}^2\) becomes: \(12 \text{ in} \times 12 \text{ in} = \) \(144 \text{ in}^2\)
  • \(1 \text{ yd}^2\) becomes: \(3 \text{ ft} \times 3 \text{ ft} = \) \(9 \text{ ft}^2\)
  • \(1 \text{ m}^2\) becomes: \(100 \text{ cm} \times 100 \text{ cm} = \) \(10{,}000 \text{ cm}^2\)
  • \(1 \text{ in}^2\) becomes: \(2.54 \text{ cm} \times 2.54 \text{ cm} = \) \(6.4516 \text{ cm}^2\)

VOLUME (Apply conversion factor THREE TIMES):

  • \(1 \text{ ft}^3\) becomes: \(12 \text{ in} \times 12 \text{ in} \times 12 \text{ in} = \) \(1728 \text{ in}^3\)
  • \(1 \text{ yd}^3\) becomes: \(3 \text{ ft} \times 3 \text{ ft} \times 3 \text{ ft} = \) \(27 \text{ ft}^3\)
  • \(1 \text{ m}^3\) becomes: \(100 \text{ cm} \times 100 \text{ cm} \times 100 \text{ cm} = \) \(1{,}000{,}000 \text{ cm}^3\)
  • \(1 \text{ in}^3\) becomes: \(2.54 \text{ cm} \times 2.54 \text{ cm} \times 2.54 \text{ cm} = \) \(16.387 \text{ cm}^3\)

Study Guide - Part 4 (Squared and Cubed Conversions)

Complete each conversion. Show your work using the long-form method!

AREA CONVERSIONS:

1. Convert 3 square feet to square inches
2. Convert 500 square inches to square feet
3. Convert 2 square yards to square feet
4. Convert 5 square meters to square centimeters
5. Convert 1 square mile to square feet (1 mile = 5280 feet)

VOLUME CONVERSIONS:

6. Convert 4 cubic feet to cubic inches
7. Convert 10,000 cubic centimeters to cubic meters
8. Convert 0.5 cubic yards to cubic feet
9. Convert 2.5 cubic meters to cubic centimeters
10. Convert 1 cubic inch to cubic centimeters (1 inch = 2.54 cm)

Answers

PART 5: Metric Conversions

The Metric System is Based on Powers of 10!

Metric Prefixes (from large to small):

Key Metric Conversion Factors

LENGTH:

  • 1 kilometer (km) = 1000 meters (m)
  • 1 meter (m) = 100 centimeters (cm)
  • 1 meter (m) = 1000 millimeters (mm)
  • 1 centimeter (cm) = 10 millimeters (mm)

MASS:

  • 1 kilogram (kg) = 1000 grams (g)
  • 1 gram (g) = 1000 milligrams (mg)

VOLUME:

  • 1 liter (L) = 1000 milliliters (mL)

Shortcut: The Ladder Method

     kilo-
      ↕ ×1000 or ÷1000
    [BASE]
      ↕ ×100 or ÷100
    centi-
      ↕ ×10 or ÷10
    milli-

Going DOWN the ladder? Multiply
Going UP the ladder? Divide

Study Guide - Part 5 (Metric Conversions)

Convert using the dimensional analysis method:

1. 4.2 km to meters
2. 850 cm to meters
3. 2500 mm to meters
4. 6.8 kg to grams
5. 450 mg to grams
6. 3.2 g to milligrams
7. 2.5 L to milliliters
8. 750 mL to liters
9. 5.6 cm to millimeters
10. 8500 g to kilograms

Answers

PART 6: Derived Units (Velocity and Density)

Derived units are combinations of base units. The two most common in chemistry are velocity and density.

Velocity = Distance ÷ Time

Common units: miles/hour (mph), meters/second (m/s), kilometers/hour (km/h)

Example: Convert 60 miles/hour to meters/second

Strategy: Convert miles→meters AND hours→seconds

\[60 \frac{\text{miles}}{\text{hour}} \times \frac{1.609 \text{ km}}{1 \text{ mile}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hour}}{3600 \text{ s}} = 26.8 \frac{\text{m}}{\text{s}}\]

Notice: We need \(3600 \text{ seconds} = 1 \text{ hour}\) (\(60 \text{ min} \times 60 \text{ sec}\))

Density = Mass ÷ Volume

Common units: g/mL, g/cm³, kg/L, lb/gallon

Example: Convert 2.7 g/mL to kg/L

\[2.7 \frac{\text{g}}{\text{mL}} \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{1000 \text{ mL}}{1 \text{ L}} = 2.7 \frac{\text{kg}}{\text{L}}\]

Notice: The mL and g conversions happen to cancel perfectly!

Study Guide - Part 6 (Derived Units)

Convert the following:

1. 45 miles/hour to feet/second (use: 1 mile = 5280 ft, 1 hour = 3600 seconds)
2. 25 meters/second to miles/hour
3. A density of 8.0 g/cm³ to kg/L (Remember: 1 cm³ = 1 mL)
4. A density of 1.2 kg/L to g/mL
5. 88 feet/second to miles/hour

Answers

PART 7: Special Conversion Factors

Special Factor #1: 1 cm³ = 1 mL

Why this is special: It connects a length-based unit (cubic centimeters) to a volume unit (milliliters)!

Example: Convert 50 cm³ to liters

\[50 \text{ cm}^3 \times \frac{1 \text{ mL}}{1 \text{ cm}^3} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 0.05 \text{ L}\]

This is incredibly useful when you calculate a volume using length measurements!

Special Factor #2: 1 mL of water = 1 g of water

Why this is special: It connects volume to mass (but ONLY for water at standard conditions)!

Example: A container holds 250 mL of water. What is the mass?

\[250 \text{ mL water} \times \frac{1 \text{ g water}}{1 \text{ mL water}} = 250 \text{ g water}\]

CAUTION: This ONLY works for water! Different liquids have different densities.

Combining Both Special Factors

Example: A cube of ice is 5 cm × 5 cm × 5 cm. What is its volume in mL? If it melts, what is the mass of water?

Step 1: Find volume in cm³

\[5 \text{ cm} \times 5 \text{ cm} \times 5 \text{ cm} = 125 \text{ cm}^3\]

Step 2: Convert to mL

\[125 \text{ cm}^3 \times \frac{1 \text{ mL}}{1 \text{ cm}^3} = 125 \text{ mL}\]

Step 3: Convert to grams (for water)

\[125 \text{ mL water} \times \frac{1 \text{ g water}}{1 \text{ mL water}} = 125 \text{ g water}\]

Study Guide - Part 7 (Special Conversions)

1. Convert 350 cm³ to mL
2. Convert 2.5 L to cm³
3. What is the mass of 500 mL of water?
4. If you have 750 g of water, what volume does it occupy in mL?
5. A rectangular container is 10 cm × 8 cm × 6 cm.
a) What is its volume in cm³?
b) What is its volume in mL?
c) If filled with water, what is the mass of the water?

Answers