Dimensional Analysis
Related Examples and Practice Problems
Topic Summary & Highlights
and Help Videos
Core Concept
Definition: A systematic method for converting units using conversion factors
Also called: Factor-label method, unit analysis
Key Principle: Conversion factors equal 1, so they don't change the actual quantity
Why Use It:
Ensures accurate unit conversions
Provides built-in error checking
Organizes complex multi-step calculations
Essential for all chemistry problem-solving
Practice Tips
Set Up Conversion Factors: Use conversion factors written as fractions (e.g., $\text{e.g., } 1 \, \text{km} = 1000 \, \text{m}$) to cancel out units and guide the calculation.
Write Units Explicitly: Always include units throughout calculations to track what cancels out and ensure the desired units remain.
Check for Unit Consistency: Confirm that the final units match the target measurement (e.g., speed in m/s\text{m/s}m/s, volume in $\text{cm}^3$).
Use a Step-by-Step Approach: Break complex conversions into smaller steps, such as converting time from hours to seconds before applying it to velocity.
Topic Overview Podcast
Topic Related Resources
|
LABORATORY
|
DEMONSTRATIONS
|
ACTIVITIES
|
VIRTUAL SIMULATIONS
|
Core Concept
The process of dimensional analysis involves using conversion factors, which are ratios of equivalent quantities expressed in different units, to convert from one set of units to another.
Here's a step-by-step approach to using dimensional analysis:
Identify the given quantity: Identify the quantity you have, along with its unit.
Determine the desired unit: Determine the unit you want to convert the given quantity to.
Set up conversion factors: Find or derive the appropriate conversion factors that relate the given unit to the desired unit. Conversion factors often come from conversion tables, unit equivalencies, or relationships derived from mathematical formulas.
Construct conversion factor chains: Use multiple conversion factors as needed to create a chain of ratios that cancel out unwanted units and leave you with the desired unit. Each conversion factor should be chosen in a way that the units cancel out appropriately.
Perform the calculation: Multiply the given quantity by the conversion factors, making sure that units cancel out correctly. The final result will be the desired quantity expressed in the desired unit.
Metric Conversions
A base unit is the basis of measurement in the sciences. The most commonly used base units are liters (L) for liquids, grams (g) for mass, and meters (m) for distance. Metric base units can be converted to other useful quantities by adding a prefix (see the common metric conversions table below). The base unit does not have a prefix, and these prefixes are not used in any other measuring system.
| Common Metric Conversions | ||||
|---|---|---|---|---|
| Prefix | Symbol | Exponential base units in given unit |
Base units in given unit |
|
| Giga | G | 10⁹ | 1,000,000,000 | |
| Mega | M | 10⁶ | 1,000,000 | |
| Kilo | k | 10³ | 1,000 | |
| Hecto | h | 10² | 100 | |
| Deca | da | 10¹ | 10 | |
| BASE UNIT | - | 10⁰ | 1 | |
| Deci | d | 10⁻¹ | .1 | |
| Centi | c | 10⁻² | .01 | |
| Milli | m | 10⁻³ | .001 | |
| Micro | µ | 10⁻⁶ | .000001 | |
| Nano | n | 10⁻⁹ | .000000001 | |
Single Step Conversion
$\boxed{\text{EXAMPLE PROBLEM}}$
How many meters are in 735 kilometers?
$735 \, \text{km} \times \frac{1000 \, \text{m}}{1 \, \text{km}} = 735 \times 1000 \, \text{m} = 735,000 \, \text{meters}$
Multi-Step Conversions
Multi-step conversions are needed when a direct conversion is not known or when multiple units are present that need to be converted. Multi-step conversions are solved the same way as one-step conversions. However, because there are multiple conversion factors, the steps are repeated as many times as needed.
$\boxed{\text{EXAMPLE PROBLEM}}$
Convert 67 miles per hour into meters per second.
$\frac{67 \, \text{miles}}{\text{hour}} \times \frac{1609.34 \, \text{meters}}{1 \, \text{mile}} \times \frac{1 \, \text{hour}}{3600 \, \text{seconds}} = \frac{67 \times 1609.34}{3600} \, \frac{\text{meters}}{\text{second}} = 29.94 \, \frac{\text{meters}}{\text{second}}$
