Dimensional Analysis

Dimensional Analysis

High School

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Learning Objectives

• Define dimensional analysis and key terms like conversion factor, base unit, and prefix.

• Recognize the seven SI base units and the meaning of common metric prefixes.

• Apply dimensional analysis to solve single and multi-step unit conversion problems.

• Convert between grams, moles, and particles using molar mass and Avogadro's number.

Key Vocabulary

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Dimensional Analysis

A problem-solving technique that uses the cancellation of units to guide you to the correct solution.

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Conversion Factor

A ratio or fraction that is equal to one, used to convert from one unit to another.

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Base Unit

A fundamental unit in the metric system on which other units of measurement are based.

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Prefix

A word or symbol placed before a unit that multiplies that unit by a specific power of 10.

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Molar Mass

The total mass in grams of one mole of a substance, calculated by adding its atomic masses.

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What is Dimensional Analysis?

• Dimensional analysis is a problem-solving method using the cancellation of units.

• It is also called the Unit Factor Method or the Label Method.

• The method uses a conversion factor to change from one unit to another.

• A conversion factor is a fraction made of different units equal to one.

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Solved Example 1
A professional basketball player is 7 feet tall. Given that 1 foot equals 12 inches, use dimensional analysis to find the player's height in inches.

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Step 1: Analyze and Sketch the Problem

• Goal: Convert the player's height from feet to inches.

• Knowns: The height is 7 feet.

• Unknown: The height in inches.

• Formula: The conversion factor is 1 ft = 12 in.

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Solved Example 1
A professional basketball player is 7 feet tall. Given that 1 foot equals 12 inches, use dimensional analysis to find the player's height in inches.

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Step 2: Solve for the Unknown

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Solved Example 1
A professional basketball player is 7 feet tall. Given that 1 foot equals 12 inches, use dimensional analysis to find the player's height in inches.

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Step 3: Evaluate the Answer

• The original unit of feet has been cancelled, leaving the desired unit of inches.

• The answer is reasonable. Since inches are smaller than feet, the numerical value for the height in inches should be larger than the value in feet.

Multiple Choice

What is the primary role of a conversion factor in dimensional analysis?

• To change the numerical value of the measurement

• To eliminate all units from the equation

• To be the final answer of the problem

• To serve as a ratio equivalent to one for converting units

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SI Units and Prefixes

SI Base Units

• The International System of Units (SI), or metric system, is built on seven fundamental base units.

• Common chemistry units include the meter (m) for length and the kilogram (kg) for mass.

• Other important units are the second (s) for time and the mole (mol) for substance amount.

SI Prefixes

• A prefix can be added to any base unit to multiply it by a power of ten.

• Common prefixes include kilo- (k) for 1,000 and centi- (c) for one-hundredth of the unit.

• Milli- (m) means one-thousandth and micro- (μ) means one-millionth of the base unit.

Multiple Choice

Which of the following is a base unit in the SI system, and which is a prefix?

• Base: gram, Prefix: centi-

• Base: kilo-, Prefix: kilogram

• Base: kilogram, Prefix: kilo-

• Base: second, Prefix: time

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The Process of Unit Conversion

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Multiple Choice

To convert 216 ounces (oz) to pounds (lb), which conversion factor setup is correct, given that 1 lb = 16 oz?

• 216 oz × $$\frac{1 \text{ lb}}{16 \text{ oz}}$$

• 216 oz × $$\frac{1 \text{ oz}}{16 \text{ lb}}$$

• 216 oz × $$\frac{16 \text{ lb}}{1 \text{ oz}}$$

• 216 oz × $$\frac{16 \text{ oz}}{1 \text{ lb}}$$

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Multi-Step Conversions

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Solved Example 4
A car is traveling at 75 miles per hour. How fast is the car traveling in feet per second? (1 mi = 5280 ft)

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Step 1: Analyze and Sketch the Problem

• Goal: Convert the speed from miles per hour to feet per second.

• Knowns: Speed = 75 mi/hr.

• Unknown: Speed in ft/s.

• Conversion Factors: 1 mi = 5280 ft; 1 hr = 60 min; 1 min = 60 s.

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Solved Example 4
A car is traveling at 75 miles per hour. How fast is the car traveling in feet per second? (1 mi = 5280 ft)

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Step 2: Solve for the Unknown

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Solved Example 4
A car is traveling at 75 miles per hour. How fast is the car traveling in feet per second? (1 mi = 5280 ft)

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Step 3: Evaluate the Answer

Multiple Choice

Using the information on the slide, what is the purpose of the '(1 kg / 2.2 lbs)' part of the equation?

• To convert the required mg dose into kg

• To cancel out the 'lbs' unit and convert the cat's weight to 'kg'

• To calculate the concentration of the drug

• To convert the final volume from mL to kg

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Calculating Molar Mass

• Molar mass is the mass of one mole of a substance.

• First, count the atoms of each element in the compound.

• Then, multiply the atom count of each element by its atomic mass.

• For H₂O, the molar mass is 2(1.01) + 16.00 = 18.02 g/mol.

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Solved Example 5
Calculate the molar mass of calcium carbonate (CaCO₃). Atomic masses are approximately Ca = 40.08 g/mol, C = 12.01 g/mol, and O = 16.00 g/mol.

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Step 1: Analyze and Sketch the Problem

• Goal: Find the molar mass of calcium carbonate (CaCO₃).

• Knowns: The formula is CaCO₃, which contains 1 atom of Calcium, 1 atom of Carbon, and 3 atoms of Oxygen.

• Unknown: The molar mass of CaCO₃.

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Solved Example 5
Calculate the molar mass of calcium carbonate (CaCO₃). Atomic masses are approximately Ca = 40.08 g/mol, C = 12.01 g/mol, and O = 16.00 g/mol.

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Step 2: Solve for the Unknown

• First, multiply the number of atoms of each element by its atomic mass.

• Ca: 1 × 40.08 g/mol = 40.08 g/mol

• C: 1 × 12.01 g/mol = 12.01 g/mol

• O: 3 × 16.00 g/mol = 48.00 g/mol

• Next, add the masses of all elements together.

• 40.08 + 12.01 + 48.00 = 100.09 g/mol

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Solved Example 5
Calculate the molar mass of calcium carbonate (CaCO₃). Atomic masses are approximately Ca = 40.08 g/mol, C = 12.01 g/mol, and O = 16.00 g/mol.

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Step 3: Evaluate the Answer

• The final answer is 100.09 g/mol, and the unit (g/mol) is correct for molar mass.

• The calculation correctly sums the masses of one calcium, one carbon, and three oxygen atoms, matching the formula CaCO₃.

Multiple Choice

Following the steps on the slide, how would you begin to calculate the molar mass for copper(II) chloride, CuCl₂?

• (1 × atomic mass of Cu) + (2 × atomic mass of Cl)

• Multiply the mass of Cu by the mass of Cl

• Add the masses of Cu and Cl

• (2 × atomic mass of Cu) + (1 × atomic mass of Cl)

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Mole Conversions in Chemistry

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Solved Example 6
How many formula units of sodium chloride (NaCl) are present in a 25.0-gram sample? The molar mass of NaCl is 58.44 g/mol.

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Step 1: Analyze and Sketch the Problem

• Goal: Find the number of formula units (particles) of NaCl.

• Knowns: Mass = 25.0 g NaCl; Molar Mass of NaCl = 58.44 g/mol; Avogadro's number = 6.02 × 10²³ formula units/mol.

• Unknown: Number of formula units = ?

• Plan: Use dimensional analysis for a two-step conversion: grams → moles → formula units.

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Solved Example 6
How many formula units of sodium chloride (NaCl) are present in a 25.0-gram sample? The molar mass of NaCl is 58.44 g/mol.

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Step 2: Solve for the Unknown

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Solved Example 6
How many formula units of sodium chloride (NaCl) are present in a 25.0-gram sample? The molar mass of NaCl is 58.44 g/mol.

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Step 3: Evaluate the Answer

Multiple Choice

What conversion factors are needed to convert a given number of grams of a substance into particles?

• Only the molar mass

• Molar mass to convert grams to moles, then Avogadro's number to convert moles to particles.

• Avogadro's number to convert grams to particles, then molar mass.

• Only Avogadro's number

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Common Misconceptions

Misconception	Correction
Conversion factors can be used in any orientation.	The factor must be set up to cancel the starting unit.
Converting from grams to particles is a single-step calculation.	It's a two-step process: grams to moles, then moles to particles.
Molar mass is the sum of the atomic masses of elements.	Multiply each atomic mass by its subscript before adding them together.

Multiple Choice

To convert a speed of 75 miles per hour (mph) into feet per second (ft/s), what would be the correct setup? (Given: 1 mile = 5280 ft, 1 hour = 3600 s)

• (75 mi/1 hr) x (5280 ft/1 mi) x (3600 s/1 hr)

• (75 mi/1 hr) x (1 mi/5280 ft) x (1 hr/3600 s)

• (75 mi/1 hr) x (5280 ft/1 mi) x (1 hr/3600 s)

• (75 mi/1 hr) x (1 hr/5280 ft) x (3600 s/1 mi)

Multiple Choice

A student is asked to determine the number of particles in 9.2 grams of MgO₂. What is the correct sequence of steps to solve this problem?

• Particles → Moles → Grams

• Grams → Moles → Particles

• Grams → Particles (in one step)

• Moles → Grams → Particles

Multiple Choice

A chemist has 3.4 moles of water (H₂O). Using the information that the molar mass of H₂O is 18.02 g/mol, analyze the problem to determine the total mass in grams.

• 3.4 moles / 18.02 g/mol ≈ 0.19 g

• 3.4 moles × 18.02 g/mol ≈ 61.3 g

• 3.4 moles + 18.02 g/mol ≈ 21.4 g

• 18.02 g/mol / 3.4 moles ≈ 5.3 g

Multiple Choice

A cat weighing 4.5 kg requires insulin at a dose of 1.5 units per kg. The insulin is supplied in a concentration of 10 units per mL. Evaluate the given information to determine how many mL of insulin are needed.

• (10 units/1 mL) x (1.5 units/1 kg) / 4.5 kg ≈ 3.3 mL

• (4.5 kg) x (1.5 units/1 kg) x (10 units/1 mL) = 67.5 mL

• (4.5 kg) x (1 kg/1.5 units) x (1 mL/10 units) = 0.3 mL

• (4.5 kg) x (1.5 units/1 kg) x (1 mL/10 units) ≈ 0.7 mL

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Summary

Poll

On a scale of 1-4, how confident are you about solving dimensional analysis problems?

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• 4