Dimensional Analysis Chemistry 101

Standard Units of Measure: SI vs English

• Only three countries don’t use the SI System: United States, Liberia, Myanmar (Burma).
• Scientists use the SI units (Système International) for measurements.

Dimensional Analysis

• Using units as a guide to problem-solving.
• Conversion factors are tools for converting units.
• Operations on units are like those with numbers:
  • $A \text{ cm } + B \text{ cm } = (A + B) \text{ cm}$
  • $A \text{ cm } – B \text{ cm } = (A – B) \text{ cm}$
  • $A \text{ cm } \times B \text{ cm } = (A\times B)(\text{cm}\times\text{cm}) = (AB) \text{ cm}^2$
  • $A \text{ cm } ÷ B \text{ cm } = (A/B)(\text{cm}/\text{cm}) = (A/B)$
• Example conversion: $1 \text{ in } = 2.54 \text{ cm}$
  • $\frac{1 \text{ in }}{1 \text{ in }} = \frac{2.54 \text{ cm}}{1 \text{ in }} = 2.54 \frac{\text{cm}}{\text{in}}$
  • $\frac{1 \text{ in }}{2.54 \text{ cm}} = \frac{2.54 \text{ cm}}{2.54 \text{ cm}} = 0.394 \frac{\text{in}}{\text{cm}}$

Conceptual Plan

• Convert feet to centimeters.
• Steps:
  1. Start with the given unit (Left Side) and identify the desired unit (Right Side).
  2. Find the equivalence relationship(s).
  3. Change equivalences into conversion factors.
• Example:
  • $ft \rightarrow in \rightarrow cm$
  • $ft * \frac{12 \text{ in}}{1 \text{ ft}} * \frac{2.54 \text{ cm}}{1 \text{ in}}$

Arranging Conversion Factors

• Arrange so the Given Unit cancels out.
  1. If the Given Unit is in the NUMERATOR, put it in the DENOMINATOR of the conversion factor.
  2. If the Given Unit is in the DENOMINATOR, put it in the NUMERATOR of the conversion factor.
• Example:
  • Convert $1.0 \text{ ft}$ to cm.
  • $1.0 \text{ ft } \times \frac{12 \text{ in }}{1 \text{ ft}} \times \frac{2.54 \text{ cm }}{1 \text{ in }} = 30.48 \text{ cm} = 3.0 \times 10^1 \text{ cm}$

Units of Weight (English)

• Pound (lb) is the basic unit.
• Avoirdupois system: $1 \text{ lb} = 16 \text{ oz}$
• $1 \text{ ton } = 2,000 \text{ lb}$ (short ton) or $2,240 \text{ lb}$ (long ton).
• Troy system (precious metals): $1 \text{ troy pound } = 12 \text{ ounces }$

Units of Length and Area (English)

• Yard (yd) is the basic unit.
• Fractions: inch (in) and foot (ft)
• Multiples: rod, furlong, and mile.
• Area: acre = 4,840 square yards.

Units of Liquid Measure (English)

• Gallon is the basic unit: divided into quarts, pints, gills.
• U.S. gallon = $231 \text{ cubic inches}$
• British imperial gallon is the volume of $10 \text{ lb}$ of water at $62°F = 277.42 \text{ cubic inches}$

Units of Dry Measure (English)

• Bushel is the basic unit, divided into pecks, dry quarts, dry pints.
• U.S. bushel = $2,150.42 \text{ cubic inches}$

Related Units in the SI System

• All units are related to the standard unit by a power of 10.
• Prefix multipliers are always the same.
• Examples:
  • Mass: $1 \text{ kg } = 1000 \text{ g}$
  • Length: $1 \text{ km } = 1000 \text{ m}$
  • Energy: $1 \text{ kJ } = 1000 \text{ J}$
  • Pressure: $1 \text{ kPa } = 1000 \text{ Pa}$
  • Power: $1 \text{ kW } = 1000 \text{ W}$

Common English Units and Their SI Equivalents

• $1 \text{ ounce (oz) } = 28.35 \text{ grams (g)}$
• $1 \text{ pound (lb) } = 453.59 \text{ grams (g)}$
• $1 \text{ kilogram (kg) } = 2.205 \text{ pounds (lb)}$
• $1 \text{ liter (L) } = 1000 \text{ cubic centimeters (cm}^3)$
• $1 \text{ U.S. gallon (gal) } = 3.785 \text{ liters (L)}$
• $1 \text{ inch (in.) } = 2.54 \text{ centimeters (cm)}$ exactly
• $1 \text{ foot (ft) } = 30.48 \text{ centimeters (cm)}$

Examples of Unit Conversions

• Frog egg diameter: $1.5 \times 10^3 \mu\text{m}$ to nanometers.
  • Plan: $\mu\text{m } \rightarrow \text{ nm}$
  • $1.5 \times 10^3 \mu\text{m } \times \frac{1 \text{ m}}{10^6 \mu\text{m}} \times \frac{10^9 \text{ nm}}{1 \text{ m}} = 1.5 \times 10^6 \text{ nm}$
• U.S. penny mass: $2.65 \text{ g}$ to pounds.
  • Plan: $\text{g } \rightarrow \text{ lbs}$
  • $2.65 \text{ g } \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{2.205 \text{ lbs}}{1 \text{ kg}} = 0.00584 \text{ lbs}$

Units Raised to Powers

• Convert cubic inches $\text{in}^3$ to cubic centimeters $\text{cm}^3$.
  • $1 \text{ in } = 2.54 \text{ cm}$
  • $(1 \text{ in})^3 = (2.54 \text{ cm})^3$
  • $1 \text{ in}^3 = 16.387 \text{ cm}^3$
• Dust level conversion: $8.0 \times 10^2 \frac{\text{kg}}{\text{km}^2}$ per day to $\frac{\text{mg}}{\text{m}^2}$ per day.
  • Plan: $\frac{\text{kg}}{\text{km}^2} \rightarrow \frac{\text{mg}}{\text{m}^2}$
  • $8.0 \times 10^2 \frac{\text{kg}}{\text{km}^2} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{1000 \text{ mg}}{1 \text{ g}} \times \frac{1 \text{ km}^2}{10^6 \text{ m}^2} = 8.0 \times 10^5 \frac{\text{mg}}{\text{m}^2}$

Equalities from Derived Units of Measure

• Density of gold: $19.30 \frac{\text{g}}{\text{cm}^3}$
• $1 \text{ cm}^3 \text{ Au } = 19.30 \text{ g Au}$
• Volume of $100.0 \text{ g Au}$.
  • Volume = $100.0 \text{ g Au } \div 19.30 \frac{\text{g}}{\text{cm}^3} = 5.181 \text{ cm}^3$

Representing Measured Numbers

• Measured numbers are obtained using measuring devices.
• Digital measurements: all displayed numbers are certain.
• Analog measurements: every digit is certain except the last, which is estimated.
  • Every measurement has a number, a unit, and an uncertainty.

Precision vs Accuracy

• Precision: reproducibility of a measurement.
• Accuracy: closeness of a measurement to its true value.
• Significant figures: the number of digits in a measurement that give meaningful information.

Rules for Significant Figures

1. All non-zero digits are significant.
2. Interior zeros are significant.
3. Leading zeros are NOT significant.
4. Trailing zeros may or may not be significant.
   • After a decimal point: significant.
   • Without a written decimal point: ambiguous (use scientific notation).
5. Exact numbers have an unlimited number of significant figures.
   • From counting, definitions, or integer values in equations.

Examples of Significant Figures

• $0.04450 \text{ m}$ has 4 significant figures.
• $5.0003 \text{ km}$ has 5 significant figures.
• $10 \text{ dm } = 1 \text{ m}$ has infinite significant figures.
• $1.000 \times 10^5 \text{ s}$ has 4 significant figures.
• $0.00002 \text{ mm}$ has 1 significant figure.
• $10,000 \text{ m}$ has 1 significant figure.

Significant Figures in Calculations

• Multiplying or Dividing: the result has the same number of significant figures as the measurement with the fewest significant figures.
• Adding or Subtracting: the result has the same number of decimal places as the measurement with the fewest decimal places.
• Rounding:
  • 0 to 4: round down.
  • 5 to 9: round up.
• Mixed Operations: follow PEMDAS, evaluate significant figures in intermediate answers, and round only at the end.