Density, Volume, Conversions, and Elements — Quick Notes

Density and Volume

• Density definition: $\rho = \frac{m}{V}$
• Common units: $\text{g cm}^{-3}$ or $\text{g mL}^{-1}$; 1 mL = 1 cm³
• Volume for irregular objects: use displacement in a liquid; V = V{final} - V{initial}
• Mass measurement: use a balance; precision tied to balance sig figs
• Example (displacement): initial volume 35.5 mL, final volume 45.0 mL → V = 9.5 mL
• Example density (irregular solid): m = 68.6 g, V = 9.5 mL → $\rho = \frac{68.6}{9.5} \approx 7.22\,\frac{g}{mL}$ → with sig figs: $7.2\,\frac{g}{mL}$

Significant Figures (Sig Figs)

• Multiplication/Division: result has as many sig figs as the factor with the fewest sig figs
• Addition/Subtraction: result limited by decimal places (least precise decimal place among inputs)
• Example: $\frac{68.6\,\text{g}}{9.5\,\text{mL}} = 7.221…\;\frac{g}{\text{mL}}$ → two sig figs (from 9.5) → $7.2\;\frac{g}{\text{mL}}$
• Conversions: treated as exact numbers; do not affect sig figs
• Mass and volume via displacement problems often yield densities of a few sig figs

Density as a Conversion Factor

• Density is a ratio: $\rho = \frac{m}{V}$, so it can convert between mass and volume
• Mass from volume: $m = \rho \cdot V$
• Volume from mass: $V = \frac{m}{\rho}$
• Inverse density: you can use density upside down to convert in the opposite direction
• Example (acetic acid): density $\rho = 1.05\;\frac{\text{g}}{\text{mL}}$, mass $m = 5.0\;\text{g}$ → volume $V = \frac{m}{\rho} = \frac{5.0}{1.05} \approx 4.76\;\text{mL}$
• Write density both ways as conversion factors:
  • $\frac{1.05\;\text{g}}{1\;\text{mL}}$
  • $\frac{1\;\text{mL}}{1.05\;\text{g}}$
• If asked for mass from a given volume using density, multiply by density; if asked for volume from mass, divide by density

Using Density to Convert Mass and Volume (Practice Ideas)

• Given density of octane: $\rho_{\text{octane}} = 0.702\;\frac{\text{g}}{\text{mL}}$
• Convert a volume to mass: $m = \rho V$
• Convert mass to volume: $V = \frac{m}{\rho}$
• Example workflow (conceptual): given a volume in mL, multiply by density to get mass in g; to get kg, convert g to kg by dividing by 1000

Examples and Quick Conversions

• Example: Convert 325 mg to g
  • Equality: $1\,\text{g} = 1000\,\text{mg}$
  • Convert: $325\;\text{mg} \times \frac{1\;\text{g}}{1000\;\text{mg}} = 0.325\;\text{g}$
  • Sig figs: 325 mg has 3 sig figs, so 0.325 g has 3 sig figs
• Example: 130 lb to kg with correct significant figures
  • Use conversion: $1\,\text{kg} = 2.21\,\text{lb}$, so $130\;\text{lb} \times \frac{1\;\text{kg}}{2.21\;\text{lb}} = 58.8\;\text{kg}$
  • Two sig figs (from 130 has 2) → $\approx 59\;\text{kg}$
• Example: 0.3 pints to liters via multi-step conversion
  • Given: 2\;\text{pints} = 1\;\text{quart}, 1.06\;\text{qt} = 1\;\text{L}
  • Convert: $0.3\;\text{pints} \times \frac{1\;\text{qt}}{2\;\text{pints}} \times \frac{1\;\text{L}}{1.06\;\text{qt}} = 0.1415\;\text{L} \approx 142\;\text{mL}$
  • Result keeps the original sig figs (two sig figs) → 0.1415 L or 142 mL
• Key strategy: chain multiple conversion factors so that undesired units cancel diagonally; exact factors don’t affect sig figs

Practice Problem Style (Summary ideas)

• How many grams of aspirin in a 325 mg tablet? (use 1 g = 1000 mg)
  • $325\;\text{mg} \times \frac{1\;\text{g}}{1000\;\text{mg}} = 0.325\;\text{g}$
• If density is given, treat as a conversion factor in both directions as needed
• When converting within metric, derive your own factors from prefixes (kilo-, milli-, centi-, etc.)

Chapter Preview: Elements and the Periodic Table

• Elements: pure substances that cannot be broken down by chemical reaction
• Each element has a symbol; 1 or 2 letters (first letter capitalized, second lower-case if present)
• Examples: Gold (Au), Carbon (C), Aluminum (Al)
• Hydrogen is one-letter symbol (H); later elements use two letters
• This chapter will focus on understanding elements and using their symbols on the periodic table