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}

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 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

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

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

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.)

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