DENSITY

1. Mass and Volume

• Mass is the amount of matter in a material; all matter has mass.

• Mass is measured in kilograms (kg) or grams (g).

• The SI unit for mass is the kilogram (kg).

• Apparatus commonly used to measure mass: triple beam balance, mass meter, electronic scale.

2. Volume

• Volume can be measured with: measuring cylinder, graduated beaker, measuring flask, burette.

• For solids, volume units include: m³, cm³, mm³.

• For liquids, volume units include: mL and L.

• Volume formula for a rectangular solid: $V = l \times w \times h$.

• The volume of an irregular object can be found by water displacement (see Section 1.4).

• 1 mL = 1 cm³ = (1 cm × 1 cm × 1 cm).

• 1 L = 1000 cm³ = (10 cm × 10 cm × 10 cm).

3. Density

• Density of a material is the amount of mass in a given volume of that material.

• Formula: $\text{density} = \frac{\text{mass}}{\text{volume}} = \frac{m}{V}$

• Common density units include: \text{g mL}^{-1} \text{ (or g·mL}^{-1}), $\text{g cm}^{-3}$, $\text{kg L}^{-1}$, etc.

• The unit for density depends on the mass and volume units used.

• Density units table (variables):

  • $m$: mass

  • $V$: volume

  • $d$: density

  • Density symbol: $d$

  • Units examples: $\text{g mL}^{-1},\; \text{g cm}^{-3},\; \text{kg L}^{-1}$

• Important note on units: for density calculations, mass and volume units must be compatible. If mass is in g, volume should be in mL or cm³; if mass is in kg, volume should be in L.

• Key relationships:

  • $d = \frac{m}{V}$

  • $V = \frac{m}{d}$

  • $m = d \times V$

4. Factors that affect density

• The density of a material depends on:

  • The nature of the particles and the strength of the forces between particles.

  • The size and type of the particles.

  • The size of the spaces (voids) between particles.

• Therefore, different materials with different internal structures will have different densities.

5. Density Calculations (worked examples)

• Example 1: If a 10 g object occupies 2.5 cm³, what is its density?

  • $d = \frac{m}{V} = \frac{10\,\text{g}}{2.5\,\text{cm}^3} = 4\,\text{g cm}^{-3}$

• Example 2: What is the mass of methanol that fills exactly a 200 mL container if the density is $0.789\ \text{g mL}^{-1}$?

  • $m = d \times V = 0.789 \times 200 = 157.8\ \text{g}$

• Example 3: Copper block density

  • Mass $m = 1896\ \text{g}$.

  • Dimensions: $l = 8.4\ \text{cm},\; w = 5.5\ \text{cm},\; h = 4.6\ \text{cm}$

  • Volume: $V = l \times w \times h = 8.4 \times 5.5 \times 4.6 = 212.52\ \text{cm}^3$

  • Density: $d = \frac{m}{V} = \frac{1896}{212.52} \approx 8.92\ \text{g cm}^{-3}$

6. Measuring density of irregular objects (water displacement method)

• Principle: When an object is submerged in water, it displaces an amount of water equal to its volume.

• This method is especially useful for irregularly shaped objects (e.g., rocks).

• Procedure (illustrative):

  • Fill a graduated cylinder with water (record initial volume).

  • Submerge the irregular object fully.

  • Record the new water level (final volume).

  • Volume of object = final volume − initial volume.

• Example 4: Rock volume by displacement

  • Initial water: 50 mL; final water after submersion: 62 mL.

  • Volume of rock = 62 − 50 = 12 mL = 12 cm³.

• Example 5: Rock density with mass and volume

  • Volume = 47 mL; Mass = 78 g.

  • Density: $d = \frac{m}{V} = \frac{78}{47} \approx 1.66\ \text{g mL}^{-1}$

7. Practice problems and worksheets (summary of problem types and solutions)

• Worksheet 1 (types of problems): density calculations using d = m/V and V = m/d with various data; checks on understanding that larger density implies more mass for the same volume.

• Example 1 (from Worksheet 1):

  • Given mass 500 g and final volume 555 mL, density $d = \frac{m}{V} = \frac{500}{555} \approx 0.90\ \text{g mL}^{-1}$

• Example 2 (Chalk):

  • Mass 50 g, density 2.6 g/mL, volume $V = \frac{m}{d} = \frac{50}{2.6} \approx 19.23\ \text{mL}$

• Example 3 (Coal vs Cork):

  • Given a volume of 100 mL, densities: coal $d<em>{coal} = 0.25\ \text{g mL}^{-1}$ and cork $d</em>{cork} = 1.50\ \text{g mL}^{-1}$

  • Masses: $m<em>{coal} = d</em>{coal} \times V = 0.25 \times 100 = 25\ \text{g}$

  • $m<em>{cork} = d</em>{cork} \times V = 1.50 \times 100 = 150\ \text{g}$

  • Conclusion: For identical volumes, the denser material has a larger mass (coal vs cork in this example).

• Worksheet 2 (additional problems): varied values for density calculations, including

  • Example: $d = \frac{m}{V},\; m = 12.9\ \text{g},\; V = 8\ \text{cm}^{3} \Rightarrow d = \frac{12.9}{8} = 1.61\ \text{g cm}^{-3}$

  • Example: $d = \frac{m}{V},\; m = 43.5\ \text{g},\; V = 50\ \text{mL} \Rightarrow d = \frac{43.5}{50} = 0.87\ \text{g mL}^{-1}$

• Worksheet 3/Examples (more practice problems):

  • Example: $m = d \times V,\; V = 15\ \text{mL},\; d = 2.5\ \text{g mL}^{-1} \Rightarrow m = 2.5 \times 15 = 37.5\ \text{g}$

  • Example: $V = \frac{m}{d},\; m = 65\ \text{g},\; d = 5.45\ \text{g mL}^{-1} \Rightarrow V = \frac{65}{5.45} \approx 11.93\ \text{mL}$

  • Example: $d = \frac{m}{V},\; m = 176\ \text{g},\; V = 4\ \text{cm}^3 \Rightarrow d = \frac{176}{4} = 44\ \text{g cm}^{-3}$ (note: ensure units are consistent; this example demonstrates formula usage)

• Worksheet 4 (more large-volume/mass figures):

  • Example: $d = \frac{m}{V},\; m = 2700\ \text{g},\; V = 10\times 15 \times 35\ \text{cm}^3 = 5250\ \text{cm}^3 \Rightarrow d \approx \frac{2700}{5250} \approx 0.51\ \text{g cm}^{-3}$

  • Example: $d = \frac{m}{V},\; m = 78\ \text{g},\; V = 60\ \text{mL} \Rightarrow d = \frac{78}{60} = 1.30\ \text{g mL}^{-1}$

  • Example: $d = \frac{m}{V},\; m = 1300\ \text{g},\; V = 743\ \text{cm}^3 \Rightarrow d \approx \frac{1300}{743} \approx 1.75\ \text{g cm}^{-3}$

8. Practical tips and key reminders

• Always ensure mass and volume units are compatible before calculating density.

  • If mass is in g, use volume in mL or cm³.

  • If mass is in kg, use volume in L.

• Remember the difference between solids and liquids in how volume is reported and measured.

• For irregular objects, water displacement gives a direct measurement of volume, which is often easier than calculating by geometry.

• Use the formulae consistently:

  • $d = \frac{m}{V}$

  • $V = \frac{m}{d}$

  • $m = d \times V$

• Density is a material property that helps identify substances and compare materials under the same conditions.

• Real-world relevance: density is used in material science, quality control, buoyancy, and identification of substances. It can indicate composition and purity when measured under standard conditions.