Density and Relative Density

Density

Density is the mass of an object per unit volume. It indicates how compact the matter in a substance is.

Formula for density

The formula to calculate density is:

$$\rho = \frac{m}{V}$$

• $\rho$ = Density (kg/m³ or g/cm³)
• $m$ = Mass of the object (kg or g)
• $V$ = Volume occupied by the object (m³ or cm³)

The SI unit of density is kilogram per cubic meter (kg/m³), but in many cases, grams per cubic centimeter (g/cm³) is used.

Density of a regular solid

For regular solids (e.g., cubes, cylinders), you can calculate the volume using geometric formulas after measuring dimensions. Then, measure mass using a balance. Finally, apply the density formula.

Density of some common substances

Substance	Density (g/cm³)
Aluminum	2.7
Copper	8.3
Gold	19.3
Iron	7.8
Lead	11.3
Glass	2.5

Example 1: Mass of glass with same volume as aluminum

Given:

• Density of aluminum, $\rho_{Al} = 2.7\, \text{g/cm}^3$
• Mass of aluminum, $m_{Al} = 5.4\, \text{g}$

Find: Mass of glass that has the same volume as the aluminum.

Step 1: Calculate volume of aluminum

Using the formula $V = \frac{m}{\rho}$:

$$V_{Al} = \frac{m_{Al}}{\rho_{Al}} = \frac{5.4\, \text{g}}{2.7\, \text{g/cm}^3} = 2\, \text{cm}^3$$

Step 2: Calculate mass of glass

Using $m = \rho \times V$, and the density of glass $\rho_{glass} = 2.5\, \text{g/cm}^3$:

$$m_{glass} = \rho_{glass} \times V_{Al} = 2.5\, \text{g/cm}^3 \times 2\, \text{cm}^3 = 5\, \text{g}$$

Density of an irregular solid

For irregular solids, volume is found using the water displacement method:

1. Measure the mass of the solid with a balance.
2. Note the initial volume of water in a measuring cylinder.
3. Immerse the solid fully and measure the new volume.
4. Calculate the volume of the solid as the difference between the final and initial water volumes.
5. Calculate density using $\rho = \frac{m}{V}$.

Example 2: Density of irregular solid

Given:

• Mass of solid, $m = 50\, \text{g}$
• Initial water volume, $V_i = 60\, \text{cm}^3$
• Final water volume after immersion, $V_f = 70\, \text{cm}^3$

Find: Density of the solid.

Step 1: Calculate volume of the solid

$$V = V_f - V_i = 70\, \text{cm}^3 - 60\, \text{cm}^3 = 10\, \text{cm}^3$$

Step 2: Calculate density

$$\rho = \frac{m}{V} = \frac{50\, \text{g}}{10\, \text{cm}^3} = 5\, \text{g/cm}^3$$

Density of liquids

To find the density of a liquid:

1. Measure the mass of an empty container ($m_1$).
2. Fill the container with a known volume of liquid and measure the combined mass ($m_2$).
3. Calculate mass of liquid as $m = m_2 - m_1$.
4. Calculate density using $\rho = \frac{m}{V}$, where $V$ is the volume of the liquid.

Example 3: Density of a liquid

Given:

• Mass of empty beaker, $m_1 = 500\, \text{g}$
• Mass of beaker + liquid, $m_2 = 600\, \text{g}$
• Volume of liquid, $V = 25\, \text{cm}^3$

Find: Density of the liquid.

Step 1: Calculate mass of liquid

$$m = m_2 - m_1 = 600\, \text{g} - 500\, \text{g} = 100\, \text{g}$$

Step 2: Calculate density

$$\rho = \frac{m}{V} = \frac{100\, \text{g}}{25\, \text{cm}^3} = 4\, \text{g/cm}^3$$

Relative density (specific gravity)

Relative Density (R.D) is the ratio of the density of a substance to the density of water:

$$R.D = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}}$$

Note: Relative Density has no unit because it is a ratio. The density of water is typically $1\, \text{g/cm}^3$ or $1000\, \text{kg/m}^3$.

Example 4: Calculate relative density

Given: Density of object = 7 g/cm³ Calculate: Relative Density.

$$R.D = \frac{7\, \text{g/cm}^3}{1\, \text{g/cm}^3} = 7$$

Example 5: Relative density of copper

Given:

• Mass of copper = 41.6 g
• Volume of copper = 5.1 cm³

Find: Relative density of copper.

Step 1: Calculate density

$$\rho = \frac{m}{V} = \frac{41.6\, \text{g}}{5.1\, \text{cm}^3} \approx 8.16\, \text{g/cm}^3$$

Step 2: Calculate relative density

$$R.D = \frac{8.16\, \text{g/cm}^3}{1\, \text{g/cm}^3} = 8.16$$

Applications of density in daily life

1. Design of Ships and Aircraft: Density helps in designing ships and planes to ensure they float or fly efficiently by balancing weight and buoyancy.
2. Identification of Gemstones: Relative density is used to distinguish real gemstones from fakes because different stones have unique densities.
3. Design of Swimming Equipment: Density is considered when designing swimming floats and life jackets to ensure they provide enough buoyancy to keep a person afloat.
4. Oil Spill Management: Density differences between oil and water allow oil to float on water, helping in the clean-up process.
5. Quality Control in Manufacturing: Density measurements ensure that materials meet required specifications, for example in metal casting and plastics.
6. Food Industry: Density is used to determine the concentration of solutions like sugar syrup and milk quality by measuring their density.