Required Practical: Determining Density

Calculating Density

Why does a small metal pebble sink, while a massive wooden log floats? It all comes down to how tightly packed the matter is inside the object. Density is the mass per unit volume of a substance. In the particle model, density depends on the spacing of atoms and molecules; solids are generally the most dense, followed by liquids, while gases are the least dense due to large intermolecular distances.

To calculate density, use the standard equation:

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

Where:

• $\rho$ (rho) = density in kilograms per metre cubed ($\text{kg/m}^3$) or grams per centimetre cubed ($\text{g/cm}^3$)
• $m$ = mass in kilograms ($\text{kg}$) or grams ($\text{g}$)
• $V$ = volume in metres cubed ($\text{m}^3$) or centimetres cubed ($\text{cm}^3$)

The formula can be rearranged to find volume ($V = \frac{m}{\rho}$) or mass ($m = \rho \times V$). You must be comfortable converting between units. Remember that $1000 \text{ g} = 1 \text{ kg}$, $1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3$, and $1 \text{ m}^3 = 1,000,000 \text{ cm}^3$.

Determining the Density of a Regular Solid

If an object has a uniform geometric shape, finding its volume is just a matter of measuring its dimensions. A regular solid is an object with a uniform geometric shape (like a cube, cylinder, or sphere) whose volume can be calculated using a mathematical formula.

• First, place the object on a digital balance to measure its mass. Ensure the balance is "tared" (set to zero) beforehand to avoid a zero error.
• Next, measure the object's dimensions using a 30 cm ruler, vernier calipers, or a micrometer. Using calipers or a micrometer provides a much higher resolution ($0.1 \text{ mm}$ or $0.01 \text{ mm}$) than a standard ruler ($1 \text{ mm}$), reducing uncertainty.
• Finally, use the appropriate geometric formula to calculate the volume (e.g., $V = \text{length} \times \text{width} \times \text{height}$ for a cuboid).

Worked Example: Regular Solid

A brass cuboid has a mass of $680 \text{ g}$ and dimensions of $2.0 \text{ cm}$, $5.0 \text{ cm}$, and $8.0 \text{ cm}$. Calculate its density.

Step 1: Calculate the volume.

• $V = 2.0 \times 5.0 \times 8.0 = 80 \text{ cm}^3$

Step 2: Substitute into the density formula.

• $\rho = \frac{680}{80}$

Step 3: Calculate the final answer with units.

• $\rho = 8.5 \text{ g/cm}^3$

Determining the Density of an Irregular Solid

You cannot simply measure the dimensions of a jagged rock with a ruler, but you can use water to reveal its volume. An irregular solid is an object without a simple geometric shape whose volume cannot be calculated by a formula. Instead, we use displacement, which is the volume of fluid pushed aside by an object when it is submerged.

• Always measure the mass of the irregular object first while it is totally dry, using a digital balance.
• Fill a Eureka can (displacement can) with water until it is level with the spout, allowing any excess to drip out until it stops.
• Place an empty measuring cylinder under the spout.
• Lower the object slowly into the water using a thin thread. This ensures the object is completely submerged without dropping it and causing a splash (which would overestimate the volume).
• Collect the displaced water. Read the volume at eye-level from the bottom of the meniscus (the curve at the liquid's surface) to avoid a parallax error.

Worked Example: Irregular Solid

An irregular rock has a mass of $96.3 \text{ g}$ and displaces $36 \text{ cm}^3$ of water. Calculate its density to 3 significant figures.

Step 1: Identify the mass and volume.

• $m = 96.3 \text{ g}$
• $V = 36 \text{ cm}^3$

Step 2: Substitute into the density formula.

• $\rho = \frac{96.3}{36}$

Step 3: Calculate the final answer with units.

• $\rho = 2.675 \text{ g/cm}^3 = 2.68 \text{ g/cm}^3$

Determining the Density of Liquids

Measuring the mass of a liquid requires a specific technique because you cannot pour it directly onto a balance without making a mess.

• Place an empty measuring cylinder on a digital balance and record its mass ($m_1$).
• Pour the liquid into the cylinder and record the new, total mass ($m_2$).
• Subtract the empty cylinder's mass from the total mass to find the mass of the liquid ($m_{\text{liquid}} = m_2 - m_1$).
• Read the volume of the liquid directly from the measuring cylinder.
• Using a larger volume of liquid (e.g., $100 \text{ cm}^3$ instead of $10 \text{ cm}^3$) improves accuracy by reducing the percentage uncertainty in your measurements.

Worked Example: Liquid Density

An empty measuring cylinder has a mass of $45.0 \text{ g}$. When $50 \text{ cm}^3$ of a liquid is added, the total mass is $95.0 \text{ g}$. Calculate the density of the liquid in $\text{kg/m}^3$.

Step 1: Calculate the mass of the liquid in grams, then convert to kg.

• $m = 95.0 - 45.0 = 50.0 \text{ g}$
• $50.0 \text{ g} = 0.050 \text{ kg}$

Step 2: Convert the volume into $\text{m}^3$.

• $50 \text{ cm}^3 = 50 \times 10^{-6} \text{ m}^3 = 0.00005 \text{ m}^3$

Step 3: Substitute into the formula.

• $\rho = \frac{0.050}{0.00005}$

Step 4: Calculate the final answer with units.

• $\rho = 1000 \text{ kg/m}^3$