Define density as mass per unit volume; recall and use the equation ρ = m / V

1.4 Density

Learning Objective

Students will be able to:

• Define density as a scalar quantity – the mass of a body per unit volume.
• Write and rearrange the fundamental relation \(\rho = \dfrac{m}{V}\).
• Carry out simple experiments to determine the density of liquids, regular solids and irregular solids (water‑displacement).
• Assess experimental uncertainties and record data systematically (AO2).
• Use density data to predict whether an object will float in a fluid, and whether one liquid will float on another immiscible liquid.

1. Definition & Formula

• Density (\(\rho\)) – the amount of mass (\(m\)) contained in a given volume (\(V\)). It is a scalar (no direction).
• Mathematical expression:
  

  \[\rho = \frac{m}{V}\]

2. Rearranging the Equation

• Mass: \(m = \rho \times V\)
• Volume: \(V = \dfrac{m}{\rho}\)

3. Units & Conversions

Unit‑consistency reminder: Before substituting numbers into \(\rho = m/V\), ensure that mass and volume are expressed in compatible units (both SI or both in the CGS system).

4. Experimental Determination of Density

4.1 Pre‑lab Safety Note

• Handle glassware (graduated cylinders, beakers, overflow cans) with care – avoid sudden impacts.
• Secure the balance on a stable surface; do not touch the weighing pan during measurements.
• When working with liquids, wear safety goggles and avoid spillage onto electrical equipment.

4.2 Checklist of Common Uncertainties

• Reading the meniscus (parallax error).
• Balance precision (typically ±0.01 g for a school balance).
• Temperature effects on liquid volume (expand/contract).
• Air bubbles trapped on irregular solids during displacement.
• Calibration of the graduated cylinder or overflow can.

4.3 What to Record (Data‑Table Template)

4.4 Density of Liquids – Graduated‑Cylinder Method

1. Weigh the clean, empty graduated cylinder and record \(m_{\text{cyl}}\).
2. Read the initial volume \(V_i\) (meniscus at the bottom of the cylinder).
3. Fill the cylinder with the liquid to a known final volume \(Vf\); record \(Vf\).
4. Weigh the cylinder + liquid to obtain \(m_{\text{cyl+liq}}\).
5. Calculate the mass of the liquid: \(m = m{\text{cyl+liq}} - m{\text{cyl}}\).
6. Determine the volume of the liquid: \(V = Vf - Vi\).
7. Compute density using \(\rho = m/V\) (apply the unit‑consistency reminder).

4.5 Density of Irregular Solids – Water‑Displacement Method

1. Weigh the solid on a balance; record its mass \(m\).
2. Fill a graduated cylinder (or overflow can) with water and note the initial volume \(V_i\).
3. Gently submerge the solid completely, ensuring no air bubbles cling to the surface.
4. Record the final volume \(V_f\).
5. Volume of the solid: \(V = Vf - Vi\).
6. Calculate density: \(\rho = m/V\) (check units).

5. Density & Buoyancy

An object will float if its average density is lower than the density of the surrounding fluid; otherwise it will sink. This follows from the balance between weight (\(mg\)) and buoyant force (\(\rho_{\text{fluid}} V g\)).

• If \(\rho{\text{object}} < \rho{\text{fluid}}\) → float.
• If \(\rho{\text{object}} > \rho{\text{fluid}}\) → sink.

5.1 Liquid‑on‑Liquid Situations

When two immiscible liquids are placed together, the liquid with the lower density will form a layer on top of the denser liquid. Example: vegetable oil (\(\rho \approx 0.92\ \text{g·cm}^{-3}\)) floats on water (\(\rho = 1.00\ \text{g·cm}^{-3}\)).

6. Typical Densities of Common Materials

7. Worked Examples

Example 1 – Regular Solid (Metal Block)

Problem: A solid block has a mass of 540 g and a volume of 80 cm³. Find its density.

Unit‑consistency reminder: Use grams and cubic centimetres (or convert both to SI).

1. \(\rho = \dfrac{m}{V}\)
2. \(\rho = \dfrac{540\ \text{g}}{80\ \text{cm}^3}\)
3. \(\rho = 6.75\ \text{g·cm}^{-3}\) (3 s.f.)
4. Convert to SI (optional): \(6.75\ \text{g·cm}^{-3} \times 1000 = 6750\ \text{kg·m}^{-3}\).

Example 2 – Liquid Density (Graduated Cylinder)

Problem: An unknown liquid fills a graduated cylinder from 25.0 cm³ to 70.0 cm³. The combined mass of cylinder + liquid is 312 g, and the empty cylinder weighs 112 g. Determine the liquid’s density.

Unit‑consistency reminder: Keep mass in grams and volume in cm³.

1. Mass of liquid: \(m = 312\ \text{g} - 112\ \text{g} = 200\ \text{g}\).
2. Volume of liquid: \(V = 70.0\ \text{cm}^3 - 25.0\ \text{cm}^3 = 45.0\ \text{cm}^3\).
3. \(\rho = \dfrac{200\ \text{g}}{45.0\ \text{cm}^3} = 4.44\ \text{g·cm}^{-3}\) (3 s.f.).

Example 3 – Predicting Floatation

Problem: A wooden block (density = 0.50 g·cm⁻³) is placed in water (density = 1.00 g·cm⁻³). Will it float?

• \(\rho{\text{wood}} < \rho{\text{water}}\) → the block floats.

8. Practice Questions

1. A wooden sphere has a mass of 0.45 kg and a volume of 0.0005 m³. Calculate its density in kg·m⁻³.
2. What volume will 250 g of a substance occupy if its density is 2.5 g·cm⁻³?
3. Identify the most dense material among aluminium (2.70 g·cm⁻³), water (1.00 g·cm⁻³) and air (0.0012 g·cm⁻³).
4. Design a simple experiment to determine the density of an irregularly shaped stone using only a balance, a beaker, and a ruler. List the steps and the calculations required.
5. A metal cylinder (density = 8.96 g·cm⁻³) is placed in oil (density = 0.92 g·cm⁻³). Predict whether it will sink or float and justify your answer.

9. Common Mistakes to Avoid

• Mixing units – always convert mass and volume to compatible units before using \(\rho = m/V\).
• Neglecting the meniscus when reading a graduated cylinder, leading to systematic volume errors.
• Forgetting to subtract the mass of the container in displacement experiments.
• Using the wrong symbol – \(\rho\) denotes density; do not confuse it with \(d\) or \(D\).
• Assuming that an object will float merely because it feels “light”. Floatation depends on density, not weight alone.

10. Box‑Note: Density of Gases (Kinetic‑Particle Model)

For an ideal gas, \(\displaystyle \rho = \frac{PM}{RT}\) where \(P\) is pressure, \(M\) the molar mass, \(R\) the gas constant and \(T\) the absolute temperature. Increasing temperature lowers the density; increasing pressure raises it.

11. Summary

Density links mass and volume through the simple scalar relation \(\rho = m/V\). Mastery of unit conversion, equation rearrangement, and basic experimental techniques enables students to determine densities of liquids, regular solids, and irregular objects, to evaluate uncertainties, and to apply this knowledge to real‑world situations such as predicting floatation of objects and liquids.