Density

Meaning of Density

Density is defined as:

The mass per unit volume of a substance.

Density describes how closely packed the particles of a substance are. Two objects may have the same volume but different densities if their masses are different.

Density as a Physical Quantity

Density is a derived physical quantity because it is obtained from two fundamental quantities:

• mass,

• volume.

It tells us:

• whether a substance is light or heavy for its size,

• how substances compare even when their sizes are different.

Mathematical Definition of Density

Density is given by the equation:

Where:

• ρ = density,

• m = mass,

• V = volume.

SI Unit of Density

The SI unit of density is:

Other commonly used units:

• g cm³ (for solids and liquids in laboratory work).

Conversion reminder:

Physical Interpretation of Density

• A high-density substance has a large mass packed into a small volume.

• A low-density substance has a small mass spread over a large volume.

Examples:

• Iron is denser than wood.

• Oil is less dense than water, so it floats on water.

[Insert diagram comparing equal volumes of a high-density and a low-density substance]

Density and Floating or Sinking (Concept Link)

Whether an object floats or sinks in a liquid depends on density, not mass alone.

• If object density > liquid density → object sinks.

• If object density < liquid density → object floats.

This idea is expanded further in later topics.

Common Examination Errors (Examiner Insight)

Students often:

• define density as mass × volume,

• omit “per unit volume” in the definition,

• give incorrect units for density,

• confuse density with weight.

A precise definition earns easy, guaranteed marks.

Exam-Style Questions (Original)

Question 1

Define density.

Question 2

State the SI unit of density.

Question 3

Two objects have the same volume but different masses.

Explain which object has the greater density.

Worked Solutions (Beyond Excellent)

Solution 1

Density is the mass per unit volume of a substance.

Solution 2

The SI unit of density is kilogram per cubic metre (kg m³).

Solution 3

The object with the greater mass has the greater density because density is equal to mass divided by volume, and the volume is the same for both objects.

End-of-Objective

A learner who has mastered this objective can:

• define density accurately,

• state and use the correct SI unit,

• explain density in terms of mass and volume,

• apply the concept qualitatively to real substances.

Experimental Determination of Density

Density is determined experimentally by:

1. Measuring the mass of the substance,

2. Measuring the volume of the substance,

3. Calculating density using:

Different methods are used depending on whether the substance is:

• a solid with regular shape,

• a solid with irregular shape,

• a liquid.

Determining the Density of a Regular Solid

Apparatus

• Beam balance or electronic balance,

• Ruler or vernier calipers,

• Regular solid (e.g. cube or rectangular block).

Procedure

1. Measure the mass of the solid using a balance.

2. Measure its dimensions using a ruler or vernier calipers.

3. Calculate the volume using the appropriate formula:
   • For a rectangular block:

   $$V = l \times w \times h$$

4. Calculate density using:

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

[Insert diagram showing measurement of dimensions of a rectangular block using a ruler]

Determining the Density of an Irregular Solid

Irregular solids cannot have their volume measured by formula. Instead, the water displacement method is used.

Apparatus

• Measuring cylinder,

• Balance,

• Water,

• Irregular solid (that does not dissolve in water).

Procedure

1. Measure the mass of the solid using a balance.

2. Pour water into a measuring cylinder and record the initial volume V1.

   $$V_1$$

3. Carefully lower the solid into the water.

4. Record the final volume V2.

   $$V_2$$

5. Calculate the volume of the solid:

1. Calculate density using:

[Insert diagram showing an irregular solid immersed in a measuring cylinder (water displacement method)]

Determining the Density of a Liquid

Apparatus

• Measuring cylinder,

• Balance,

• Liquid sample.

Procedure

1. Measure the mass of the empty measuring cylinder.

2. Pour the liquid into the cylinder and measure its volume.

3. Measure the mass of the cylinder plus liquid.

4. Calculate the mass of the liquid:

1. Calculate density using:

[Insert diagram showing mass and volume measurement of a liquid using a measuring cylinder and balance]

Precautions for Accurate Results

• Zero the balance before measuring mass.

• Read the measuring cylinder at eye level to avoid parallax error.

• Ensure the solid is fully submerged without trapping air bubbles.

• Record all readings with correct units.

Common Examination Errors (Examiner Insight)

Students often:

• forget to subtract the mass of the container,

• use incorrect units for volume,

• fail to calculate displaced volume correctly,

• state procedures without mentioning measurements.

Clear step-by-step explanation earns full practical marks.

Exam-Style Questions (Original)

Question 1

Describe an experiment to determine the density of an irregular solid.

Question 2

Explain how the volume of a liquid is measured when determining its density.

Question 3

A solid has a mass of 120 g and a volume of 40 cm³.

Calculate its density.

Worked Solutions (Beyond Excellent)

Solution 1

Measure the mass of the solid using a balance. Pour water into a measuring cylinder and record the initial volume. Lower the solid into the water and record the final volume. The difference between the two readings gives the volume of the solid. Density is then calculated by dividing the mass by the volume.

Solution 2

The liquid is poured into a measuring cylinder, and its volume is read at eye level from the graduated scale. This volume is then used together with the measured mass to calculate density.

Solution 3

Given:

• $m = 120\ \text{g}$

• $V = 40\ \text{cm}^3$

The density of the solid is 3 g cm³.

End-of-Objective

A learner who has mastered this objective can:

• select suitable methods for solids and liquids,

• measure mass and volume accurately,

• apply the density formula correctly,

• explain experimental procedures clearly and logically.

The Density Equation

Density ((p)) relates mass ((m)) and volume ((V)):

Where:

• ρ = density

• m = mass

• V = volume

This equation applies to solids, liquids, and gases (at this level, mainly solids and liquids).

Units Used in Calculations

Common unit sets used at BGCSE level:

• SI units:
  • mass → kg
  • volume → m³
  • density → kg m³

• Laboratory units:
  • mass → g
  • volume → cm³
  • density → g cm³

Units must be consistent within a calculation.

Rearranging the Density Equation (Exam-Critical Skill)

From:

We obtain:

These rearrangements allow calculation of any one quantity when the other two are known.

[Insert triangle diagram showing m at the top, ρ and V at the bottom]

Step-by-Step Calculation Strategy

When solving density problems:

1. Write down the given values.

2. Write the correct formula.

3. Rearrange if necessary.

4. Substitute values with units.

5. Calculate carefully.

6. State the final answer with correct unit.

Clear working earns method marks even if the final answer is incorrect.

Interpreting Results Physically

• Larger density → more mass packed into a given volume.

• Smaller density → less mass for the same volume.

• Comparing densities explains floating and sinking behaviour.

Common Examination Errors (Examiner Insight)

Students often:

• use the wrong rearrangement,

• mix units (e.g. g with m³),

• omit units in final answers,

• write the formula incorrectly as ρ = V/m.

Accuracy and clarity are essential.

Exam-Style Questions (Original)

Question 1

State the equation used to calculate density.

Question 2

A block has a mass of 600 g and a volume of 200 cm³.

Calculate its density.

Question 3

A liquid has a density of 800 kg m³ and a volume of 0.005 m³.

Calculate the mass of the liquid.

Question 4

A metal has a mass of 540 g and a density of 9 g cm³.

Calculate its volume.

Worked Solutions (Beyond Excellent)

Solution 1

Solution 2

Given:

• $m = 600\ \text{g}$

• $V = 200\ \text{cm}^3$

Solution 3

Given:

• $\rho = 800\ \text{kg m}^{3}$

• $V = 0.005\ \text{m}^3$

Solution 4

Given:

• $m = 540\ \text{g}$

• $\rho = 9\ \text{g cm}^{3}$

End-of-Objective

A learner who has mastered this objective can:

• use the density equation correctly,

• rearrange the formula confidently,

• carry out accurate calculations with correct units,

• interpret numerical answers physically.

Why Measuring the Density of Air Is Difficult

Air is a gas, which means:

• it has a very small mass,

• it spreads out and cannot be seen,

• its mass cannot be measured directly without a container.

To determine the density of air, we therefore measure:

• the mass of air inside a container, and

• the volume of that container.

Principle of the Experiment

The experiment is based on:

The mass of air is found by measuring the difference in mass between:

• a container filled with air, and

• the same container with air removed.

Apparatus Required

• Balloon or flask with stopper

• Electronic balance

• Measuring cylinder or ruler (for volume measurement)

• Vacuum pump (or hand pump, if available)

Experimental Method (Step-by-Step)

Step 1: Measuring the Mass with Air

• Inflate a balloon and tie it securely or seal air inside a flask.

• Place it on an electronic balance.

• Record the mass m1.

  $$m_1$$

[Insert diagram showing an inflated balloon placed on an electronic balance]

Step 2: Removing the Air

• Carefully remove the air from the balloon (or evacuate the flask using a pump).

• Ensure the container remains the same.

• Measure the mass again and record it as m2.

  $$m_2$$

[Insert diagram showing deflated balloon on the balance]

Step 3: Determining the Mass of Air

This small difference represents the mass of the air previously inside the container.

Step 4: Measuring the Volume of Air

• Measure the volume of the inflated balloon by:
  • submerging it in water and measuring displaced water, or
  • approximating the balloon as a sphere and calculating its volume.

[Insert diagram showing volume measurement of balloon by water displacement]

Calculating the Density of Air

Once mass and volume are known:

The density of air obtained is typically around:

under normal laboratory conditions.

Precautions and Sources of Error

• Use a sensitive balance (air has very small mass).

• Ensure no air escapes during weighing.

• Avoid drafts or air currents affecting the balance.

• Measure volume as accurately as possible.

7. Why Results May Vary

The density of air depends on:

• temperature,

• atmospheric pressure,

• humidity.

This explains why experimental values may differ slightly from the accepted value.

Common Examination Errors (Examiner Insight)

Students often:

• forget to subtract the two masses,

• describe only weighing without mentioning volume,

• state the formula without explaining measurements,

• confuse density of air with density of the container.

A clear, logical description earns full marks.

Exam-Style Questions (Original)

Question 1

Describe an experiment to determine the density of air.

Question 2

Explain why a sensitive balance is required in this experiment.

Question 3

State two factors that affect the density of air.

Worked Solutions (Beyond Excellent)

Solution 1

Weigh a container filled with air and record the mass. Remove the air and weigh the container again. The difference in mass gives the mass of air. Measure the volume of the container and calculate density by dividing the mass of air by the volume.

Solution 2

The mass of air is very small, so a sensitive balance is needed to measure the small difference in mass accurately.

Solution 3

The density of air is affected by temperature and pressure. (Humidity also affects density.)

End-of-Objective

A learner who has mastered this objective can:

• describe a practical method to find the density of air,

• explain how mass and volume are obtained,

• apply the density equation correctly,

• identify sources of error and variation.

What a Hydrometer Is

A hydrometer is a calibrated instrument used to measure the density (or relative density) of liquids directly.

It consists of:

• a sealed glass tube,

• a weighted bulb at the bottom,

• a graduated stem at the top.

The hydrometer floats upright when placed in a liquid.

Principle of Operation (Physics Behind the Instrument)

A hydrometer works on the principle of upthrust (buoyancy):

A floating object displaces a volume of liquid whose weight is equal to the weight of the object.

Because different liquids have different densities:

• a hydrometer floats higher in a denser liquid,

• a hydrometer sinks deeper in a less dense liquid.

This change in floating depth allows the density to be read directly from the scale.

Using a Hydrometer (Step-by-Step Procedure)

Apparatus

• Hydrometer

• Tall measuring cylinder

• Liquid sample

Procedure

1. Pour the liquid into a clean, tall measuring cylinder.

2. Lower the hydrometer gently into the liquid.

3. Ensure the hydrometer:
   • floats freely,
   • does not touch the sides or bottom.

4. Allow the hydrometer to come to rest.

5. Read the scale at eye level where the liquid surface touches the stem.

[Insert diagram showing a hydrometer floating in a measuring cylinder, with eye-level reading indicated]

Reading the Hydrometer Scale Correctly

• The reading is taken at the liquid surface on the stem.

• For most school hydrometers, the scale is marked in:
  • g cm³, or
  • relative density (no unit).

Avoid parallax error by ensuring the eye is level with the meniscus.

Interpreting Hydrometer Readings

• Higher reading → liquid is denser.

• Lower reading → liquid is less dense.

Examples:

• Salt water has a higher density than pure water.

• Oil has a lower density than water.

[Insert diagram comparing hydrometer positions in water and oil]

Advantages of Using a Hydrometer

• Direct reading without calculation,

• Quick and convenient,

• Suitable for liquids that cannot be easily poured for mass measurement.

7. Precautions and Sources of Error

• Ensure the hydrometer is clean and dry before use.

• Remove air bubbles sticking to the hydrometer.

• Read the scale at eye level.

• Ensure the liquid is still (no movement).

Common Examination Errors (Examiner Insight)

Students often:

• confuse hydrometer readings with mass or volume,

• read the scale from above or below eye level,

• forget to mention buoyancy or upthrust,

• state that the hydrometer sinks more in denser liquids.

Correct physical explanation earns full marks.

Exam-Style Questions (Original)

Question 1

Name the instrument used to measure the density of liquids directly.

Question 2

Describe how a hydrometer is used to measure the density of a liquid.

Question 3

Explain why a hydrometer floats higher in salt water than in pure water.

Worked Solutions (Beyond Excellent)

Solution 1

A hydrometer is used to measure the density of liquids directly.

Solution 2

The liquid is poured into a measuring cylinder, and the hydrometer is gently lowered into it. The hydrometer is allowed to float freely, and the density is read at eye level from the scale where the liquid surface touches the stem.

Solution 3

Salt water is denser than pure water, so a smaller volume of salt water is needed to provide the upthrust equal to the weight of the hydrometer. As a result, the hydrometer floats higher.

End-of-Objective

A learner who has mastered this objective can:

• describe and use a hydrometer correctly,

• explain the buoyancy principle involved,

• read density values accurately,

• interpret results for different liquids.