STEM Calculators

Slide presentation

Density of Irregular Solids

General Chemistry • Matter, Its Properties, and Measurement

Open calculator Open questions
Slide 1 of 1

Matter, properties, and measurement

Irregular solids need displacement to reveal their volume

A cube or cylinder can be measured with a ruler, but a rock, metal fragment, or unknown solid often has no simple geometric formula. Water displacement lets the object’s volume be measured directly.

Learning target

  • Use mass and displaced volume to calculate density.
  • Explain how water displacement measures irregular volume.
  • Connect laboratory readings to \(d = m/V\).
  • Evaluate accuracy, units, and significant figures.
Irregular solid density overview A balance measures mass and a graduated cylinder measures displaced water volume for an irregular solid. measure mass balance reading in grams measure displaced volume final volume − initial volume mass and volume

Why it matters

Water displacement makes real lab objects measurable

Many solid samples in chemistry are not perfect blocks or spheres. Displacement is useful because it turns an irregular shape into a measurable volume change in a graduated cylinder.

Identification

Compare unknown solids

A measured density can help compare an unknown solid with possible materials such as metals, minerals, or plastics.

Laboratory skill

Use readings correctly

The method requires careful initial and final volume readings, plus correct subtraction.

Measurement check

Evaluate accuracy

Air bubbles, splashing, parallax, and incomplete submersion can change the displaced volume.

Irregular shape
No simple length × width × height formula applies.

use

Water displacement
The volume of water displaced equals the object’s volume.

Core concept

The object’s volume equals the rise in water level

When a solid is fully submerged and does not dissolve or react, it pushes water out of the space it occupies. The increase in measured water volume is the volume of the solid.

Initial and final water volume model Two graduated cylinders show initial water level and higher final water level after adding an irregular solid. Initial volume water only Final volume water + solid water level rises rise = solid volume

Displacement relationship

The displaced volume is found by subtracting the initial water volume from the final water volume.

\[ V_{\text{solid}} = V_{\text{final}} - V_{\text{initial}} \]

Once the volume is known, density can be calculated using the measured mass of the solid.

Vocabulary and units

Every density measurement needs mass, volume, and units

The water displacement method combines a balance reading with two graduated-cylinder readings.

Quantity or term Meaning Common units Laboratory note
Mass, \(m\) Amount of matter in the irregular solid. g Measure on a balance before placing the solid into water.
Initial volume, \(V_{\text{initial}}\) Water volume before the solid is added. mL Read the bottom of the meniscus at eye level.
Final volume, \(V_{\text{final}}\) Water volume after the solid is fully submerged. mL Make sure the object is completely underwater and no water is lost.
Displaced volume The volume of the irregular solid. mL or cm3 For water displacement, \(1\ \text{mL} = 1\ \text{cm}^{3}\).

Unit habit

If mass is measured in grams and displaced volume in milliliters, the density unit is g/mL. For solids, g/cm3 is equivalent when volume is from water displacement.

Main relationship

Density uses the displaced volume, not the final water volume

The final cylinder reading includes the original water and the object’s volume. The object’s volume is only the difference between final and initial readings.

Density of an irregular solid

\[ d = \frac{m}{V_{\text{final}} - V_{\text{initial}}} \]
Measure mass

\(m\) comes from the balance.

Find volume

\(V_{\text{solid}} = V_{\text{final}} - V_{\text{initial}}\)

Calculate density

\(d = m/V_{\text{solid}}\)

Correct

Use displaced volume

If water rises from 40.0 mL to 47.2 mL, the solid’s volume is 7.2 mL.

Incorrect

Do not use total final volume

The final reading is not the solid’s volume. It includes the starting water plus the solid.

Interactive displacement model

Adjust mass and water displacement to calculate density

Move the sliders. The cylinder shows initial water level, final water level, displaced volume, and the resulting density.

Final volume 47.2 mL
Density 7.50 g/mL
Material range dense solid

The solid displaces 7.2 mL of water, so a 54.0 g sample has a density of 7.50 g/mL.

Interactive water displacement simulation A graduated cylinder shows initial water level, final water level after adding an irregular solid, and density readouts. initial 40.0 mL 80 mL 50 mL 20 mL relative density high low 7.50 g/mL final volume = 47.2 mL displaced volume = 7.2 mL

Dynamic relationship

For a fixed mass, larger displaced volume means lower density

If two irregular objects have the same mass, the one that displaces more water has a larger volume and therefore a lower density.

Choose the fixed mass

For a 50 g object, density decreases as displaced volume increases because the same mass occupies more space.

Density versus displaced volume graph Three inverse curves show density decreasing as displaced volume increases for different fixed masses. displaced volume density high low small large 90 g 50 g 20 g same displaced volume comparison

Worked example

Calculate density from water displacement

An irregular metal sample has a mass of \(54.0\ \text{g}\). A graduated cylinder initially contains \(40.0\ \text{mL}\) of water. After the metal is submerged, the final volume is \(47.2\ \text{mL}\). Find the density.

  1. Identify the mass.

    The balance gives \(m = 54.0\ \text{g}\).

  2. Find the displaced volume.

    \(V_{\text{solid}} = V_{\text{final}} - V_{\text{initial}} = 47.2\ \text{mL} - 40.0\ \text{mL} = 7.2\ \text{mL}\).

  3. Use the density formula.

    \(d = m/V = 54.0\ \text{g} / 7.2\ \text{mL}\).

  4. Calculate the density.

    \(d = 7.5\ \text{g/mL}\).

  5. Check significant figures.

    The displaced volume has 2 significant figures, so the final density should be reported with 2 significant figures.

Final answer

The density of the irregular metal sample is \(7.5\ \text{g/mL}\), equivalent to \(7.5\ \text{g/cm}^{3}\).

Common misconception

The final cylinder reading is not the solid’s volume

A frequent error is to divide mass by the final water reading. That gives a density that is too small because the final reading includes water that was already in the cylinder.

Mistake

“The final reading is \(47.2\ \text{mL}\), so the object’s volume is \(47.2\ \text{mL}\).”

Correction

The object’s volume is the increase in water level: \(47.2\ \text{mL} - 40.0\ \text{mL} = 7.2\ \text{mL}\).

Final volume misconception A comparison shows the wrong method using final volume and the correct method using displaced volume. Incorrect 54.0 g / 47.2 mL Uses water + object volume. Density is underestimated. Correct 54.0 g / 7.2 mL Uses only displaced volume. Density matches the solid. Always subtract initial volume before using the density formula.

Practice check

Use displacement to calculate density

A small irregular solid has a mass of \(32.6\ \text{g}\). The water level in a graduated cylinder rises from \(28.4\ \text{mL}\) to \(34.1\ \text{mL}\) when the object is fully submerged.

Question

What is the volume of the solid, and what is its density? Report the density with the correct significant figures.

Show answer

Displaced volume: \(34.1\ \text{mL} - 28.4\ \text{mL} = 5.7\ \text{mL}\).

Density: \(d = 32.6\ \text{g} / 5.7\ \text{mL} = 5.719...\ \text{g/mL}\).

The displaced volume has 2 significant figures, so the density is \(5.7\ \text{g/mL}\).

Reasoning check

The density is greater than water’s density, so this solid would be expected to sink if it does not trap air and does not react with water.

Apply the topic

Use displacement reasoning before calculating

A strong solution explains how the volume was found, not only how the density was calculated. Always identify whether the volume came from geometry or water displacement.

Open calculator Density of Irregular Solids Calculator

Calculate density using mass, initial volume, final volume, and displaced volume.

 Practice questions Density of Irregular Solids Questions

Practice displacement, units, significant figures, and lab-measurement reasoning.

How to apply this topic

Measure mass first. Read the initial water volume. Submerge the object completely. Read the final volume. Subtract to find object volume. Then calculate density and check units.

Final summary

The essential takeaways

Irregular objects need measured volume.

Water displacement finds volume when geometry formulas are not practical.

Displaced volume is a difference.

Use \(V_{\text{solid}} = V_{\text{final}} - V_{\text{initial}}\).

Density is mass divided by volume.

Use \(d = m/V\), where \(V\) is the solid’s displaced volume.

Use units carefully.

For water displacement, mL and cm3 are equivalent volume units.

Measurement technique affects accuracy.

Avoid parallax, air bubbles, splashing, and incomplete submersion.

Round from measured values.

Significant figures in the final density come from the limiting measurement.