7.2 Measuring Densities of Different Substances

In the previous section, we learned that density is mass per unit volume (ρ = m/V). To find the density of a substance, we need to measure both its mass and its volume. However, the method for measuring volume depends on whether the substance is a liquid, a regularly shaped solid, or an irregularly shaped solid. In this section, we will explore the practical methods for measuring the densities of different types of substances.

Definition

Measuring Density is the process of determining the density of a substance by experimentally measuring its mass and volume and then calculating the ratio of mass to volume.

Main Idea

The main idea is that the method for measuring density depends on the type of substance (liquid, regular solid, or irregular solid). For liquids, you measure the volume using a measuring cylinder and the mass using a balance. For regularly shaped solids, you can calculate the volume using geometric formulas (e.g., V = l × w × h for a cuboid, V = s³ for a cube). For irregularly shaped solids, you use the displacement method to measure volume. In all cases, the density is calculated as mass divided by volume.

Core Concept

The text describes three methods for measuring density:

1. Density of Liquids The text states: "To measure density of liquids, we need a graduated measuring cylinder to measure volume of liquid, a balance to measure mass of liquids. We will measure density of liquid using following steps:"

• Place the empty measuring cylinder on a balance and measure its mass.
• Add liquid to the measuring cylinder and measure its volume.
• Subtract the mass of the empty cylinder from the mass of the cylinder and liquid to get the mass of the liquid.
• Measure the volume of the liquid from the measuring cylinder.
• Calculate the density: ρ = Mass of liquidVolume of liquid.

2. Density of Regularly Shaped Solids The text states: "To find density of regularly shaped solids (like solid cubes, solid cuboids, solid sphere etc.), we will follow these steps:"

• Cuboid: V = length × width × height
• Cube: V = (side length)³
• Sphere: V = 43π r³

3. Density of Irregularly Shaped Solids (Displacement Method) The text states: "We can find the density of irregular shaped solid objects like stones which can sink in the water. We need graduated measuring cylinder to measure volume and balance to measure mass of object. We will follow these steps to find its density:"

• Find the mass of the irregularly shaped object.
• Add some water to a measuring cylinder and measure the initial volume (V_i).
• Tie a thread to the object and lower it into the measuring cylinder.
• The water level rises. Measure the final volume (V_f).
• Calculate the volume of the object: V = V_f - V_i.
• Calculate the density: ρ = Mass of objectVolume of object.

Detailed Explanation

Measuring density is a common laboratory procedure that requires careful measurement of both mass and volume. The method chosen depends on the nature of the substance.

Density of Liquids Measuring the density of a liquid is straightforward. The key is to account for the mass of the container (the measuring cylinder). By measuring the mass of the empty cylinder first, you can subtract it from the total mass to find the mass of the liquid alone.

• Mass of empty cylinder: 50 g
• Mass of cylinder + liquid: 150 g
• Mass of liquid: 150 - 50 = 100 g
• Volume of liquid: 100 mL
• Density: 100 g / 100 mL = 1 g/mL

Density of Regularly Shaped Solids For regularly shaped solids, the volume can be calculated using geometric formulas. This is the easiest method because it does not require a measuring cylinder.

• A cube of side 5 cm has a volume of 5³ = 125 cm³.
• If its mass is 500 g, its density is 500125 = 4 g/cm³.

Density of Irregularly Shaped Solids (Displacement Method) For irregularly shaped solids (like stones), the volume cannot be calculated using a simple formula. Instead, we use the displacement method, which is based on the fact that an object submerged in a fluid displaces a volume of fluid equal to its own volume.

• Mass of rock: 80.52 g
• Initial volume of water: 18 mL
• Final volume of water: 46 mL
• Volume of rock: 46 - 18 = 28 mL = 28 cm³
• Density: 80.5228 ≈ 2.88 g/cm³ (Note: The text's calculation gives 3.22 g/cm³ using 25 cm³, which appears to be a typo in the source. I will follow the correct calculation based on the provided values: 80.5228 = 2.88 g/cm³.)

• Mass measurement: Use a balance (electronic or physical) for accurate mass measurement.
• Volume measurement: Use a measuring cylinder for liquids and irregular solids. Read the meniscus at eye level to avoid parallax error.
• Units: Ensure mass and volume are in compatible units (e.g., grams and cm³, or kilograms and m³).
• Temperature: Density changes with temperature, so measurements should be taken at a known temperature (usually room temperature).

Intermediate Level

In physics, measuring density requires precise measurements of mass and volume. For liquids, the mass is found by subtracting the mass of the empty container from the mass of the container with the liquid. The volume is read directly from a measuring cylinder, taking care to read the bottom of the meniscus at eye level.

For regularly shaped solids, volume is calculated using geometric formulas (e.g., V = s³ for a cube, V = 43π r³ for a sphere). The mass is measured using a balance.

For irregularly shaped solids, the displacement method is used. The object is submerged in water in a measuring cylinder, and the volume of water displaced (final volume minus initial volume) equals the volume of the object. This method works for objects that sink in water and do not dissolve or absorb water.

Density is then calculated as ρ = m/V. It is important to ensure that the units of mass and volume are consistent (e.g., g and cm³, or kg and m³).

Important Points

• - Density of liquids: Measure mass (using a balance) and volume (using a measuring cylinder).
• - Density of regular solids: Measure mass (balance) and volume (using geometric formulas).
• - Density of irregular solids: Measure mass (balance) and volume (displacement method).
• - Displacement method: Volume of object = Final volume of water - Initial volume of water.
• - Meniscus: Read the bottom of the meniscus at eye level.
• - Units: Ensure consistent units (g/cm³ or kg/m³).

Remember

• - Liquids: Mass (balance) + Volume (cylinder).
• - Regular Solids: Mass (balance) + Volume (geometry).
• - Irregular Solids: Mass (balance) + Volume (displacement).
• - Displacement: V_object = V_final - V_initial.
• - Density formula: ρ = m/V.

Common Mistakes

• - Forgetting to subtract the mass of the container: When measuring the mass of a liquid, always subtract the mass of the empty container.
• - Reading the wrong part of the meniscus: Read the bottom of the meniscus (for water) at eye level.
• - Using inconsistent units: Ensure mass and volume are in compatible units.
• - Not drying the irregular solid: If the solid is wet when weighed, the mass will be too high. Dry it first.

Quick Revision

To measure the density of a substance, you need to determine both its mass and its volume. For liquids, measure the mass using a balance and the volume using a measuring cylinder. For regularly shaped solids, calculate the volume using geometric formulas. For irregularly shaped solids, use the displacement method: submerge the object in water in a measuring cylinder and measure the volume of water displaced. Then, calculate density as mass divided by volume (ρ = m/V).

60-Second Revision

• Liquid: Mass (balance) + Volume (cylinder).
• Regular Solid: Mass (balance) + Volume (geometry).
• Irregular Solid: Mass (balance) + Volume (displacement).
• Displacement: V = V_f - V_i.
• Density: ρ = m/V.
• Read meniscus at eye level.

Frequently Asked Questions (FAQs)

Q1. How do you measure the density of a liquid? Answer: Measure the mass of an empty measuring cylinder. Add the liquid and measure the total mass. Subtract the mass of the empty cylinder to get the mass of the liquid. Read the volume of the liquid from the cylinder. Calculate density as mass/volume.

Q2. How do you measure the density of a regularly shaped solid? Answer: Measure the mass using a balance. Calculate the volume using the appropriate geometric formula (e.g., V = s³ for a cube). Calculate density as mass/volume.

Q3. How do you measure the density of an irregularly shaped solid? Answer: Measure the mass using a balance. Use the displacement method: put water in a measuring cylinder, note the initial volume, submerge the object, and note the final volume. The volume of the object is the difference between the final and initial volumes. Calculate density as mass/volume.

Q4. What is the displacement method? Answer: The displacement method is a technique for measuring the volume of an irregularly shaped object. The object is submerged in a fluid (usually water) in a measuring cylinder, and the volume of fluid displaced (the rise in water level) equals the volume of the object.

• Q5. Why is it important to read the meniscus at eye level?
• Answer: Reading the meniscus at eye level avoids parallax error, which can cause the volume reading to be too high or too low.

• Q6. What is the density of water?
• Answer: The density of pure water is approximately 1 g/cm³ (or 1000 kg/m³) at 4°C.

Q7. How do you measure the mass of a liquid? Answer: Measure the mass of the empty container, then measure the mass of the container with the liquid. Subtract the mass of the empty container from the total mass to get the mass of the liquid.

Multiple Choice Questions (MCQs)

• Q1. To measure the density of a liquid, you need:
• A. A ruler and a stopwatch
• B. A measuring cylinder and a balance
• C. A thermometer and a barometer
• D. A voltmeter and an ammeter

• Correct Answer: B
• Explanation: The text states that to measure the density of liquids, we need a graduated measuring cylinder and a balance.

• Q2. The volume of a regularly shaped solid is calculated using:
• A. A measuring cylinder
• B. Geometric formulas
• C. The displacement method
• D. A thermometer

• Correct Answer: B
• Explanation: The text states that for regularly shaped solids, the volume is calculated using formulas according to the shape of the object.

• Q3. The displacement method is used to measure the volume of:
• A. Liquids
• B. Regularly shaped solids
• C. Irregularly shaped solids
• D. Gases

• Correct Answer: C
• Explanation: The text states that the displacement method is used for irregularly shaped solid objects like stones.

• Q4. In the displacement method, the volume of the object is equal to:
• A. The initial volume of water
• B. The final volume of water
• C. The difference between the final and initial volumes
• D. The sum of the initial and final volumes

• Correct Answer: C
• Explanation: The text states that the volume of the object is found by subtracting the initial volume from the final volume.

• Q5. When measuring the mass of a liquid, you must:
• A. Add the mass of the container
• B. Subtract the mass of the empty container
• C. Ignore the mass of the container
• D. Multiply the mass by the volume

• Correct Answer: B
• Explanation: The mass of the liquid is the mass of the cylinder and liquid minus the mass of the empty cylinder.

• Q6. The formula for density is:
• A. ρ = m × V
• B. ρ = m / V
• C. ρ = V / m
• D. ρ = m + V

• Correct Answer: B
• Explanation: Density is mass per unit volume, ρ = m/V.

• Q7. A rock has a mass of 80 g. Its volume, measured by displacement, is 20 cm³. Its density is:
• A. 2 g/cm³
• B. 4 g/cm³
• C. 8 g/cm³
• D. 16 g/cm³

• Correct Answer: B
• Explanation: ρ = 8020 = 4 g/cm³.

• Q8. Which of the following is NOT needed to measure the density of an irregular solid?
• A. A balance
• B. A measuring cylinder
• C. A ruler
• D. Water

• Correct Answer: C
• Explanation: A ruler is not needed for an irregular solid because the volume is measured by displacement, not by geometric measurement.

• Q9. The meniscus of water in a measuring cylinder is:
• A. Flat
• B. Curved upward
• C. Curved downward
• D. Straight

• Correct Answer: C
• Explanation: (This is a standard fact.) The meniscus of water is curved downward (concave). It should be read at the bottom of the curve.

• Q10. A liquid has a mass of 150 g and a volume of 100 mL. Its density is:
• A. 0.5 g/mL
• B. 1.0 g/mL
• C. 1.5 g/mL
• D. 2.0 g/mL

• Correct Answer: C
• Explanation: ρ = 150100 = 1.5 g/mL.

Frequently Asked Questions (FAQs)

Q1. Why is the displacement method not suitable for all solids? Answer: The displacement method is not suitable for solids that float (they won't submerge), dissolve in water (they will affect the volume reading), or absorb water (their mass will change). For such solids, alternative methods (e.g., using a liquid in which they are insoluble) may be needed.

Q2. How do you measure the density of a gas? Answer: Measuring the density of a gas is more complex. It typically involves measuring the mass of a known volume of gas at a known pressure and temperature, using a gas syringe or a flask of known volume.

• Q3. What is the density of a substance if its mass is 200 g and its volume is 50 cm³?
• Answer: ρ = 20050 = 4 g/cm³.

• Q4. Why is it important to use a balance with a small least count?
• Answer: A balance with a small least count (e.g., 0.01 g) allows for more precise mass measurements, which leads to a more accurate density calculation.

Q5. How does temperature affect the density of a liquid? Answer: In general, heating a liquid causes it to expand (volume increases), so its density decreases. Cooling causes it to contract (volume decreases), so its density increases.

• Q6. What is the density of a cube with side 3 cm and mass 81 g?
• Answer: Volume = 3³ = 27 cm³. Density = 8127 = 3 g/cm³.

Q7. What is the difference between mass and density? Answer: Mass is the amount of matter in an object (measured in kg or g). Density is the mass per unit volume (measured in kg/m³ or g/cm³). Mass is an extensive property (depends on size), while density is an intensive property (does not depend on size).

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