Question

In each of the following pairs, which has the greater volume? a. \(1.0 \mathrm{~kg}\) of feathers or \(1.0 \mathrm{~kg}\) of lead b. \(100 \mathrm{~g}\) of gold or \(100 \mathrm{~g}\) of water c. \(1.0 \mathrm{~L}\) of copper or \(1.0 \mathrm{~L}\) of mercury

Short answer

a. Feathers have a greater volume. b. Water has a greater volume. c. Copper and Mercury have the same volume.

Step-by-step solution

Gather density values

To determine the volume, we need the density values for each substance. Here are their densities: - Feathers: \(1.2 \mathrm{~kg/m^3}\) - Lead: \(11,340 \mathrm{~kg/m^3}\) - Gold: \(19,320 \mathrm{~kg/m^3}\) - Water: \(1,000 \mathrm{~kg/m^3}\) - Copper: \(8,960 \mathrm{~kg/m^3}\) - Mercury: \(13,534 \mathrm{~kg/m^3}\)

Calculate the volumes

Now, we can use the densities to find the volumes: a. Feathers, \(V_f = \frac{1.0\mathrm{~kg}}{1.2 \mathrm{~kg/m^3}} = 0.833\mathrm{~m^3}\), and Lead, \(V_l = \frac{1.0\mathrm{~kg}}{11,340 \mathrm{~kg/m^3}} = 8.81 \times 10^{-5}\mathrm{~m^3}\). b. Gold, \(V_g = \frac{0.1\mathrm{~kg}}{19,320 \mathrm{~kg/m^3}} = 5.17 \times 10^{-6}\mathrm{~m^3}\), and Water, \(V_w = \frac{0.1\mathrm{~kg}}{1,000 \mathrm{~kg/m^3}} = 0.0001\mathrm{~m^3}\).Note that we converted 100g of gold and water to 0.1kg each. c. Copper, \(V_c = 1.0\mathrm{~L} = 0.001\mathrm{~m^3}\) (since 1 L = 0.001 m³), and Mercury, \(V_m = 1.0\mathrm{~L} = 0.001\mathrm{~m^3}\).

Compare the volumes

Lastly, we'll compare the volumes: a. Feathers have a volume of \(0.833\mathrm{~m^3}\), while Lead has a volume of \(8.81 \times 10^{-5}\mathrm{~m^3}\). Feathers have a greater volume than lead. b. Gold has a volume of \(5.17 \times 10^{-6}\mathrm{~m^3}\), while Water has a volume of \(0.0001\mathrm{~m^3}\). Water has a greater volume than gold. c. Both Copper and Mercury have a volume of \(0.001\mathrm{~m^3}\), so their volumes are equal. Final answers: a. Feathers have a greater volume. b. Water has a greater volume. c. Copper and Mercury have the same volume.

Key concepts

Volume Calculation

Calculating volume is an essential step in understanding the relationship between mass and density. Volume is defined as the amount of three-dimensional space an object occupies. To find the volume, we need to know the object's mass and density:

• Mass is the amount of matter in an object, typically measured in kilograms (kg) or grams (g).
• Density is the mass per unit volume and is measured in kilograms per cubic meter (kg/m³).

The formula to calculate volume when you have mass and density is:\[ V = \frac{m}{\rho} \]where \( V \) is volume, \( m \) is mass, and \( \rho \) is density. In practical terms, this means that for the same mass, a substance with a lower density will occupy more volume.

Mass-to-Volume Conversion

Converting mass to volume requires using the density of a substance. This process is essential when comparing how different materials behave under the same mass. The conversion involves using the formula:

• Consider the mass you have.
• Look up the density of the material.
• Use the formula \( V = \frac{m}{\rho} \) to calculate the volume.

Take the exercise example where we compare 1.0 kg of feathers and 1.0 kg of lead:

• Feathers: With a density of 1.2 kg/m³, the volume is \( \frac{1.0 \text{ kg}}{1.2 \text{ kg/m}^3} = 0.833 \text{ m}^3 \).
• Lead: With a density of 11,340 kg/m³, the volume is \( \frac{1.0 \text{ kg}}{11,340 \text{ kg/m}^3} = 8.81 \times 10^{-5} \text{ m}^3 \).

This highlights how a lower density results in a greater volume for an equivalent mass.

Comparison of Volumes

Comparing volumes allows us to better understand the physical differences among materials for a given mass or volume. This concept is crucial when evaluating material properties for practical applications. In the exercise, comparing volumes helps determine which of the two elements in each pair has a greater capacity to occupy space:

• Feathers vs Lead: Even though both have the same mass, feathers take up much more space due to their lower density.
• Gold vs Water: Here, despite having the same mass, water has a greater volume than gold, again due to density differences.
• Copper vs Mercury: Given the same volume, both elements occupy the same physical space, highlighting that their densities do not influence their volume in this case.

Understanding these comparisons helps in fields such as material science and engineering, where managing space and material choice is key.