Question

(a) Three spheres of equal size are composed of aluminum \(\left(\right.\) density \(\left.=2.70 \mathrm{~g} / \mathrm{cm}^{3}\right),\) silver \(\left(\right.\) density \(\left.=10.49 \mathrm{~g} / \mathrm{cm}^{3}\right),\) and nickel (density \(\left.=8.90 \mathrm{~g} / \mathrm{cm}^{3}\right)\). List the spheres from lightest to (b) Three cubes of equal mass are composed of gold \(\left(\right.\) density \(\left.=19.32 \mathrm{~g} / \mathrm{cm}^{3}\right)\), platinum (density \(\left.=21.45 \mathrm{~g} / \mathrm{cm}^{3}\right)\) and lead (density \(\left.=11.35 \mathrm{~g} / \mathrm{cm}^{3}\right)\). List the cubes from smallest to largest. [Section 1.4]

Short answer

(a) The order of spheres from lightest to heaviest based on density is: aluminum, nickel, and silver. (b) The order of cubes from smallest to largest volume based on equal mass and reciprocal densities is: platinum, gold, and lead.

Step-by-step solution

Since the spheres have the same volume, we can use the mass and density relationship: mass = density × volume. To compare their masses, we can ignore the volume factor and simply compare their densities because volume is the same for all spheres. 2. Compare the densities

We are given the densities: aluminum (2.70 g/cm³), silver (10.49 g/cm³), and nickel (8.90 g/cm³). Comparing the densities, we find the order is aluminum < nickel < silver. 3. List the spheres from lightest to heaviest

Based on density comparison, the order from lightest to heaviest is: aluminum, nickel, and silver. #Part (b)#: 1. Determine the relationship between mass, density, and volume for equal mass cubes

Since the cubes have the same mass, we can use the mass and density relationship: mass = density × volume. To compare their volumes, we can rearrange the formula to: volume = mass / density. In this case, we can ignore the mass factor and just compare the reciprocal of their densities because mass is the same for all cubes. 2. Compare the reciprocal of the densities

We are given the densities: gold (19.32 g/cm³), platinum (21.45 g/cm³), and lead (11.35 g/cm³). Taking the reciprocal of the densities, we have: gold (1/19.32), platinum (1/21.45), and lead (1/11.35). Comparing these values, we find the order is platinum > gold > lead. 3. List the cubes from smallest to largest

Based on the reciprocal density comparison, the order from smallest to largest volume is: platinum, gold, and lead.

Key concepts

Mass and Volume Relationship

Understanding the connection between mass and volume is essential in many scientific calculations. Mass is the measure of the amount of matter in an object, while volume describes the space that the object occupies.
The relationship between mass, volume, and density is expressed by the formula: \( \text{mass} = \text{density} \times \text{volume} \).
This equation shows that if you know any two of these quantities, you can find the third. In scenarios where items have the same volume, like the spheres in the problem, their mass is directly proportional to their density.
Thus, knowing the density value of a material allows you to compare masses without calculating the actual mass. On the other hand, if you hold mass constant, as in the cube scenario, you can uncover relative volumes by comparing densities.

Metal Densities

Metals are often characterized by their densities, which is a critical physical property. Density is defined as mass per unit volume, expressed as \( \text{g/cm}^3 \) in many standard contexts.
Let's consider the metals mentioned:

• Aluminum: Lightweight, with a density of 2.70 g/cm³.
• Silver: Dense compared to aluminum, at 10.49 g/cm³.
• Nickel: Falls between the two, with a density of 8.90 g/cm³.
• Gold: Very dense at 19.32 g/cm³.
• Platinum: Even denser than gold, with 21.45 g/cm³.
• Lead: Relatively dense, at 11.35 g/cm³.

These densities influence how the metals behave in applications and how we perceive their weight when dealing with objects of similar size or mass.

Density Comparison

Comparing densities is a practical approach when determining how metals relate to each other in terms of mass and volume, especially when one of the physical characteristics (mass or volume) is held constant.
In our exercise, for items of the same volume like spheres, denser metals like silver will weigh more than less dense ones like aluminum.
When dealing with equal mass objects like cubes, we use the reciprocal of their densities to find which occupies more space. This is because denser materials will occupy less volume for a given mass. Thus:

• Platinum, being the densest, has a small volume compared to the others for the same mass.
• Lead has the largest volume among the group, indicating it's the least dense of those compared when holding mass constant.

Physical Properties of Metals

Metals have distinct physical properties that can determine their use in different applications. Density is one such property, but others include:

• Conductivity: Most metals are excellent conductors of electricity and heat, making them ideal for wiring and cookware.
• Malability: Many metals can be stretched and shaped without breaking.
• Strength: Metals like nickel are chosen for their ability to withstand tension and compression.
• Luster: Metals like silver and gold are valued for their shiny appearances.

The interplay of these properties with density provides a complete picture of why certain metals are preferred for specific tasks or environments. Each property, in conjunction with density, dictates how a metal is exploited in practical and industrial settings.