Problem 7 - Chemistry The Central Science In Si Units

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 heaviest. (b) Three cubes of equal mass are composed of gold (density \(\left.=19.32 \mathrm{~g} / \mathrm{cm}^{3}\right),\) platinum (density \(\left.=21.45 \mathrm{~g} / \mathrm{cm}^{3}\right),\) and lead \(\left(\right.\) density \(\left.=11.35 \mathrm{~g} / \mathrm{cm}^{3}\right)\) List the cubes from smallest to largest.

Step-by-Step Solution

Answer

(a) Aluminum, Nickel, Silver. (b) Platinum, Gold, Lead.

Step 1: Understanding the Densities of Spheres

We need to order the aluminum, silver, and nickel spheres by weight. The weight depends on the density since all spheres have the same size. Aluminum has a density of 2.70 g/cm³, silver 10.49 g/cm³, and nickel 8.90 g/cm³.

Step 1: Comparing Densities for Spheres

By comparing densities, we find that aluminum, with the smallest density, is lightest. Nickel, with a higher density than aluminum but lower than silver, is heavier than aluminum but lighter than silver. Silver, with the highest density, is the heaviest.

Step 2: Sorting Spheres by Weight

The order from lightest to heaviest is: Aluminum, Nickel, Silver.

Step 4: Understanding the Densities of Cubes

The cubes have equal mass, which means the cube with the highest density will have the smallest volume. Conversely, the cube with the lowest density will have the largest volume. The densities are gold: 19.32 g/cm³, platinum: 21.45 g/cm³, and lead: 11.35 g/cm³.

Step 3: Comparing Densities for Cubes

Since platinum has the highest density, its cube will occupy the smallest volume. Gold follows with a medium density, meaning its volume is larger than platinum but smaller than lead. Lead, with the lowest density, will occupy the largest volume.

Step 4: Sorting Cubes by Volume

The order from the smallest to largest volume is: Platinum, Gold, Lead.

Key Concepts

Mass

When we talk about mass, we're addressing how much matter an object contains. Mass is an intrinsic property of all physical objects. It does not change with location, unlike weight, which varies with gravitational forces. Mass is usually measured in kilograms (kg) or grams (g).
In this exercise, the concept of mass is used to determine the order of the cubes, which all have the same mass but differ in volume due to their varying densities.
Remember, mass is not the same as weight, but they are proportional in a constant gravitational field. Mass helps us understand the "amount" of material we have, which is crucial in many scientific calculations.

• Mass is the amount of matter in an object.
• Mass does not change based on location.
• Measured in kilograms or grams.

Volume

Volume refers to the amount of space that an object occupies. In simple terms, it's the space inside the shape or object.
Volume is often measured in cubic centimeters (cm³), cubic meters (m³), or liters (L). For shapes like spheres and cubes, mathematical formulas can calculate volume based on dimensions.
In this exercise, the volume is crucial when considering the cubes with equal mass. The material with lower density covers more space, resulting in a larger volume. On the contrary, a material with higher density occupies less space.

• Volume is the measure of space an object occupies.
• Common units: cubic centimeters, cubic meters.
• Calculated using specific formulas for different shapes.

Material Comparison

Comparing materials based on density allows us to understand how tightly packed an object's molecules are. Materials with higher densities have more tightly packed molecules compared to those with lower densities.
In this exercise, comparing aluminum, silver, and nickel helps us determine which sphere is lightest or heaviest based on density.
The same principle applies when comparing gold, platinum, and lead cubes. Higher density materials have molecules packed more closely together, resulting in less volume for the same mass.

• Density shows how tightly packed material molecules are.
• High density means more tightly packed molecules.
• Materials with the same mass can have different volumes due to density differences.

Spheres and Cubes

Spheres and cubes are two common three-dimensional shapes. Understanding their properties is essential for calculating volume or comparing materials based on density.**Spheres:**The volume of a sphere is calculated by the formula \(V = \frac{4}{3} \pi r^3\) where \(r\) is the radius. Materials like aluminum, silver, and nickel can be compared when formed into spheres to determine weight differences based on density.**Cubes:**The volume of a cube is determined using \(V = a^3\), with \(a\) being the length of a side. In this scenario, identical mass cubes made of gold, platinum, and lead can be compared. Their volume differences are tied directly to their densities.

• Spherical volumes: \(V = \frac{4}{3} \pi r^3\).
• Cubic volumes: \(V = a^3\).
• Shape affects how we calculate and compare material properties.

Weight Ordering

Weight ordering involves arranging objects based on their weight. This concept is often related to understanding an object's density and volume when masses are constant among items compared. When comparing the spheres of aluminum, silver, and nickel, weight ordering helps deduce which sphere is lightest or heaviest. Order is established by examining densities since all spheres share the same volume but different masses due to varied densities. Similarly, cubes of the same mass from different materials, like gold and platinum, have volumes inversely related to their densities, making weight ordering a bit different. We focus on volume sorting here because the mass is constant.

• Order based on weight using density and volume concepts.
• Weight ordering relies heavily on density comparison when size is constant.
• Volume-based ordering applies when masses are fixed, as in the cubic example.