Determine the number of atoms in 1 cm³ of copper.

An automobile company displays a die-cast model of its first car, made from 9.35 kg of iron. To celebrate its hundredth year in business, a worker will recast the model in solid gold from the original dies. What mass of gold is needed to make the new model?

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• Reference
• Question
• Explain Question
• Explain Solve
• Summary
• Solve

Reference

The problem is from "Serway & Jewett Physics For Scientists And Engineers With Modern Physics 9ed". This comprehensive textbook covers various topics in physics, including mechanics, thermodynamics, electromagnetism, and modern physics. It is widely used in undergraduate physics courses and provides a solid foundation for students in scientific and engineering fields.

Question

An automobile company displays a die-cast model of its first car, made from 9.35 kg of iron. To celebrate its hundredth year in business, a worker will recast the model in solid gold from the original dies. What mass of gold is needed to make the new model?

Explain Question

This question deals with the concept of density and its relation to mass and volume. We need to find the mass of gold required to create a model with the same volume as the original iron model. The key points to consider are:

• The original model is made of iron and has a mass of 9.35 kg.
• The new model will be made of gold.
• Both models have the same volume (as they use the same dies).
• We need to calculate the mass of gold required.

To solve this problem, we'll need to use the density formula and the densities of iron and gold.

Explain Solve

Let's break down the solution into steps:

1. Recall the density formula: ρ = m / V, where ρ is density, m is mass, and V is volume.
2. Express the volume of the iron model using its mass and density.
3. Use the fact that the volume of the gold model will be the same as the iron model.
4. Set up an equation relating the densities and masses of both models.
5. Solve for the mass of gold required.

Step 1: We use ρ = m / V for both iron and gold models.
Step 2: For iron, V = 9.35 kg / ρ_iron
Step 3: V_gold = V_iron
Step 4: ρ_gold / ρ_iron = m_gold / 9.35 kg
Step 5: m_gold = (ρ_gold / ρ_iron) × 9.35 kg

Now, let's plug in the values:
ρ_iron = 7.87 × 10³ kg/m³
ρ_gold = 19.3 × 10³ kg/m³

m_gold = (19.3 × 10³ kg/m³ / 7.87 × 10³ kg/m³) × 9.35 kg
m_gold = 2.45 × 9.35 kg
m_gold = 22.9 kg

Summary

To solve this problem, we used the concept of density and the fact that both models have the same volume. We set up a ratio of densities equal to the ratio of masses, which allowed us to calculate the mass of gold needed. The key was recognizing that the volume remains constant while the density changes, leading to a change in mass.

Solve

Let V represent the volume of the model, the same in ρ = m/V, for both. Then ρ_iron = 9.35 kg/V and ρ_gold = m_gold/V. Next, ρ_gold/ρ_iron = m_gold/9.35 kg and m_gold = (ρ_gold/ρ_iron) × 9.35 kg = (19.3 × 10³ kg/m³ / 7.87 × 10³ kg/m³) × 9.35 kg = 22.9 kg

To answer briefly and in an organized way:

1. Use the density formula: ρ = m/V
2. Set up the equation: ρ_gold/ρ_iron = m_gold/m_iron
3. Substitute known values: 19.3 × 10³ kg/m³ / 7.87 × 10³ kg/m³ = m_gold / 9.35 kg
4. Solve for m_gold: m_gold = 22.9 kg

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