Three Cubes Of Equal Mass Are Composed Of Gold (density 19.32 G/cm3), Platinum (density = 21.45 G/cm3), And Lead (density 11.35 G/cm3). List The Cubes From Smallest To Largest. How Many Atoms Are Contained In The Largest Cube?
Understanding the Relationship Between Density and Volume
When dealing with objects of equal mass but different densities, it's essential to consider the relationship between density and volume. Density is defined as the mass per unit volume of a substance, and it's calculated using the formula: density = mass/volume. Since the three cubes have equal masses, the one with the highest density will have the smallest volume, and the one with the lowest density will have the largest volume.
Calculating the Volume of Each Cube
To determine the order of the cubes from smallest to largest, we need to calculate their volumes. We can use the formula: volume = mass/density. Since the masses are equal, we can set up a proportion to compare the volumes of the cubes.
- Gold cube: volume = mass/density = (mass) / 19.32 g/cm^3
- Platinum cube: volume = mass/density = (mass) / 21.45 g/cm^3
- Lead cube: volume = mass/density = (mass) / 11.35 g/cm^3
Since the masses are equal, we can simplify the proportions by canceling out the mass term. This leaves us with:
- Gold cube: volume ∝ 1 / 19.32 g/cm^3
- Platinum cube: volume ∝ 1 / 21.45 g/cm^3
- Lead cube: volume ∝ 1 / 11.35 g/cm^3
To compare the volumes, we can calculate the reciprocals of the densities:
- Gold cube: volume ∝ 1 / 19.32 g/cm^3 ≈ 0.0517 cm^3
- Platinum cube: volume ∝ 1 / 21.45 g/cm^3 ≈ 0.0466 cm^3
- Lead cube: volume ∝ 1 / 11.35 g/cm^3 ≈ 0.0884 cm^3
Based on these calculations, the order of the cubes from smallest to largest is:
- Platinum cube
- Gold cube
- Lead cube
Calculating the Number of Atoms in the Largest Cube
To calculate the number of atoms in the largest cube, we need to know the number of atoms per unit volume of lead. This value is known as the number density of lead. The number density of lead is approximately 5.86 x 10^28 atoms/m^3.
We can use this value to calculate the number of atoms in the largest cube:
Number of atoms = number density x volume
First, we need to convert the volume of the lead cube from cubic centimeters to cubic meters:
Volume (m^3) = 0.0884 cm^3 x (1 m / 100 cm)^3 ≈ 8.84 x 10^-8 m^3
Now, we can calculate the number of atoms:
Number of atoms = 5.86 x 10^28 atoms/m^3 x 8.84 x 10^-8 m^3 ≈ 5.18 x 10^21 atoms
Therefore, the largest cube contains approximately 5.18 x 10^21 atoms.
Conclusion
In conclusion, the order of the cubes from smallest to largest is platinum, gold, and lead. The largest cube, composed of lead, contains approximately 5.18 x 10^21 atoms. This calculation demonstrates the relationship between density and volume and highlights the importance of considering these factors when working with objects of equal mass but different densities.
Frequently Asked Questions About the Relationship Between Density and Volume
Q: What is density, and how is it related to volume?
A: Density is a measure of the mass per unit volume of a substance. It's calculated using the formula: density = mass/volume. Since the three cubes have equal masses, the one with the highest density will have the smallest volume, and the one with the lowest density will have the largest volume.
Q: Why is it important to consider the relationship between density and volume when working with objects of equal mass but different densities?
A: When dealing with objects of equal mass but different densities, it's essential to consider the relationship between density and volume. This is because the density of a substance determines its volume, and vice versa. By understanding this relationship, you can accurately predict the volume of an object based on its density and mass.
Q: How do you calculate the volume of an object based on its density and mass?
A: To calculate the volume of an object, you can use the formula: volume = mass/density. Since the masses are equal, you can set up a proportion to compare the volumes of the cubes.
Q: What is the order of the cubes from smallest to largest?
A: Based on the calculations, the order of the cubes from smallest to largest is platinum, gold, and lead.
Q: How many atoms are contained in the largest cube?
A: To calculate the number of atoms in the largest cube, we need to know the number of atoms per unit volume of lead. This value is known as the number density of lead. The number density of lead is approximately 5.86 x 10^28 atoms/m^3. We can use this value to calculate the number of atoms in the largest cube.
Q: What is the number density of lead?
A: The number density of lead is approximately 5.86 x 10^28 atoms/m^3.
Q: How do you convert the volume of the lead cube from cubic centimeters to cubic meters?
A: To convert the volume of the lead cube from cubic centimeters to cubic meters, you can multiply the volume in cubic centimeters by (1 m / 100 cm)^3.
Q: What is the volume of the lead cube in cubic meters?
A: The volume of the lead cube in cubic meters is approximately 8.84 x 10^-8 m^3.
Q: How do you calculate the number of atoms in the largest cube?
A: To calculate the number of atoms in the largest cube, you can multiply the number density of lead by the volume of the lead cube in cubic meters.
Q: What is the number of atoms in the largest cube?
A: The number of atoms in the largest cube is approximately 5.18 x 10^21 atoms.
Q: What is the significance of understanding the relationship between density and volume?
A: Understanding the relationship between density and volume is crucial in various fields, including physics, chemistry, and engineering. It allows you to accurately predict the behavior of objects and materials, which is essential for designing and optimizing systems.
Q: Can you provide examples of real-world applications of understanding the relationship between density and volume?
A: Yes, understanding the relationship between density and volume has numerous real-world applications, including:
- Designing ships and submarines that can operate efficiently in different environments
- Developing materials with specific properties for use in construction, aerospace, and other industries
- Optimizing the performance of engines and other mechanical systems
- Predicting the behavior of fluids and gases in various applications
By understanding the relationship between density and volume, you can unlock new possibilities and improve the performance of various systems and materials.