Question

Consider a 1.5 in. diameter solid sphere of sterling silver, which is a uniform mixture (an alloy) of \(7.5 \%\) copper and \(92.5 \%\) silver, by mass. The densities of the metals are: copper \(=8.96 \mathrm{~g} / \mathrm{cm}^{3} ;\) silver \(=10.5 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate the mass of the sphere.

Short answer

The mass of the sterling silver sphere is approximately 300 grams.

Step-by-step solution

Calculate the Volume of the Sphere

The first step in finding the mass of the sphere is to calculate its volume. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Given the diameter is 1.5 inches, first convert this to radius (0.75 inches), and then convert inches to centimeters (1 inch = 2.54 cm). This gives \( r = 0.75 \times 2.54 = 1.905 \) cm. Now calculate the volume: \( V = \frac{4}{3} \pi (1.905)^3 \approx 29.047 \) cm³.

Determine the Mass Contribution of Each Element

The mass of the sphere is a combination of silver and copper. First, calculate the total mass using the average density of the alloy. The average density \( \rho \) is given by the sum of the components weighted by their percentage in the alloy. Thus, \( \rho = 0.925 \times 10.5 + 0.075 \times 8.96 = 10.332 \; \text{g/cm³} \). Multiply the volume by this density to find the total mass: \( m = 29.047 \times 10.332 \approx 299.968 \) grams.

Key concepts

Alloys

Alloys are fascinating in the world of materials because they are mixtures of metals and offer a combination of properties. In simple terms, an alloy is a blend of two or more elements, at least one of which is a metal. Alloys often have improved characteristics compared to pure metals. For instance, they might be stronger, more resistant to corrosion, or have a different color or texture.
For the case of sterling silver, it's a well-known alloy that consists primarily of silver combined with a small amount of copper. This combination maintains the beautiful appearance of silver but enhances its durability. Sterling silver is widely used in making jewelry and silverware due to these balanced properties.
When working with alloys, it's important to know their composition, which is usually expressed as a percentage of the total mass. Understanding these percentages helps in predicting the alloy's properties and behavior under different circumstances.

• Alloys often have a mix of metals and non-metals.
• They provide enhanced properties compared to pure metals.
• Sterling silver is an alloy of silver and copper.

Density Calculation

Density is a fundamental concept in chemistry that describes how much mass a substance has in a given volume. The formula for density is \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] and it is usually expressed in units like g/cm³.
When dealing with alloys, calculating the density follows a weighted average approach. The overall density of an alloy depends on the densities of its components and their respective proportions. This allows us to calculate an average density that realistically represents the mixture.
Using the example of sterling silver: we calculate its density by weighing the density of silver at 92.5% and copper at 7.5%. This results in an average density that can then be used to compute the total mass of objects made from the alloy.

• Density is mass per unit volume.
• For alloys, a weighted average is used to determine density.
• Knowing densities helps predict how an alloy will behave in different situations.

Volume of a Sphere

Understanding how to calculate the volume of a sphere is essential, especially in situations where you need to determine the mass or density of spherical objects. The volume of a sphere is given by the formula \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.
To find the volume of a sphere, you need to first determine its radius. Remember, the radius is half of the diameter, and it’s often necessary to convert units into a consistent measurement system. For example, converting inches to centimeters if your density is in g/cm³ can be crucial for accurate calculations.
By knowing its volume, we can further calculate the total mass of the sphere if we also have the density of the material. This is particularly relevant in alloy-based spheres, where understanding both volume and density plays a crucial role in computations.

• Volume of a sphere uses the formula \( V = \frac{4}{3} \pi r^3 \).
• Ensure radius is in correct units for calculation consistency.
• Volume calculations help in determining mass when density is known.