Question

Brass is a copper-zinc alloy. What is the mass in grams of a brass cylinder having a length of \(1.62\) in. and a diameter of \(0.514\) in. if the composition of the brass is \(67.0 \%\) copper and \(33.0 \%\) zinc by mass? The density of copper is \(8.92 \mathrm{~g} / \mathrm{cm}^{3}\), and the density of zinc is \(7.14 \mathrm{~g} / \mathrm{cm}^{3}\). Assume that the density of the brass varies linearly with composition.

Short answer

The mass of the brass cylinder is approximately 45.89 grams.

Step-by-step solution

Convert Dimensions to Metric Units

First, convert the dimensions of the brass cylinder from inches to centimeters, because the density is in grams per cubic centimeter. Use the conversion factor: 1 inch = 2.54 cm.Length of the cylinder in cm:\[ 1.62 \text{ in} \times 2.54 \text{ cm/in} = 4.1148 \text{ cm} \]Diameter of the cylinder in cm:\[ 0.514 \text{ in} \times 2.54 \text{ cm/in} = 1.30636 \text{ cm} \]

Calculate Cylinder's Volume

The volume \( V \) of a cylinder is given by:\[ V = \pi r^2 h \]where \( r \) is the radius and \( h \) is the height (length).First, find the radius in cm:\[ r = \frac{1.30636}{2} = 0.65318 \text{ cm} \]Next, calculate the volume:\[ V = \pi \times (0.65318)^2 \times 4.1148 \approx 5.509 \text{ cm}^3 \]

Determine Brass Density

Assume the density \( \rho \) of brass is a weighted average of the densities of copper and zinc. Use the given composition percentages:\[ \rho_{brass} = 0.67 \times 8.92 + 0.33 \times 7.14 \]Calculate:\[ \rho_{brass} = 5.9764 + 2.3562 = 8.3326 \text{ g/cm}^3 \]

Calculate Mass of the Cylinder

Now, use the formula for mass based on density and volume:\[ \text{Mass} = \text{Density} \times \text{Volume} \]Substitute the values:\[ \text{Mass} = 8.3326 \times 5.509 \approx 45.89 \text{ grams} \]

Key concepts

Understanding Brass Density

Brass is a metal alloy made primarily of copper and zinc. The density of brass is not a fixed number because it depends on the proportion of copper and zinc present in the alloy.
The relationship between the density of brass and its composition is assumed to be linear, meaning it is directly affected by the densities of its constituent metals.
Copper has a density of \( 8.92 \ \text{g/cm}^3 \), while zinc has a density of \( 7.14 \ \text{g/cm}^3 \).

• If brass contains more copper, it will have a density closer to 8.92 g/cm³.
• If it contains more zinc, the density will approach 7.14 g/cm³.

To find the density of a particular brass composition, we calculate the weighted average of the densities of copper and zinc. In our example, where brass consists of 67% copper and 33% zinc, the density is calculated using the formula: \[ \rho_{brass} = 0.67 \times 8.92 + 0.33 \times 7.14 \] Once calculated, this weighted average density is used to find other properties like mass.

Volume Calculation of a Cylinder

Calculating the volume of a cylinder involves determining its base area and multiplying by its height. The basic formula for a cylinder's volume is: \[ V = \pi r^2 h \] Here, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
Given dimensions are typically in different units, such as inches, which need to be converted into centimeters or other metric units first.
The steps for calculating the volume of our brass cylinder are:

• Convert the height and diameter from inches to centimeters since density is provided in \( \text{g/cm}^3 \).
• Find the radius by halving the diameter.
• Use the formula with the converted dimensions to find the volume.

Once properly calculated, the volume is used in further equations to determine the mass of the cylinder.

Metric Unit Conversion

Metric unit conversion is a fundamental step in solving problems with international unit standards. Often, measurements provided in inches need to be converted into centimeters, especially when dealing with densities given in \( \text{g/cm}^3 \).
The conversion factor between inches and centimeters is \( 1 \text{ inch} = 2.54 \text{ cm} \), which is crucial for consistent measurements.
Let's convert a few measurements:

• To convert 1.62 inches to centimeters: \[ 1.62 \times 2.54 = 4.1148 \ cm \]
• The diameter of 0.514 inches becomes: \[ 0.514 \times 2.54 = 1.30636 \ cm \]

This conversion ensures uniformity in calculations and is essential when working through exercises involving density and volume.

Understanding Weighted Average Density

Weighted average density takes into account the proportion of each component in a mixture or alloy.
In a copper-zinc alloy like brass, knowing the proportion by mass for each element helps calculate the overall density of the material.
The formula for calculating weighted average density in a two-component alloy is:

• \( \rho = \text{(fraction of component 1)} \times \text{density of component 1} \)
• \( + \text{(fraction of component 2)} \times \text{density of component 2} \)

This method is applicable whenever the property of a composite can be calculated from its individual constituents.
In the example provided, the composition is understood as:

• 67% copper translates to 0.67 in the formula
• 33% zinc translates to 0.33

Applying their respective densities results in the brass density, allowing for further calculations in problems involving mass and volume.