Question

Monel metal is a corrosion-resistant copper-nickel alloy used in the electronics industry. A particular alloy with a density of \(8.80 \mathrm{g} / \mathrm{cm}^{3}\) and containing \(0.022 \%\) Si by mass is used to make a rectangular plate \(15.0 \mathrm{cm}\) long, \(12.5 \mathrm{cm}\) wide, \(3.00 \mathrm{mm}\) thick, and has a \(2.50 \mathrm{cm}\) diameter hole drilled through its center. How many silicon- 30 atoms are found in this plate? The mass of a silicon- 30 atom is \(29.97376 \mathrm{u}\) and the percent natural abundance of silicon- 30 is 3.10\%.

Short answer

To find the number of Silicon-30 atoms, first the total and the real volume of the plate should be calculated. By using the density of the alloy, the total mass of the plate can be found and then the mass of Silicon is calculated from the percentage by mass. Then, using Avogadro's number, the number of Silicon-30 atoms can be determined. Lastly, adjust the total number of atoms by the percent natural abundance of Silicon-30.

Step-by-step solution

Calculate the volume of the metal plate

The first step is calculating the total, then the real volume of the plate, taking into account the hole in the center. The volume of a rectangular plate can be obtained using the formula length * width * height. However, as there is a hole drilled in the plate, this hole's volume needs to be subtracted from the total. The volume of the hole can be calculated using the formula for a volume of a cylinder, \(\pi r^{2} h\), where r is the radius and h is the height. The radius is half the diameter. Also, note that all measurements should be converted to the same units, in this case, cm.

Calculate the mass of the metal plate and the mass of Silicon

The mass of an object can be found if we know its volume and density. Here, the density of the alloy and the volume that has been calculated will allow us to find the mass of the alloy. To find the mass of Silicon in the plate, use the percentage by mass of Silicon and the total mass of the plate.

Determine the number of Silicon-30 atoms

To find the number of atoms in the given mass of Silicon, it's necessary to use Avogadro's number. Avogadro's number tells us that one mole of any substance contains \(6.022 * 10^{23}\) entities. Convert the mass of Silicon to moles using the atomic mass of Silicon-30, then use Avogadro's number to find the number of atoms. However, since only 3.10% of Silicon is Silicon-30, the number of atoms should be adjusted accordingly.

Key concepts

Density and Volume Calculations

Understanding density and volume is crucial for solving problems related to material composition and structural dimensions.
Density is defined as mass per unit volume, commonly expressed as \( \text{g/cm}^3 \). Knowing the density of a material allows us to determine the total mass of an object if we know its volume, using the relation: \[ \text{Mass} = \text{Density} \times \text{Volume} \].
The volume of a rectangular object, like a metal plate, is calculated by multiplying its length, width, and height. For the plate in the exercise, this means: \[ V_{\text{total}} = \text{Length} \times \text{Width} \times \text{Height} \].
When a hole is present, its volume, calculated using the formula for a cylinder \( V_{\text{hole}} = \pi r^2 h \), must be subtracted from the total volume of the plate. This adjusted volume gives the true volume of the metal in the object, which can then be used to calculate its mass using the given density.

Mass Percentage Composition

Mass percentage composition tells us how much of a material's mass comes from a specific component. It is calculated using the equation: \[ \text{Mass percentage} = \left( \frac{\text{Mass of component}}{\text{Total mass of material}} \right) \times 100 \text{%} \].
In this context, it tells us how much silicon is present in the metal plate. The metal plate's total mass can be found by using its volume (adjusted for the hole) and the density of the alloy.
Finding the mass of silicon involves multiplying the total mass of the plate by the mass percentage of silicon. This value is essential for subsequent calculations, such as determining the number of silicon atoms present.

Avogadro's Number

Avogadro's Number, \(6.022 \times 10^{23}\), is a fundamental constant in chemistry, representing the number of atoms, ions, or molecules in one mole of a substance.
In this scenario, knowing the mass of silicon allows us to convert it into moles. First, calculate the number of moles of silicon using: \[ \text{Moles of Silicon} = \frac{\text{Mass of Silicon}}{\text{Molar mass of Silicon-30}} \].
Once the number of moles is known, Avogadro's Number helps convert this to the actual number of silicon-30 atoms. Since only 3.10% of silicon naturally occurs as silicon-30, we adjust the result to account for this isotopic abundance by multiplying: \[ \text{Number of Silicon-30 atoms} = \text{Total atoms from moles} \times 0.031 \].
This calculation provides the quantity of silicon-30 atoms, which is a crucial step in solving the problem.

Corrosion-resistant Alloys

Corrosion-resistant alloys are mixtures designed to withstand chemical degradation from environmental interactions. The Monel alloy, a blend of copper and nickel, is an excellent example due to its effective corrosion resistance.
This characteristic is vital for industries like electronics, where material longevity and reliability are paramount. Such alloys often incorporate other elements, like silicon, to further enhance properties, though silicon's role is more often in controlling alloy structures and adding wear resistance.
Understanding the composition and benefits of corrosion-resistant alloys aids in selecting the right materials for specific applications. The presence of a small mass percentage of silicon in the Monel alloy contributes to its stability and performance, making it suitable for demanding environments.