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Chapter 2: Problem 82

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

Expert verified
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

01

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.
02

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.
03

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.

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