For a BCC iron-carbon alloy that contains \(0.1\) wt \(\%\) C, calculate the fraction of unit cells that contain carbon atoms.

In a body-centered cubic (BCC) unit cell, there are 2 atoms per unit cell: one at each corner of the unit cell and one at its center.

We are given that the alloy contains 0.1 wt% carbon. To find the number of carbon atoms in the alloy, we first need to find the weight of carbon in the alloy. First, let's assume that the total weight of the alloy is 100g. In this case, the total weight of carbon in the alloy would be: Weight of carbon = (0.1/100) * 100g = 0.1g To find the number of carbon atoms, we need to know the molar mass of carbon which is 12.01 g/mol. Using this information, we can find the moles of carbon present in the alloy as follows: Moles of carbon = (Weight of carbon) / (Molar mass of carbon) = (0.1g) / (12.01 g/mol) = 0.008325 mol To calculate the number of carbon atoms, use Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol: Number of carbon atoms = (Moles of carbon) * (Avogadro's number) = 0.008325 mol * \(6.022 \times 10^{23}\) atoms/mol = \(5.01 \times 10^{21}\)

The alloy is made up of iron and carbon atoms. Since we have assumed a total weight of 100g, the weight of iron in the alloy can be found as follows: Weight of iron = Total weight - Weight of carbon = 100g - 0.1g = 99.9g The molar mass of iron is 55.85 g/mol. We can find the moles of iron in the alloy as follows: Moles of iron = (Weight of iron) / (Molar mass of iron) = (99.9g) / (55.85 g/mol) = 1.788 mol Now, using Avogadro's number, we can calculate the number of iron atoms in the alloy: Number of iron atoms = (Moles of iron) * (Avogadro's number) = 1.788 mol * \(6.022 \times 10^{23}\) atoms/mol = \(1.077 \times 10^{24}\)

In a BCC unit cell, the ratio of iron atoms to unit cells is 2:1. Therefore, we can find the number of unit cells in our alloy as follows: Number of unit cells = Number of iron atoms / 2 = \((1.077 \times 10^{24}\) atoms) / 2 = \(5.385 \times 10^{23}\) Now, we can calculate the fraction of unit cells containing carbon atoms by dividing the number of carbon atoms by the total number of unit cells: Fraction of unit cells containing carbon atoms = (Number of carbon atoms) / (Number of unit cells) = \((5.01 \times 10^{21}\)) / (\(5.385 \times 10^{23}\)) = \(9.3 \times 10^{-3}\) Therefore, the fraction of unit cells containing carbon atoms is approximately \(9.3 \times 10^{-3}\).