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Chapter 4: Problem 24

Germanium forms a substitutional solid solution with silicon. Compute the weight percent of germanium that must be added to silicon to yield an alloy that contains \(2.43 \times 10^{21} \mathrm{Ge}\) atoms per cubic centimeter. The densities of pure Ge and \(\mathrm{Si}\) are 5.32 and \(2.33 \mathrm{g} / \mathrm{cm}^{3}\) respectively .

Short Answer

Expert verified
Answer: To find the weight percent of Ge in the Si alloy, follow these steps: 1. Calculate the volume fraction of Ge in the alloy using the given number of Ge atoms per cubic centimeter, the densities of Ge and Si, and Avogadro's number. 2. Determine the total concentration of atoms in the alloy by adding the numbers of Ge and Si atoms in 1 cubic centimeter of the alloy. 3. Find the weight fraction of Ge in the alloy using the molecular weights of Ge and Si and the volume fraction of Ge. 4. Convert the weight fraction to weight percent by multiplying it by 100. After completing these steps, you will have calculated the weight percent of Ge in the Si alloy.

Step by step solution

01

Find the volume fraction of Ge in the alloy

Since we are given the number of Ge atoms per cubic centimeter in the alloy, we can use the densities of pure Ge and Si to find the volume fraction of Ge in the alloy. Density of Ge = 5.32 g/cm³ Density of Si = 2.33 g/cm³ The given number of Ge atoms per cubic centimeter is \(2.43 \times 10^{21} \mathrm{Ge/cm^3}\). We can determine the volume fraction of Ge in the alloy as follows: Volume fraction of Ge = \(\frac{2.43 \times 10^{21} \mathrm{Ge/cm^3}}{(5.32 \mathrm{g/cm^3}) N_A}\) Where \(N_A\) is Avogadro's number (\(6.022 \times 10^{23} \mathrm{atoms/mol}\)).
02

Calculate the total concentration of atoms in the alloy

Now we need to find the total concentration of atoms in the alloy. We can do this by adding the numbers of Ge and Si atoms in \(1 \mathrm{cm^3}\) of the alloy. The total concentration of atoms = \(\frac{2.43 \times 10^{21} \mathrm{Ge/cm^3}}{\mathrm{Volume\:fraction\:of\:Ge}} + \frac{(2.33 \mathrm{g/cm^3}) N_A}{(5.32 \mathrm{g/cm^3})}\)
03

Determine the weight fraction of Ge in the alloy

Once we have the total concentration of atoms in the alloy, we can determine the weight fraction of Ge using the molecular weights: Molecular weight of Ge = 72.63 g/mol Molecular weight of Si = 28.08 g/mol Weight fraction of Ge = \(\frac{\mathrm{Molecular\:weight\:of\:Ge} \times \mathrm{Volume\:fraction\:of\:Ge}}{\mathrm{Molecular\:weight\:of\:Ge} \times \mathrm{Volume\:fraction\:of\:Ge} + \mathrm{Molecular\:weight\:of\:Si} \times (1-\mathrm{Volume\:fraction\:of\:Ge})}\) Plug in the values and solve the equation to get the weight fraction of Ge in the alloy.
04

Convert the weight fraction to weight percent of Ge in the alloy

Once you have obtained the weight fraction of Ge, you can multiply it by 100 to convert it to weight percent: Weight percent of Ge = \(\mathrm{Weight\:fraction\: of\:Ge} \times 100\) Calculate the weight percent of Ge in the silicon alloy by substituting the weight fraction of Ge into the equation.

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