Final answer:
To find the weight percentage of germanium in the alloy, we calculate the mass of germanium atoms, determine the alloy's total mass, and then find the ratio of germanium mass to the total alloy mass. The percentage of germanium in the alloy is approximately 11.71%.
Step-by-step explanation:
To compute the weight percentage of germanium that must be added to silicon to yield an alloy containing 2.43 × 1021 germanium atoms per cubic centimeter, we first need to determine the mass of these atoms and then find the total mass of the alloy.
The atomic mass of germanium (Ge) is approximately 72.63 g/mol. Using Avogadro's number (6.022 × 1023 atoms/mol), we can find the mass of germanium in the sample:
Mass of Ge = (72.63 g/mol) × (2.43 E21 atoms / 6.022 E23 atoms/mol) = 0.292g.
Knowing that the density of germanium is 5.32 g/cm3, the volume occupied by the germanium atoms in the alloy can be calculated:
Volume of Ge = Mass of Ge / Density of Ge = 0.292g / 5.32g/cm3 = 0.0549cm3.
To find the total mass of the alloy, we add the mass of the silicon needed to fill the rest of the volume. Assuming no volume change upon alloying, the mass of silicon is determined by its density (2.33 g/cm3) and the remaining volume:
Volume of Si = 1 cm3 - Volume of Ge = 1 cm3 - 0.0549cm3 = 0.9451cm3.
Mass of Si = Volume of Si × Density of Si = 0.9451cm3 × 2.33g/cm3 = 2.202g.
Now calculate the total mass of the alloy and the weight percentage of germanium:
The total mass of alloy = Mass of Ge + Mass of Si = 0.292g + 2.202g = 2.494g.
Weight percentage of Ge = (Mass of Ge / Total mass of alloy) × 100 = (0.292g / 2.494g) × 100 = 11.71%.