Question

An alloy made up of iron \((52.6 \%)\), nickel \((38.0 \%)\), cobalt \((8.06 \%)\), and molybdenum \((1.34 \%)\) has a density of \(7.68 \mathrm{~g} /\) \(\mathrm{cm}^{3}\). How many molybdenum atoms are there in a block of the alloy measuring \(13.0 \mathrm{~cm} \times 22.0 \mathrm{~cm} \times 17.5 \mathrm{~cm}\) ?

Short answer

Answer: There are approximately \(3.23 \times 10^{24}\) molybdenum atoms in the alloy block.

Step-by-step solution

Find the volume of the alloy block

Calculate the volume of the block by multiplying the three dimensions given: \(Volume = Length \times Width \times Height\) \(Volume = 13.0 \mathrm{~cm} \times 22.0 \mathrm{~cm} \times 17.5 \mathrm{~cm} = 5005 \mathrm{~cm^3}\)

Find the mass of the alloy block

Calculate the mass of the alloy block by multiplying the density by the volume: \(Mass = Density \times Volume\) \(Mass = 7.68 \mathrm{~g/cm^3} \times 5005 \mathrm{~cm^3} = 38438.4 \mathrm{~g}\)

Calculate the mass of molybdenum in the alloy block

Find the mass of molybdenum in the alloy block using its percentage in the alloy: \(Mass_{molybdenum} = Mass_{alloy} \times \frac{Percentage_{molybdenum}}{100}\) \(Mass_{molybdenum} = 38438.4 \mathrm{~g} \times \frac{1.34}{100} = 515.07496 \mathrm{~g}\)

Convert the mass of molybdenum to moles

Use the molar mass of molybdenum to convert the mass of molybdenum to moles: \(Moles_{molybdenum} = \frac{Mass_{molybdenum}}{Molar_{mass}}\) \(Moles_{molybdenum} = \frac{515.07496 \mathrm{~g}}{95.94 \mathrm{~g/mol}} \approx 5.367 \mathrm{~moles}\)

Calculate the number of molybdenum atoms

Use Avogadro's number to find the number of molybdenum atoms: \(Number_{atoms} = Moles_{molybdenum} \times Avogadro's_{number}\) \(Number_{atoms} = 5.367 \mathrm{~moles} \times 6.022 \times 10^{23} \mathrm{atoms/mol} \approx 3.23 \times 10^{24} \mathrm{atoms}\) There are approximately \(3.23 \times 10^{24}\) molybdenum atoms in the alloy block.