Question

Gold has an atomic weight of 196.97 g/mol, and a density of 19.3 g/cm3. Approximately how many atoms are in a spherical gold nanoparticle 10 nm in diameter?

Answer 1

Answer: The number of atoms in spherical gold nano particle are $26.918\times 10^4$

Explanation:

• To calculate the volume of sphere, we use the equation:

$V=\frac{4}{3}\pi r^3$

where,

r = radius of the sphere = $10nm=10^{-6}cm$    (Conversion factor:  $1nm=10^{-7}cm$ )

Putting values in above equation, we get:

$V=\frac{4}{3}\times 3.14\times (10^{-6})^3\\\\V=4.19\times 10^{-18}cm^3$

• To calculate mass of of the substance, we use the equation:

$\text{Density of substance}=\frac{\text{Mass of substance}}{\text{Volume of substance}}$

Volume of gold = $4.19\times 10^{-18}cm^3$

Density of gold = $19.3g/cm^3$

Putting values in above equation, we get:

$19.3g/cm^3=\frac{\text{Mass of gold}}{4.19\times 10^{-18}}\\\\\text{Mass of gold}=8.0867\times 10^{-17}g$

• To calculate the number of moles, we use the equation:

$\text{Number of moles}=\frac{\text{Given mass}}{\text{Molar mass}}$

Mass of gold = $8.8067\times 10^{-17}g$

Molar mass of gold = 196.97 g/mol

Putting values in above equation, we get:

$\text{Moles of gold}=\frac{8.8067\times 10^{-17}g}{196.97g/mol}=4.47\times 10^{-9}moles$

• According to mole concept:

1 mole of an element contains $6.022\time 10^{23}$  number of atom

So, $4.47\times 10^{-19}$ moles of gold will contain = $4.47\times 10^{-19}\times 6.022\times 10^{23}=26.918\times 10^4$ number of atoms.

Hence, the number of atoms in spherical gold nano particle are $26.918\times 10^4$