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* 10^4`

Step-by-step explanation:

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

`V=(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=(4)/(3)* 3.14* (10^(-6))^3\\\\V=4.19* 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* 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* 10^(-18)}\\\\\text{Mass of gold}=8.0867* 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* 10^(-17)g`

Molar mass of gold = 196.97 g/mol

Putting values in above equation, we get:

`\text{Moles of gold}=(8.8067* 10^(-17)g)/(196.97g/mol)=4.47* 10^(-9)moles`

• According to mole concept:

1 mole of an element contains
`6.022\time 10^(23)` number of atom

So,
`4.47* 10^(-19)` moles of gold will contain =
`4.47* 10^(-19)* 6.022* 10^(23)=26.918* 10^4` number of atoms.

Hence, the number of atoms in spherical gold nano particle are
`26.918* 10^4`