Estimate the number of gold atoms in one of your gold nanoparticles given its di

Question

Estimate the number of gold atoms in one of your gold nanoparticles given its diameter. Show your calculations and use the following assumptions: 3 pts The atomic radius for a gold atom is 135 picometers. The gold atoms and the nanoparticles are both spheres with a volume of 4/3 r3, where r is the radius. Only 74% of the total volume of the nanoparticle is physically occupied by gold atoms and the rest is “empty space” between atoms. (NOTE: The maximum “packing efficiency” of spheres in any crystal lattice is 74%.) Diameter of your selected gold nanoparticle 10 nm

Explanation / Answer

Solution:-

Atomic radii for gold atom = 135 pm , volume = 4/3 r3 ( r = radius )

diameter of gold nanopartical = 10 nm then radii = diameter/2 = 10/2 = 5nm = 5000 pm { 1 nm = 1000 pm }

packing efficency = 74%

so first we will calculate the volume of gold nano partical

volume of gold nano partical = 4/3 r3 ( r = radius = 5000 pm ) = 4/3 * 3.14 * (5000 pm)3 = 5.23*1011 pm3

packing efficency is 74% so volume occupied by gold atom is 0.74 times the volume of gold nano partical

so volume occupied by gold atom = 0.74 * volume of the gold nanoparticle  = 0.74 * 5.23*1011 = 3.87 * 10 11 pm3---1

volume occupied by gold atom = number of gold atom * volume of one gold atom

= n * 4/3 r3 ( r = radius = 135 pm ) = n * 4/3 * 3.14 * (135 pm)3 = n*1.03*107 pm3----2

now equilate both the volume calculated --1 and ----2

so volume occupied by gold atom = 3.87 * 10 11 pm3 = n*1.03*107 pm3 then n= 3.87 * 10 11 pm3 /(1.03*107 pm3) n = (3.87 * 10 11 pm3 )/(1.03*107 pm3) = 3.7 * 104 gold atom

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