Avogadro\'s number (6.023 x 10^23) is a pure (unitless) number which serves as a

Question

Avogadro's number (6.023 x 10^23) is a pure (unitless) number which serves as a good standard for measuring the number of molecules in ideal gases at STP.

A) What is the volume, in cubic kilometers, of Avogadro's number of sand grains, if each grain is a cube with an edge length of 0.65 mm and the cuves are densely packed (with no air between them).

B) How long, in kilometers, would a beach have to be for this sand to cover it to a depth of 10.0 m? Assume a beach is 100.0 m wide, and you can neglect the air spaces between the grains

Explanation / Answer

Given,

A)No of sand grain = N = A = 6.023 x 1023 ; a = 0.65 mm = 6.5 x 10-4 m

We know that, Volume of the cube = a3 (a is the length of one side)

Volume of one cubical shaped sand grain = V = a3 = (6.5 x 10-4)3 = 274.63 x 10-12 m3

Volume of N no of sand grains = N x V =  6.023 x 1023 x  (274.63 x 10-12 ) = 1.654 x 1014 m3

Hence, volume = V = 165400 km3

B)depth = d = 10 m ; width = w = 100 m.

we need to find the length of the beach. Let it be L.

If we assume the beach as the box to accomodate V volume of sand, we can determine the required length as follows:

L x w x d = Volume of sand

L = Volume of sand / w x d

L = 1.654 x 1014 m3 / 100 x 10 = 1.654 x 1011 m

Hence, the length of beach required = L = 1.654 x 1011 m = 1.654 x 108 km.

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