Physics conversion problem?

Question

I am having a lot of trouble understanding how to do this problem... An explanation would be very helpful. For some reason I cannot get the right solution from what the book has.

Grains of fine California beach sand are approximately spheres with an average radius of 55 µm and are made of silicon dioxide. A solid cube of silicon dioxide with a volume of 1.00 m3 has a mass of 2600 kg. What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube 1.50 m on an edge?

Answer 1

There are online calculation algerithims for this.

the best thing to do is google search for one of them keep looking they are out there.

Answer 2

Grains of fine California beach sand are approximately spheres with an average radius of 55 µm and are made of silicon dioxide. A solid cube of silicon dioxide with a volume of 1.00 m3 has a mass of 2600 kg. What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube 1.50 m on an edge?

a lot of steps, you just have to keep nibbling at it.

First, calculate the density in kg/m³

d = 2600/1 = 2600 kg/m³

m = unknown mass of sand

total volume of sand = m/2600

volume of one grain = V = ⁴/₃πr³

V = ⁴/₃π(0.000055)³ = 6.97e-13 m³

number of grains, divide the volumes

(does not account for packing factor)

N = m/2600/6.97e-13 = m/1.812e-9

Surface area of one grain = A = 4πr²

A = 4π(0.000055)² = 3.80e-8

Surface area of the cube is

AC = 6*1.5² = 13.5 m²

from this the number of grains is

N = 13.5 / 3.80e-8 = 3.5e8

now go back to the other value for N, in terms of m

N = 3.5e8 = m/1.812e-9

m = 6.4e17 kg

is that right? I don't know, seems high. Did I make an arith. mistake somewhere? probably.

.

Answer 3

1) First, find the surface area of the cube:

A = 6s² = 6 * (1.5 m)² = 13.5 m²

2) Next, find the surface area of one sand grain:

A = 4πr² = 4π * (55 µm)² = 37994 µm²

3) How many grains of sand equal the area of the cube.

13.5 m²/ 37,994 µm² * (1,000,000 µm/m)² = 355,319,261 grains of sand

4) Find the volume of one grain of sand.

V = (4/3)πr³ = (4/3)π(55 µm)³ = 696557 µm³

5) How many grains of sand is in that cube weigh 2600 kg.

1.00 m³/696557 µm³ * (1,000,000 µm/m)³ = 1,435,633,377,519 pieces of sand weighs 2600 kg

6) How much does each piece of sand weigh.

2600 kg/1,435,633,377,519 pieces * 1000 g/kg = 1.81 E -6 g

7) How much does 355,319,261 grains (from #3) of sand weigh?

1.81 E -6 g * 355,319,261 = 644 g