Question

Grains of fine california beach sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide, which has a density of 2.6 × 103 kg/m3. 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 1.1 m on an edge?

Answer 1

To find the mass of sand grains that would have the same total surface area as a cube, we can calculate the surface area of the cube and then determine the number of sand grains needed to cover that surface area. Finally, we can multiply the number of sand grains by the mass of a single grain to find the total mass of the sand grains.

Step-by-step explanation:

To find the mass of sand grains that would have a total surface area equal to the surface area of a cube, we need to determine the surface area of the cube and then calculate the mass of sand grains. The surface area of a cube can be found using the formula: Surface Area = 6× (Side Length)^2. In this case, the side length of the cube is given as 1.1 m.

Once we have the surface area of the cube, we can calculate the number of sand grains required to have the same surface area. Since each sand grain is approximately a sphere with an average radius of 50 μm, we can calculate the surface area of a single sand grain using the formula: Surface Area = 4 × π × (Radius)^2. Then, we can divide the total surface area of the cube by the surface area of a single sand grain to find the number of sand grains required.

we can find the mass of the sand grains by multiplying the number of sand grains by the mass of a single grain. The density of silicon dioxide, which the sand grains are made of, is given as 2.6 × 10^3 kg/m^3.

Answer 2

The average radius(r) of each grain is r = 50 nanometers = 50*10^-6 meters

Since it is spherical, so

Volume=(4/3)*pi*r^3

V= (4/3)*pi*(50*10^-6)^3

V=5.23599*10^-13 m^3

We are given the Density(ρ) =2600kg/m^3

We know that:

Density(p) = mass(m)/volume(V)

m = ρV

So the mass of a single grain is:

m = 5.23599*10^-13 * 2600 = 1.361357*10^-9 kg

The surface area of a grain is:

a = 4*pi*r^2

a = 4*pi*(50*10^-6)^2

a = 3.14*10^-8 m^2

Since we know the surface area and mass of a grain, the conversion factor is:

1.361357*10^-9 kg / 3.14*10^-8 m^2

Find the Surface area of the cube:

cube = 6a^2

cube = 6*1.1^2 = 7.26m^2

multiply this by the converions ratio to get:

total mass of sand grains = (7.26 m^2 * 1.361357*10^-9 kg) / (3.14*10^-8 m^2)

total mass of sand grains = 0.3148 kg = 314.80 g