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 2.4 × 10^3 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 0.9 m on an edge?
To find the mass of sand grains with a total surface area equal to the surface area of a cube, we'll need to calculate the surface area of a cube and then determine the number of sand grains required to cover that surface area.
Let's start by calculating the surface area of the cube:
The surface area of a cube is given by the formula:
Surface Area = 6 * (side length)^2
Given that the side length of the cube is 0.9 m, we can calculate the surface area as follows:
Surface Area of Cube = 6 * (0.9 m)^2
Surface Area of Cube = 6 * 0.81 m^2
Surface Area of Cube = 4.86 m^2
Now we need to calculate the total surface area of the sand grains, which are approximately spheres. The surface area of a sphere is given by the formula:
Surface Area of Sphere = 4 * π * (radius)^2
Given that the average radius of the sand grains is 50 μm (or 50 x 10^-6 m), we can calculate the surface area of one sand grain as follows:
Surface Area of One Sand Grain = 4 * π * (50 x 10^-6 m)^2
Surface Area of One Sand Grain = 4 * π * (2.5 x 10^-9 m^2)
Surface Area of One Sand Grain ≈ 1.57 x 10^-8 m^2
Now, we can find the number of sand grains required to cover the surface area of the cube:
Number of Sand Grains = Surface Area of Cube / Surface Area of One Sand Grain
Number of Sand Grains = (4.86 m^2) / (1.57 x 10^-8 m^2)
Finally, to calculate the mass of the sand grains, we need to multiply the number of sand grains by the mass of a single sand grain. Given that the sand grains are made of silicon dioxide with a density of 2.4 x 10^3 kg/m^3, we can assume each sand grain has a mass equal to its volume multiplied by the density. Since the sand grains are approximately spheres, the volume of a sphere is given by the formula:
Volume of Sphere = (4/3) * π * (radius)^3
The mass of a sand grain is then calculated as follows:
Mass of One Sand Grain = Density * Volume of Sphere
Mass of One Sand Grain = (2.4 x 10^3 kg/m^3) * [(4/3) * π * (50 x 10^-6 m)^3]
With the number of sand grains and the mass of a single sand grain calculated, we can find the total mass of the sand grains:
Total Mass of Sand Grains = Number of Sand Grains * Mass of One Sand Grain
By substituting the values and performing the calculations, you should be able to find the mass of sand grains required to have a total surface area equal to the surface area of the given cube.