Grains of fine California beach sand are approximately spheres with an average radius of 50 and are made of silicon dioxide. A solid cube of silicon dioxide with a volume of 1.00 m 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 1.00 m on an edge?
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 calculate the surface area of the cube and then determine the number of sand grains required to cover the same area.
Step 1: Calculate the surface area of the cube
A cube has six equal square faces. Since the length of each side of the cube is 1.00 m, the surface area can be calculated by multiplying the length of one side by itself and then multiplying the result by 6.
Surface Area of Cube = 1.00 * 1.00 * 6 = 6.00 m²
Step 2: Calculate the total surface area of the sand grains
Since the sand grains are spherical, the total surface area can be calculated by multiplying the surface area of a single grain by the total number of grains.
The surface area of a sphere is given by the formula: Surface Area of Sphere = 4πr², where r is the radius of the sphere.
Given that the average radius of the sand grains is 50, the surface area of a single sand grain can be calculated as follows:
Surface Area of a Single Grain = 4π * 50²
Step 3: Determine the number of sand grains required
To find the number of sand grains required to cover the same surface area as the cube, we divide the total surface area of the cube by the surface area of a single grain.
Number of Grains = Surface Area of Cube / Surface Area of a Single Grain
Step 4: Calculate the mass of the sand grains
To find the mass of the sand grains that would have the total surface area equal to the cube, we multiply the number of grains by the mass of each grain.
We need additional information to calculate the mass of a single sand grain. If the density of silicon dioxide is provided, we can use it to find the mass.