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 × 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 0.9 m on an edge?
To find the mass of the sand grains with a total surface area equal to the surface area of a cube, we need to calculate the total surface area of the sand grains first.
The surface area of a sphere can be calculated using the formula: A = 4πr², where A is the surface area and r is the radius of the sphere.
Given that the average radius of the sand grains is 50 μm (which is 50 × 10^(-6) m), we can calculate the surface area of each sand grain using the formula above.
Surface area of one sand grain = 4π(50 × 10^(-6))^2
Now, we need to find the number of sand grains that would cover the surface area of the cube. The surface area of a cube is given by A = 6s², where A is the surface area and s is the length of one side of the cube.
Given that the length of one side of the cube is 0.9 m, we can calculate the surface area of the cube using the formula above.
Surface area of the cube = 6(0.9)²
Now, to find the total mass of the sand grains, we need to know the number of sand grains that would cover the surface area of the cube. We can calculate this by dividing the surface area of the cube by the surface area of one sand grain.
Number of sand grains = Surface area of the cube / Surface area of one sand grain
Finally, to find the mass, we multiply the number of sand grains by the mass of each sand grain. The mass of each sand grain can be calculated using the formula: mass = density x volume.
Given that the density of silicon dioxide is 2.4 × 10^3 kg/m³, and the volume of a sphere is given by V = (4/3)πr³, we can calculate the mass of one sand grain using the formula above.
Mass of one sand grain = density x volume = (2.4 × 10^3)(4/3)π(50 × 10^(-6))^3
Now, we can calculate the mass of sand grains that would have a total surface area equal to the surface area of the cube by multiplying the number of sand grains by the mass of one sand grain.
Mass of sand grains = Number of sand grains x Mass of one sand grain
By following these steps and plugging in the given values into the appropriate formulas, you can find the answer to the question.