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 1.1 m on an edge?
The area of that cube would be 1.1^3 = 1.331 m^3.
If that is the total surface area of sand grains with radius 50*10^-6 m, the number N of sand grains would be given by
N*4*pi*(50^10^-6)^2 = 1.331
N = 4.23*10^7 grains
Each sand grain has a mass m given by
(4/3)*pi*(50*10^-6)^3*2400 = 1.273^10^-10 kg/grain
The mass of the N sand grains with an area of 1.331 m^3 is
5.4*10^-3 kg = 5.4 grams
Check my thinking and calculations.
To find the mass of the sand grains, we first need to calculate the total surface area of all the individual spheres and then equate it to the surface area of the cube.
Let's start by calculating the surface area of one sphere:
The surface area of a sphere is given by the formula: A = 4πr^2, where "r" is the radius of the sphere.
In this case, the radius of the sand grains is given as 50 μm (micrometers) or 50 × 10^-6 meters.
Calculating the surface area of one sand grain:
A = 4π(50 × 10^-6)^2
A = 4π(2.5 × 10^-9)
A ≈ 3.14 × 10^-8 m^2
Now, we need to calculate the surface area of the cube:
The surface area of a cube is given by the formula: A = 6s^2, where "s" is the length of each side of the cube.
In this case, the length of each side of the cube is given as 1.1 m.
Calculating the surface area of the cube:
A = 6(1.1)^2
A = 6(1.21)
A = 7.26 m^2
Since we want the total surface area of all the sand grains to be equal to the surface area of the cube, we can set up the following equation:
Number of sand grains × Surface area of one sand grain = Surface area of the cube
Let "n" be the number of sand grains.
n × 3.14 × 10^-8 m^2 = 7.26 m^2
Solving for "n":
n ≈ 7.26 m^2 ÷ 3.14 × 10^-8 m^2
n ≈ 2.31 × 10^8 sand grains
Now, we can calculate the total mass of the sand grains by multiplying the number of sand grains by the mass of one sand grain.
The density of silicon dioxide is given as 2.4 × 10^3 kg/m^3.
Mass of sand grains = Number of sand grains × Mass of one sand grain
Mass of sand grains = 2.31 × 10^8 × (4/3 × π × (50 × 10^-6)^3 × 2.4 × 10^3)
Mass of sand grains ≈ 0.000742 kg or 0.742 grams
Therefore, the mass of sand grains that would have a total surface area equal to the surface area of the cube is approximately 0.742 grams.