Final answer:
The total number of particles in the powders is 9.217 x 10^8 particles. The percentage increase in the total surface area is -92.5%. The bulk density of the powders is 8.955 g/cm³.
Step-by-step explanation:
To determine the total number of particles in the powders, we need to calculate the volume of the cube and then divide it by the volume of a single particle. The volume of the cube is calculated by taking the cube of the side length, so V = (30 cm)^3 = 27,000 cm^3. The volume of a single particle can be calculated using the diameter of the particle, which is 75 microns = 0.075 mm. The volume of a sphere is given by V = (4/3)πr^3, where r is the radius. Converting the diameter to radius, we have r = 0.075 mm / 2 = 0.0375 mm. Converting to cm, we have r = 0.0375 mm * (1 cm / 10 mm) = 0.00375 cm. Substituting the values into the formula, we get V = (4/3) * π * (0.00375 cm)^3 = 2.933 x 10^-5 cm^3. Now, we can calculate the total number of particles by dividing the volume of the cube by the volume of a single particle: N = 27,000 cm^3 / 2.933 x 10^-5 cm^3 = 9.217 x 10^8 particles.
To calculate the percentage increase in the total surface area, we need to compare the surface area of the solid cube to the surface area of the powders. The surface area of a solid cube is given by A = 6 * (side)^2 = 6 * (30 cm)^2 = 5,400 cm^2. The surface area of a single particle can be calculated using the diameter, which is 75 microns = 0.075 mm. The surface area of a sphere is given by A = 4πr^2, where r is the radius. Converting the diameter to radius, we have r = 0.0375 mm / 2 = 0.01875 mm. Converting to cm, we have r = 0.01875 mm * (1 cm / 10 mm) = 0.001875 cm. Substituting the value into the formula, we get A = 4 * π * (0.001875 cm)^2 = 4.415 x 10^-5 cm^2. Now, we can calculate the total surface area of the powders by multiplying the surface area of a single particle by the total number of particles: A = 4.415 x 10^-5 cm^2 * 9.217 x 10^8 particles = 4.065 x 10^4 cm^2. To calculate the percentage increase, we divide the difference between the surface areas of the powders and the solid cube by the surface area of the solid cube and multiply by 100: % Increase = [(4.065 x 10^4 cm^2 - 5,400 cm^2) / 5,400 cm^2] * 100 = -92.5%.
To calculate the bulk density of the powders, we need to know the volume of the powders and the mass of the powders. The volume of the powders can be calculated by multiplying the volume of a single particle by the total number of particles: V = 2.933 x 10^-5 cm^3 * 9.217 x 10^8 particles = 270 cm^3. The mass of the powders can be calculated using the density of copper, which is 8.95 g/cm³: Mass = Density * Volume = 8.95 g/cm³ * 270 cm^3 = 2416.5 g. Finally, we can calculate the bulk density by dividing the mass of the powders by the volume of the powders: Bulk Density = Mass / Volume = 2416.5 g / 270 cm^3 = 8.955 g/cm³.