How many kilograms of sand have a total surface area of 6 m^2?

[icesalmon]

Homework Statement

As part of the initial treatment for waste water from an industrial site, you will be running the afluent through a sand filter. The filtering ability of the sand in this filter is roughly proportional to the surface area of the sand enclosed. The grains of fine sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide. A solid cube of this material with a volume of 1 m³ has a mass of 2500 kg. You have determined that the total surface area (the total area of all the individual grains) is equal to the surface area of a cube 1 meters on an edge.

What is the mass of the sand in kilograms

Homework Equations

Volume of sphere
Surface Area of Cube

The Attempt at a Solution

The Volume of a sphere is (4pir³)/3 and the surface area of a cube is 4a² and the surface area of a cube 1 meter on an edge is 1m³ I also have the number of particles in a sphere (4/3)pi(50μm)³ and I have 6m²/(4pi(50μm)²)

I don't know what this information is doing geometrically or what it has to do with finding the mass of the sand but it was written down and I needed help filling in the blanks.

 

[voko]

Since you have the total surface area, and you know it is the surface of a number of spheres with the given radius, you can find the number of spheres. Then, can you find the total volume of the spheres?

 

[SteamKing]

Are you sure you have the correct formula for surface area?

 

[icesalmon]

for a cube it is 6*(a)(a) where a is the length of the edge and my a = 1m -> 6meters²
for the sphere I am using the volume and the surface area, both of which I am fairly sure are correct.
/edit you were right, my formulas were switched up. my mistake

 

[icesalmon]

Since you have the total surface area, and you know it is the surface of a number of spheres with the given radius, you can find the number of spheres. Then, can you find the total volume of the spheres?

i'm not sure how to do this part, and which surface area are you talking about here, the cube or the sphere? when you say spheres you mean grains of sand, right?

I feel I should multiply the surface area of one of the figures by the number of spheres to find the total volume of the spheres, so I have the surface area and I need the number of spheres. Which I think I can find with each sphere has an average radius of 50μm and I find out how many of them cover the surface area of the sphere/cube by multiplying by the surface area of that object so either 4/3pi(50μm)³ or for the cube with 50μm*1m³ maybe?

 

[voko]

The total area of the grains of sand is given to be equal to the area of the 1m cube. This is what you should use to find the number of grains.

 

[icesalmon]

okay, so I have 6(50x10^-3m)

 

[voko]

okay, so I have 6(50x10^-3m)

What is this?

 

[icesalmon]

the surface area of a cube 1m on an edge multiplied by the average radius of one grain of SiO2

 

[voko]

the surface area of a cube 1m on an edge multiplied by the average radius of one grain of SiO2

And what does that mean, anyway?

Note also that the radius of grains is 50 micrometers, no millimeters as you have it.

 

[SteamKing]

If you can calculate the surface area of 1 grain of sand, can't you then figure out how many grains of sand it takes to equal 6 m^2? I don't know what multiplying the surface area of a 1m cube by the radius of a grain of sand gives you.

 

[icesalmon]

I wasn't sure either, so how many times does 50 micrometers go into 6 square meters
so I've got 6m²/(50x10^-6m) = 120,000m
Should I multiply this value by the mass of each sand grain?

 

[voko]

Stop trying things at random. Think: what is the number of grains so that their total surface are is 6 square meters? What is the surface area of one grain?

 

[Chestermiller]

What is the volume of 1 grain of sand if its radius is 50 microns?
What is the mass of 1 grain of sand, given that its density is 2500 kg per cubic meter?
What is the surface area of 1 grain of sand if its radius is 50 microns?
How many sand grains is required if their total surface area is 6 m^2?
What is the mass of these sand grains?

 

[icesalmon]

Stop trying things at random. Think: what is the number of grains so that their total surface are is 6 square meters? What is the surface area of one grain?

the surface area of one grain, since they are spheres with radius 50microns, is (4pi(50x10^-6)²

 

[icesalmon]

What is the volume of 1 grain of sand if its radius is 50 microns?
What is the mass of 1 grain of sand, given that its density is 2500 kg per cubic meter?
What is the surface area of 1 grain of sand if its radius is 50 microns?
How many sand grains is required if their total surface area is 6 m^2?
What is the mass of these sand grains?

the volume of one grain of sand is (4pi(50x10^-6)³)/3
I have no idea how to find out the mass of 1 grain of sand if it's density is 2500kg/m³
the surface area of 1 grain of sand is 4pi(50x10^-6)²

 

[Enigman]

What is density?

 

[icesalmon]

mass/unit volume but that mass is not per 50microns it is for 1 cubic meter, so I think I have to use dimensional analysis to convert cubic meters to 50 microns which I don't have any idea how to do because I have different powers of meters.

 

[Enigman]

How many cubic microns= 1 cubic meter? :) google it up...

 

[icesalmon]

1x10¹⁸

 

[Enigman]

mass/volume=density
plug and chug.

 

[icesalmon]

m/1x10¹⁸ = 2500kg/m³
m = (2500kg/m³)*1x10¹⁸microns³

 

[Enigman]

m/1x10¹⁸ = 2500kg/m³
m = (2500kg/m³)*1x10¹⁸microns³

Not in micron³ its 1x10^-18m³

 

[icesalmon]

I thought I found how many cubic microns were in 1 cubic meter. Why aren't those the units that I'm using?

 

[SteamKing]

Think about the volume of 1 grain of sand. It's 4*pi*(50*10^-6)^3. What are the units of this volume?

 

[Enigman]

I thought I found how many cubic microns were in 1 cubic meter. Why aren't those the units that I'm using?

Then you also found that 1 cubic micron = 1x10^ -18 meters
If you use it in microns wouldn't
m = (2500kg/m³)*1x10¹⁸microns³
be in units of kg*micron³/m³
On the other hand:
m = (2500kg/m³)*1x10^-18m³
the m³ cancels out in numerator and denominator leaving kg behind.

 

[icesalmon]

so the mass of one grain of sand of SiO₂ is 2.5x10^-15kg
now I need to figure out how many grains of sand with a volume of (4pi(50microns)^3)/3 go into one cubic meter

 

[icesalmon]

so I've got 4(2500x10^-18m^3)/3

 

[SteamKing]

so I've got 4(2500x10^-18m^3)/3

Something isn't right. R = 50 microns = 50*10^-6 meters

Volume = (4/3)pi*R^3 = (4/3)*pi*(50*10^-6)^3 m^3 / grain of sand

Surface Area = 4*pi*R^2 = 4 * pi * (50*10^-6)^2 m^2 / grain of sand

Sand has a density of 2500 kg / m^3

You want to find out:

1. how many grains of sand have a total surface area of 6 m^2, and then
2. how many kg of sand this is equivalent to.

Knowing the surface area of 1 grain of sand, how many grains have a total surface area of 6 m^2?

Knowing how many grains of sand, what is the total volume of sand?

Knowing the total volume of the sand, what is its mass?

Answer these questions, and you will have solved the problem from the OP.