Volume of a grain of sand

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Chapter 2

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Retired, collecting minerals and stamps, growing flowers and vegetables, and when the spirit moves, toying with technology
How do you find the volume of a grain?
How many grains of sand are in a cubic meter?
How do you measure the volume of a sand particle?
How big is a grain of sand in meters?

Finding the volume of a sand grain:

The formula for the volume of a sphere is:

V = (4π/3)R3,
where R = radius of sphere.

R = 0.25mm = 2.5 x 10-1mm
= (2.5 x 10-1mm)(1m/103mm)
= 2.5 x 10-4m.

(4π/3)R3 ~= 4.18 x (2.5 x 10-4m)3
= 4.18 x (15.6 x 10-12m3
= 6.48 x 10-11m3.

(Review the Scientific Notation tutorial for help here.)

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Bibliographic Entry	Result (w/surrounding text)	Standardized Result
Earth Science. Englewood Cliffs: Prentice Hall, 1987: 569.	"Mineral: Quartz, Specific Gravity: 2.65"	N/A
"Sand." World Book Encyclopedia. Chicago: World Book, 2000.	"Scientists define sand as grains that measure from 1/400 inch (0.06 millimeter) to 1/12 inch (2.1 millimeters) in diameter."	0.30 μg–13 mg
Hess, Kunze, Leslie, Letro, Millage, Sharp, & Theodore Snow. Earth Science: Geology, the Environment, and the Universe. Columbus, OH: Glencoe McGraw Hill, 2002.	"Sand may be sub-divided into five categories according to grain size: (1) Very coarse (2 mm–1 mm) (2) Coarse (1 mm–0.5 mm) (3) Medium (0.5 mm–0.25 mm) (4) Fine (0.25 mm–0.10 mm) and (5) Fine sand (0.10 mm–0.05 mm)."	0.17 μg–11 mg
Berman, Bob. "Much Ado About Nothing: Time to let go of matter — emptiness is what rules the universe." Discover. 23,7 (July 2002).	"But if you extended a one-inch wide tube all the way from Earth to Vega and scooped up every bit of matter within, the contents would weigh just one-millionth of an ounce, roughly equal to a grain of sand."	28 μg
Corbett, Dan, Stafford, Kate, and Wright, Patrick. Introduction to String Theory.	"Strings' minimum energies are actually whole-number multiples of the Planck energy (roughly 1000 kilowatt-hours), which, translated into mass, yields the Planck mass (ten billion billion times that of a proton; roughly 1/100 of 1/000 of a gram; about the mass of a grain of sand)."	10 μg

Sand is a type of sediment produced by the breakdown of various types of rocks. Once parted from the original source rock, the material is then eroded and transported via wind, water, or even ice. Sand often lands at the deposits of lakes or rivers as sand dunes or may ultimately reach the depths of the sea where it may harden into sedimentary rock. Sand varies in size, color, and composition.

The composition of sand depends largely on the source material. Mineral sands are old beach sands that are made up of the important minerals, rutile, ilmenite, zircon, and monazite. Since these minerals are heavy they are also known as "heavy minerals". Most beach sands, however, are composed of grains of the mineral quartz. The relative density (also known as specific gravity) of quartz is 2.65. Relative density refers to the weight of a mineral relative to an equal volume of water.

Since the grain size of sand may vary, the mass of the individual grain of sand may vary according to its size as well as its composition. Using the ranges of diameters, I obtained the volume of the sand grain by using the formula for the volume of a sphere

volume = 4/3π·radius3

Since we know the density of quartz, using the density formula

density = mass/volume

I obtained the mass of a grain of sand for both ranges of diameters.

diameter	=	0.060 mm	diameter	=	2.10 mm
volume	=	1.13 × 10−13 m3	volume	=	4.85 × 10−9 m3
mass	=	3.00 × 10−10 kg → 0.30 μg	mass	=	1.28 × 10−5 kg → 13 mg

Marina Theodoris -- 2003

Did you ever wonder how many grains of sand were on a beach? Or perhaps the simpler question of how many grains of sand fit into a 20 ml collecting vial like the one in the cover photo? I imagine you probably have not, but I did.

To get started, we know that 20 ml is equal to 20 cubic centimeters. Now we also will make three assumptions:

assume a medium-grained, well-sorted sand with a grain diameter of 0.33 mm (all grains equal)
assume well rounded spherical grains
assume hexagonal close packing (HCP) of the grains where alternating layers of grains sit on top of each other. Each sphere will touch 12 others, 6 within its layer and 3 in the layers above and below. With this packing of grains, the grains will comprise 60% of the volume in the vial. The rest will be pore space. Put another way, the porosity in the sand will be 40%

Now we are ready to do some math. There are multiple ways to perform this calculation, but let’s work it two different ways. Why, you ask, do we do it two ways? Well, because we can. And because math with a purpose is fun!

First method: Convert the problem into a standard cube and work only with length dimensions. This simple method requires only the first two assumptions.

Each edge of a cube with a volume of 20 cubic centimeters will be 2.714 cm long. (or the cube root of 20). Yes, I do know I violated significant digit rules with this conversion and others).
This allows for 82.2 cubes with a diameter of 0.33 mm to align along each edge (27.14 mm/0.33 mm = 82.2)
2 x 82.2 x82.2 is ~555,000
There are about 555,000 tiny cubes with an edge length of 0.33 mm in a 20 cubic centimeter or 20 ml volume. Each could hold one spherical grain if they were stacked directly above each other.

Of course, they would never stack that way so …

Second method: Construct the problem to account for the spherical grain shape that is more commensurate with actual sand and incorporating the third assumption.

The volume of a sphere is equal to 4/3 r3 where π is 3.1416 and r is the sphere radius (in this case 0.166 mm). The volume of one sand grain is therefore 0.192 cubic millimeters.
There are 20,000 cubic millimeters in 20 cubic centimeters (or ml).
So there is room for ~1,040,000 sand grains in that volume (20,000/0.0192).
BUT wait, we assumed HCP for these grains and 40% porosity. Therefore, we must multiply our calculated grain count by the percentage of the volume that would be occupied by grains. (60% of 1,040,000 = 624,000.

The larger number (624,000 grains in a 20ml volume) is likely a more accurate assessment because it incorporates the concept that the grains will fill the space more effectively that just sitting atop the center of the one below. In reality all the grains are not the same size or shape and these size variations generally allow for closer packing as smaller grains settle into spaces between larger grains. However, I do not think I plan to count sand grains in any of my 20 ml sample vials to verify this.

20 ml vial of medium-grained garnet-rich sand from Hamlin Beach on Lake Ontario. Imagine that, over half a million grains of sand!

Remember, sand grain size ranges from 0.06mm (very fine) to 2.0 mm (very coarse). There would be a large difference in the number of sand grains in your 20 ml sample vial depending on the grain size of the sand. I will let you calculate this difference. Anyone who does can report on their results at the next club meeting.

I will also leave it to you to calculate how many 20ml vials it would take to remove all the sand from your favorite beach!

Who said math was useless?

Retired, collecting minerals and stamps, growing flowers and vegetables, and when the spirit moves, toying with technology

How do you find the volume of a grain?

Finding the volume of a sand grain: The formula for the volume of a sphere is: V = (4π/3)R³, where R = radius of sphere.

How many grains of sand are in a cubic meter?

If the diameter of the grains is on average 1 mm, this would give 1.30 × 10⁹ (1,300 million) grains in one cubic metre of sand.

How do you measure the volume of a sand particle?

Use the funnel to pour a sample of dry sand into the graduated cylinder. You want it to be about half-way full (5-6 mL), but the exact amount is not critical. Read the actual volume of sand using the markings on the graduated cylinder. Record the measured volume in your lab notebook.