(a) What is the volume (in km^3 ) of Avogadro's number of sand grains if each grain is a cube an

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(a) What is the volume (in km^3 ) of Avogadro's number of...

(a) What is the volume (in $\mathrm{km}^{3}$ ) of Avogadro's number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages $100 \mathrm{m}$ in width and $10.0 \mathrm{m}$ in depth? Neglect air spaces between grains.

Key Concepts

Volume Distribution Along a Geometric Shape

This concept involves understanding how a total volume can be distributed over a given shape or area to determine dimensions such as length or thickness. When a volume is spread along a structure with known cross-sectional dimensions, it is necessary to divide the total volume by the cross-sectional area to compute an extended linear dimension.

Avogadro's Number

Avogadro's number is a constant that represents the number of constituent particles (such as atoms, molecules, or grains) in one mole of a substance, approximately 6.022 x 10^23. In problems like this, it is used to scale up properties measured for one unit (such as the volume of one sand grain) to a bulk quantity containing an extremely large number of the units.

Volume of a Cube

The volume of a cube is found by raising the length of one of its sides to the third power. This fundamental geometric formula is applicable to any cubic structure and is crucial for determining the amount of space occupied by a single object before scaling the results to a larger quantity.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another, ensuring consistency and accuracy in calculations. In this context, converting lengths and volumes between units such as millimeters, meters, and kilometers is essential to ensure that all parts of the problem are expressed in the same measurement system.

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