Moles are simply another unit of measurement, like a meter, or a foot, or a dozen. A mole is defined as 6.02 X 10^23 of that particular item. You can have a mole of apples, a mole of grains of sand, whatever you want, but moles are usually used to measure atoms. The weight for a single mole of atoms is listed below each element on the Periodic Table, typically called by the name of "Atomic Weight".
A mole is also referred to as Avogadro's number, named after the Italian physics professor who discovered moles, Amadeo Avogadro. Avogadro theorized that equal volumes of different gases at equal temperatures contained an equal amount of atoms. His theory would later go on to be proved, and the estimated number of atoms in that measurement of gas would be thereon called Avogadro's number, known to us today as a mole.
Moles are highly important today in chemistry. They allow us to easily abbreviate large quantities of atoms without having to write out the entire number. A drop of water the size of the period at the end of this sentence contains approximately 10 trillion molecules; however, it is much simpler to use moles to communicate these numbers rather than continuously speaking in trillions and trillions of molecules.
Sources
-http://scidiv.bellevuecollege.edu/bg/moles.html
-http://www.visionlearning.com/library/module_viewer.php?mid=53
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