These terms' apparently inconsistent useage can seem confusing due to what seems to be an interchangeability of similar or synonymous terms. For example, mass and weight are being used interchangeably. After all, you are making the same type of calculations to be able to "do business" in moles for comparison of "apples to oranges." However, the differences denote subtleties accounted for more precisely by using slightly different words depending upon which phenomenon is being considered. I'm assuming you know or can look up some things I'm taking for granted as already within your understanding in this matter.
Molar mass is the mass of a substance (in grams/mol), which is numerically = formula weight (in amu) For example, add the atomic weights of a chemical formula's constituent atoms, and the units for this summed amount will be in g/mol.
If the chemical formula is the symbol of an element, then the formula weight = atomic weight of that element. If the chemical formula is for a molecule, its formula weight is also called the molecular weight. Here's where there's a problem in attempting to use "one size fits all" terminology for everything, when it doesn't fit everything. Since ionic substances aren't precisely described by the term "molecules," as they're more like 3D conglomerations of ions, we use "formula units" for such things. As you can start to see, the phrase "formula weight" is also used in this case to be less imprecise.
Hope this helps! It has helped me to take some time to revisit this. - Will
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