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Moles and avogadro’s number

It is often convenient to measure the amount of substance with a unit on a more human scale than molecules. The SI unit for this purpose was developed by the Italian scientist Amedeo Avogadro (1776–1856). (He worked from the hypothesis that equal volumes of gas at equal pressure and temperature contain equal numbers of molecules, independent of the type of gas. As mentioned above, this hypothesis has been confirmed when the ideal gas approximation applies.) A mole    (abbreviated mol) is defined as the amount of any substance that contains as many molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. (Technically, we should say “formula units,” not “molecules,” but this distinction is irrelevant for our purposes.) The number of molecules in one mole is called Avogadro’s number     ( N A ) , and the value of Avogadro’s number is now known to be

N A = 6.02 × 10 23 mol −1 .

We can now write N = N A n , where n represents the number of moles of a substance.

Avogadro’s number relates the mass of an amount of substance in grams to the number of protons and neutrons in an atom or molecule (12 for a carbon-12 atom), which roughly determine its mass. It’s natural to define a unit of mass such that the mass of an atom is approximately equal to its number of neutrons and protons. The unit of that kind accepted for use with the SI is the unified atomic mass unit (u) , also called the dalton . Specifically, a carbon-12 atom has a mass of exactly 12 u, so that its molar mass M in grams per mole is numerically equal to the mass of one carbon-12 atom in u. That equality holds for any substance. In other words, N A is not only the conversion from numbers of molecules to moles, but it is also the conversion from u to grams: 6.02 × 10 23 u = 1 g . See [link] .

The illustration shows relatively flat land with a solitary mountain, labeled “Mt. Everest for scale”, and blue sky well above the mountain top. A double-headed vertical arrow, labeled “table tennis balls”, stretches between the land and the sky.
How big is a mole? On a macroscopic level, Avogadro’s number of table tennis balls would cover Earth to a depth of about 40 km.

Now letting m s stand for the mass of a sample of a substance, we have m s = n M . Letting m stand for the mass of a molecule, we have M = N A m .

Check Your Understanding The recommended daily amount of vitamin B 3 or niacin, C 6 NH 5 O 2 , for women who are not pregnant or nursing, is 14 mg. Find the number of molecules of niacin in that amount.

We first need to calculate the molar mass (the mass of one mole) of niacin. To do this, we must multiply the number of atoms of each element in the molecule by the element’s molar mass.
( 6 mol of carbon ) ( 12.0 g/mol ) + ( 5 mol hydrogen ) ( 1.0 g/mol ) + ( 1 mol nitrogen ) ( 14 g/mol ) + ( 2 mol oxygen ) ( 16.0 g/mol ) = 123 g/mol
Then we need to calculate the number of moles in 14 mg.
( 14 mg 123 g/mol ) ( 1 g 1000 mg ) = 1.14 × 10 −4 mol .
Then, we use Avogadro’s number to calculate the number of molecules:
N = n N A = ( 1.14 × 10 −4 mol ) ( 6.02 × 10 23 molecules / mol ) = 6.85 × 10 19 molecules .

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Check Your Understanding The density of air in a classroom ( p = 1.00 atm and T = 20 ° C) is 1.28 kg/m 3 . At what pressure is the density 0.600 kg/m 3 if the temperature is kept constant?

The density of a gas is equal to a constant, the average molecular mass, times the number density N / V . From the ideal gas law, p V = N k B T , we see that N / V = p / k B T . Therefore, at constant temperature, if the density and, consequently, the number density are reduced by half, the pressure must also be reduced by half, and p f = 0.500 atm .

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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