The number of particles in a given chemical substance is an important quantity in the analysis of chemical and nuclear reactions. However, in most practical situations, the precise number is impossible to count, and an estimate of the number is typically a large enough number to make calculations more cumbersome than desired. To make calculations simpler, chemists introduced a new unit, the mole, defined as the amount of substance that contains the same number of elementary particles (atoms, molecules, ions, electrons) as there are atoms in 12 grams of carbon-12, carbon's most common isotope. The actual number (of atoms in 12 grams of carbon-12) has been experimentally determined to be 6.02214129×1023, and this is known as Avogadro's number. The amount of a substance in a chemical is an expression of how many moles the substance contains. Avogadro's constant, equal to Avogadro's number divided by one mole, or 6.02214129×1023 mol−1, is a factor which allows conversion between the expression of a quantity in terms of number of particles, and its expression in terms of amount of substance. The molar mass of a substance is its mass divided by the amount of substance. The molar mass of carbon-12 is 12 g /mol, by definition. For other substances the molar mass is equal to the substance's mean molecular mass, where the average is taken over all the isotopes that make up that substance.
Avogadro's constant can be used as a conversion factor between mass, amount of substance and number of particles (usually atoms or molecules) as summarized by the following diagram:
It just so happens that Avogadro's number is very close to (within .38% of) a power of 2, namely 279= 604462909807314587353088. This allows for simple rough approximations in nuclear physics when considering chain reaction growth rates or particle decay.
Example: Sugar molecules
In the following example, the amount of sugar (C12H22O11) (that is, the number of moles of sugar) is known. With this information, how can you determine the total number of C12H22O11 molecules present?
Calculate the total number of C12H22O11 molecules in 1.5 mol of sugar:
# of C12H22O11 molecules= amount of substance ⋅ Avogadro's constant# of C12H22O11 molecules = 1.5 mol ⋅ 6.022 ⋅ 1023 mol−1# of C12H22O11 molecules = 9.03 ⋅ 1023
In the following example, the mass of the snowflake is known. With this information, how can you determine the number of water molecules present in the particular snowflake?
a) Calculate the molar mass of H2O:
mmH2O = 2 ⋅ mmH + 1 ⋅ mmO mmH2O = 2 ⋅ 1.01 gmol+ 1 ⋅ 16.00 gmol mmH2O = 18.02 gmol
b) Calculate the number of H2O molecules in 5.0 g of water:
* First determine the amount of substance *
nH2O = mH2O mmH2O nH2O = 5.0 g18.02 gmol nH2O = 0.28 mol
* Now determine the number of molecules as shown in the previous example *
# of H2O molecules= amount of substance ⋅ Avogadro's constant # of H2O molecules = 0.277 mol ⋅ 6.022 ⋅ 1023 mol−1 # of H2O molecules = 1.67 ⋅ 1023
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