# Avogadro’s Number and the Molar Mass of an Element

####
**Avogadro’s
Number (N**_{A})

_{A})

** Atomic mass
units provide a relative scale for the masses of the elements.

** But because atoms
have such small masses, no usable scale can be devised to weigh them in calibrated
units of atomic mass units.

** In any real
situation, we deal with macroscopic samples containing enormous numbers of
atoms. Therefore, it is convenient to have a special unit to describe a very
large number of atoms.

** The idea of
a unit to denote a particular number of objects is not new. For example, the
pair (2 items), the dozen (12 items), and the gross (144 items) are all
familiar units. Chemists measure atoms and molecules in moles.

**The mole (mol):**is the amount of a substance that contains as many elementary entities (atoms, molecules, or other particles) as there are atom exactly 12 g (or 0.012 kg) of the carbon-12 isotope.

** The actual
number of atoms in 12 g of carbon-12 is determined experimentally. This number is
called Avogadro’s number (N

_{A}), in honor of the Italian scientist Amedeo Avogadro. The currently accepted value is
** Generally,
we round Avogadro’s number to 6.022 × 10

^{23}. Thus, just as one dozen oranges contains 12 oranges, 1 mole of hydrogen atoms contains 6.022 × 10^{23}H atoms.
** The
following Figure shows samples containing 1 mole each of several common
elements.

(One mole each
of several common elements. Carbon (black charcoal powder), sulfur (yellow
powder), iron (as nails), copper (wires), and mercury (shiny liquid metal).

** The enormity
of Avogadro’s number is difficult to imagine. For example, spreading 6.022 × 10

^{23 }oranges over the entire surface of Earth would produce a layer 9 mi into space! Because atoms (and molecules) are so tiny, we need a huge number to study them in manageable quantities.####
**The
Molar Mass of an Element**

** We have seen
that 1 mole of carbon-12 atoms has a mass of exactly 12 g and contains 6.022 × 10

^{23}atoms. This mass of carbon-12 is its molar mass (M),**The molar mass**

**(M):**is the mass (in grams or kilograms) of 1 mole of units (such as atoms or molecules) of a substance.

** Note that
the molar mass of carbon-12 (in grams) is numerically equal to its atomic mass
in amu. Likewise, the atomic mass of sodium (Na) is 22.99 amu and its molar
mass is 22.99 g; the atomic mass of phosphorus is 30.97 amu and its molar mass
is 30.97 g; and so on. If we know the atomic mass of an element, we also know
its molar mass.

** Knowing the
molar mass and Avogadro’s number, we can calculate the mass of a single atom in
grams. For example, we know the molar mass of carbon-12 is 12.00 g and there
are 6.022 × 10

^{23}carbon-12 atoms in 1 mole of the substance; therefore, the mass of one carbon-12 atom is given by:
** We can use
the preceding result to determine the relationship between atomic mass units
and grams. Because the mass of every carbon-12 atom is exactly 12 amu, the
number of atomic mass units equivalent to 1 gram is:

This example
shows that Avogadro’s number can be used to convert from the atomic mass units
to mass in grams and vice versa.

** The
following figure shows The relationships between mass (m in grams) of an
element and number of moles of an element (n) and between number of moles of an
element and number of atoms (N) of an element. μ is the molar mass (g/mol) of
the element and N

_{A}is Avogadro’s number
** The notions
of Avogadro’s number and molar mass enable us to carry out conversions between
mass and moles of atoms and between moles and number of atoms (Figure up) . We will employ the following conversion factors in the calculations:

where X
represents the symbol of an element. Using the proper conversion factors we can
convert one quantity to another.

####
**Solved
problems**

**Problem (1): Zinc (Zn) is a silvery metal that is used in making brass (with copper) and in plating**

**iron to prevent corrosion. How many moles of Zn are there in 45.9 g of Zn?**

**Strategy:**

We are trying to solve for moles of Zn. What
conversion factor do we need

**to convert between grams and moles? Arrange the appropriate conversion factor so that****grams cancel and the unit mol is obtained for your answer.****Solution:**

The conversion
factor needed to convert between grams and moles is the molar mass. In the periodic
table (see inside front cover) we see that the molar mass of Zn is 65.39 g.
This can be expressed as:

From this
equality, we can write the two conversion factors:

The conversion factor on the
left is the correct one. Grams will cancel, leaving the unit of mol for the
answer. The number of moles of Zn is:

Thus, there is
0.702 mole of Zn in 45.9 g of Zn.

**Check:**

Because 45.9 g
is less than the molar mass of Zn, we expect the result to be less than 1
mole.

**Problem (2): Sulfur (S) is a nonmetallic element that is present in coal. When coal is burned, sulfur is converted to sulfur dioxide and eventually to sulfuric acid, which gives rise to the acid rain phenomenon. How many atoms are in 25.1 g of S?**

**Strategy**:

The question asks for atoms of sulfur. We
cannot convert directly from grams to atoms of sulfur. What unit do we need to
convert grams of sulfur to in order to convert to atoms? What does Avogadro’s
number represent?

**Solution:**

We need two conversions: first from grams to
moles and then from moles to number of particles (atoms). The first step is similar
to problem (1). Because

the conversion
factor is:

Avogadro’s
number is the key to the second step. We have:

and the
conversion factors are:

The conversion
factor on the left is the one we need because it has the number of S atoms in
the numerator. We can solve the problem by first calculating the number of
moles contained in 25.1 g of S, and then calculating the number of S atoms from
the number of moles of S:

We can combine
these conversions in one step as follows:

Thus, there are
4.71 × 10

^{23}atoms of S in 25.1 g of S.**Check:**

Should 25.1 g S contain fewer than Avogadro’s
number of atoms? What mass of S would contain Avogadro’s number of atoms?

**Problem (3): Silver (Ag) is a precious metal used mainly in jewelry. What is the mass (in grams) of one Ag atom?**

**Strategy:**

The question asks for the mass of one Ag atom.
How many Ag atoms are in 1 mole of Ag and what is the molar mass of Ag?

**Solution:**

Because 1 mole of Ag atom contains 6.022 × 10

^{23}Ag atoms and has a mass of 107.9 g, we can calculate the mass of one Ag atom as follows:**Check:**

Because 6.022 × 10

^{23}atoms of Ag have a mass 107.9 g, one atom of Ag should have a significantly smaller mass.**Reference:**

*General Chemistry: The Essential Concepts / Raymond Chang , Jason Overby. (sixth edition)*

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