### Avogadro Number Calculations II How Many Atoms or Molecules?

The value I will use for Avogadro's Number is 6.022 x 1023 mol¯1.

Types of problems you might be asked look something like these:

0.450 mole of Fe contains how many atoms? (Example #1)
0.200 mole of H2O contains how many molecules? (Example #2)

0.450 gram of Fe contains how many atoms? (Example #3)
0.200 gram of H2O contains how many molecules? (Example #4)

When the word gram replaces mole, you have a related set of problems which requires one more step.

And, two more:

0.200 mole of H2O contains how many atoms?
0.200 gram of H2O contains how many atoms?

When the word gram replaces mole, you have a related set of problems which requires one more step. In addition, the two just above will have even another step, one to determine the number of atoms once you know the number of molecules.

Here is a graphic of the procedure steps: Pick the box of the data you are given in the problem and follow the steps toward the box containing what you are asked for in the problem.

Example #1: 0.450 mole of Fe contains how many atoms?

Solution:

Start from the box labeled "Moles of Substance" and move (to the right) to the box labeled "Number of Atoms or Molecules." What do you have to do to get there? That's right - multiply by Avogadro's Number.

0.450 mol x 6.022 x 1023 mol¯1 = see below for answer

Example #2: 0.200 mole of H2O contains how many molecules?

Solution:

Start at the same box as Example #1.

0.200 mol x 6.022 x 1023 mol¯1 = see below for answer

### The answers (including units) to Examples #1 and #2

The unit on Avogadro's Number might look a bit weird. It is mol¯1 and you would say "per mole" out loud. The question then is WHAT per mole?

The answer is that it depends on the problem. In the first example, I used iron, an element. Almost all elements come in the form of individual atoms, so the correct numerator with most elements is "atoms." (The exceptions would be the diatomic elements plus P4 and S8.)

So, doing the calculation and rounding off to three sig figs, we get 2.71 x 1023 atoms. Notice "atoms" never gets written until the end. It is assumed to be there in the case of elements. If you wrote Avogadro's Number with the unit atoms/mol in the problem, you would be correct.

The same type of discussion applies to substances which are molecular in nature, such as water. So the numerator I would use in example #2 is "molecule" and the answer is 1.20 x 1023 molecules.

Once again, the numerator part of Avogadro's Number depends on what is in the problem. Other possible numerators include "formula units," ions, or electrons. These, of course, are all specific to a given problem. When a general word is used, the most common one is "entities," as in 6.022 x 1023 entities/mol.

Keep this in mind: the "atoms" or "molecules" part of the unit is often omitted and simply understood to be present. However, it will often show up in the answer. Like this:

0.450 mol x 6.022 x 1023 mol¯1 = 2.71 x 1023 atoms

It's not that a mistake was made, it's that the "atoms" part of atoms per mole was simply assumed to be there.

Example #3: 0.450 gram of Fe contains how many atoms?

Example #4: 0.200 gram of H2O contains how many molecules?

Look at the solution steps in the image above and you'll see we have to go from grams (on the left of the image above) across to the right through moles and then to how many atoms or molecules.

Solution to Example #3:

Step One (grams ---> moles): 0.450 g / 55.85 g/mol = 0.0080573 mol

Step Two (moles ---> how many): (0.0080573 mol) (6.022 x 1023 atoms/mol) = 4.85 x 1021 atoms

Solution to Example #4:

Step One: 0.200 g / 18.015 g/mol = 0.01110186 mol

Step Two: (0.01110186 mol) (6.022 x 1023 molecules/mol) = 6.68 x 1021 molecules

Example #5: Calculate the number of molecules in 1.058 mole of H2O

Solution:

(1.058 mol) (6.022 x 1023 mol¯1) = 6.371 x 1023 molecules

Example #6: Calculate the number of atoms in 0.750 mole of Fe

Solution:

(0.750 mol) (6.022 x 1023 mol¯1) = 4.52 x 1023 atoms (to three sig figs)

Example #7: Calculate the number of molecules in 1.058 gram of H2O

Solution:

(1.058 g / 18.015 g/mol) (6.022 x 1023 molecules/mole)

Here is the solution set up in dimensional analysis style:

 1 mol 6.022 x 1023 1.058 g x ––––––––– x –––––––––– = 3.537 x 1022 molecules (to four sig figs) 18.015 g 1 mol ↑ grams to moles ↑ ↑ moles to ↑molecules

Example #8: Calculate the number of atoms in 0.750 gram of Fe

(0.750 gram divided by 55.85 g/mole) x 6.022 x 1023atoms/mole

 1 mol 6.022 x 1023 0.750 g x ––––––––– x –––––––––– = 8.09 x 1021 atoms (to three sig figs) 55.85 g 1 mol

Example #9: Which contains more molecules: 10.0 grams of O2 or 50.0 grams of iodine, I2?

Solution:

Basically, this is just two two-step problems in one sentence. Convert each gram value to its mole equivalent. Then, multiply the mole value by Avogadro's Number. Finally, compare these last two values and pick the larger value. That is the one with more molecules.

 1 mol 6.022 x 1023 10.0 g x ––––––––– x –––––––––– = number of O2 molecules 31.998 g 1 mol

 1 mol 6.022 x 1023 50.0 g x ––––––––– x –––––––––– = number of I2 molecules 253.8 g 1 mol

Example #10: 18.0 g of H2O is present. (a) How many oxygen atoms are present? (b) How many hydrogen atoms are present?

Solution:

1) Convert grams to moles:

18.0 g / 18.0 g/mol = 1.00 mol

2) Convert moles to molecules:

(1.00 mol) (6.02 x 1023 mol¯1) = 6.02 x 1023 molecules

3) Determine number of atoms of oxygen present:

(6.02 x 1023 molecules) (1 O atom / 1 H2O molecule) = 6.02 x 1023 O atoms

4) Determine number of atoms of hydrogen present:

(6.02 x 1023 molecules) (2 H atoms / 1 H2O molecule) = 1.20 x 1024 H atoms (to three sig figs)

Notice that there is an additional step (as seen in step 3 for O and step 4 for H). You multiply the number of molecules times how many of that atom are present in the molecule. In one molecule of H2O, there are 2 atoms of H and 1 atom of O.

Sometimes, you will be asked for the total atoms present in the sample. Do it this way:

(6.02 x 1023 molecules) (3 atoms/molecule) = 1.81 x 1024 atoms (to three sig figs)

The 3 represents the total atoms in one molecule of water: one O atom and two H atoms.

Example #11: Which of the following contains the greatest number of hydrogen atoms?

(a) 1 mol of C6H12O6
(b) 2 mol of (NH4)2CO3
(c) 4 mol of H2O
(d) 5 mol of CH3COOH

Solution:

1) Each mole of molecules contains N number of molecules, where N equals Avogadro's Number. How many molecules are in each answer:

(a) 1 x N = N
(b) 2 x N = 2N
(c) 4 x N = 4N
(d) N x 5 = 5N

2) Each N times the number of hydrogen atoms in a formula equals the total number of hydrogen atoms in the sample:

(a) N x 12 = 12N
(b) 2N x 8 = 16N
(c) 4N x 2 = 8N
(d) 5N x 4 = 20N

Example #12: How many oxygen atoms are in 27.2 L of N2O5 at STP?

Solution:

1) Given STP, we can use molar volume:

27.2 L / 22.414 L/mol = 1.21353 mol

2) There are five moles of O atoms in one mole of N2O5:

(1.21353 mol N2O5) (5 mol O / 1 mol N2O5) = 6.06765 mol O

(6.06765 mol O) (6.022 x 1023 atoms O / mole O) = 3.65 x 1024 atoms O (to three sig figs)

Example #13: How many carbon atoms are in 0.850 mol of acetaminophen, C8H9NO2?

Solution:

1) There are 8 moles of C in every mole of acetaminophen:

(0.850 mol C8H9NO2) (8 mol C / mol C8H9NO2) = 6.80 mol C

(6.80 mol C) (6.022 x 1023 atoms C / mole C) = 4.09 x 1024 atoms C (to three sig figs)

Bonus Example: A sample of C3H8 has 2.96 x 1024 H atoms.

(a) How many carbon atoms does the sample contain?
(b) What is the total mass of the sample?

Solution to (a):

1) The ratio between C and H is 3 to 8, so this:

 3 y ––––––– = –––––––––––––––– 8 2.96 x 1024 H atoms

2) will tell us the number of carbon atoms present:

y = 1.11 x 1024 carbon atoms

3) By the way, the above ratio and proportion can also be written like this:

3 is to 8 as y is to 2.96 x 1024

Be sure you understand that the two different ways to present the ratio and proportion mean the same thing.

Solution to (b) using hydrogen:

1) Determine the moles of C3H8 present.

2.96 x 1024 / 8 = 3.70 x 1023 molecules of C3H8