# Determining Avogadro's Number

- Avogadro's Number Problems
- Determining Avogadro's Number
- How Big Is Avogadro's Number
- Determining Avogadro's Number Lab

Which of the following will determine the number of moles in a sample Dividing the mass of the sample by Avogadro’s number Multiplying the mass of the sample by Avogadro’s number Dividing the number of molecules in the sample by Avogadro’s number Multiplying the number of molecules in the sample by Avogadro’s number. The number of atoms or molecules (n) in a mass (m) of a pure material having atomic or molecular weight (M) is easily computed from the following equation using Avogadro's number (NA = 6.022×10 23 atoms or molecules per gram-mole): M mN n A (1) In some situations, the atomic number density (N), which is the concentration of atoms or molecules per. This number (Avogadro's number) is 6.023 X 10 23. It is the number of molecules of any gas present in a volume of 22.41 L and is the same for the lightest gas (hydrogen) as for a heavy gas such as carbon dioxide or bromine. Avogadro's number is one of the fundamental constants of chemistry.

Avogadro’s number is considered one of the few fundamental constants in chemistry. By definition, it is the number of Carbon atoms in exactly 12 grams of carbon. A mole of any substance contains an extremely large number of particles and will always be equal to. This mass to calculate Avogadro’s number. B) During the course of the electrolysis, the current decreases significantly. You include all these data in your calculation of Avogadro’s number. C) You used twice as much sulfuric acid to do your electrolysis. D) You did not allow your anode to dry completely before taking its mass at the end of the.

### Avogadro Number Calculations II

How Many Atoms or Molecules?

The value I will use for Avogadro's Number is 6.022 x 10^{23} 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 H

_{2}O contains how many molecules? (Example #2)

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

0.200 gram of H_{2}O 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 H_{2}O contains how many atoms?

0.200 gram of H_{2}O 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:**

0.450 mol x 6.022 x 10^{23} mol¯^{1} = see below for answer

**Example #2:** 0.200 mole of H_{2}O contains how many molecules?

**Solution:**

0.200 mol x 6.022 x 10^{23} 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 P_{4} and S_{8}.)

So, doing the calculation and rounding off to three sig figs, we get 2.71 x 10^{23} 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 10^{23} 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 10^{23} 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 10^{23}mol¯^{1}= 2.71 x 10^{23}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 H_{2}O 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 Two (moles ---> how many): (0.0080573 mol) (6.022 x 10^{23} atoms/mol) = 4.85 x 10^{21} atoms

**Solution to Example #4:**

Step Two: (0.01110186 mol) (6.022 x 10^{23} molecules/mol) = 6.68 x 10^{21} molecules

**Example #5:** Calculate the number of molecules in 1.058 mole of H_{2}O

**Solution:**

(1.058 mol) (6.022 x 10^{23}mol¯^{1}) = 6.371 x 10^{23}molecules

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

**Solution:**

(0.750 mol) (6.022 x 10^{23}mol¯^{1}) = 4.52 x 10^{23}atoms (to three sig figs)

**Example #7:** Calculate the number of molecules in 1.058 gram of H_{2}O

**Solution:**

^{23}molecules/mole)

Here is the solution set up in dimensional analysis style:

1 mol | 6.022 x 10^{23} | |||

1.058 g x | ––––––––– | x | –––––––––– | = 3.537 x 10^{22} 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

^{23}atoms/mole

1 mol | 6.022 x 10^{23} | |||

0.750 g x | ––––––––– | x | –––––––––– | = 8.09 x 10^{21} atoms (to three sig figs) |

55.85 g | 1 mol |

**Example #9:** Which contains more molecules: 10.0 grams of O_{2} or 50.0 grams of iodine, I_{2}?

**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 10^{23} | |||

10.0 g x | ––––––––– | x | –––––––––– | = number of O_{2} molecules |

31.998 g | 1 mol |

1 mol | 6.022 x 10^{23} | |||

50.0 g x | ––––––––– | x | –––––––––– | = number of I_{2} molecules |

253.8 g | 1 mol |

**Example #10:** 18.0 g of H_{2}O 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 10^{23}mol¯^{1}) = 6.02 x 10^{23}molecules

3) Determine number of atoms of oxygen present:

(6.02 x 10^{23}molecules) (1 O atom / 1 H_{2}O molecule) = 6.02 x 10^{23}O atoms

4) Determine number of atoms of hydrogen present:

(6.02 x 10^{23}molecules) (2 H atoms / 1 H_{2}O molecule) = 1.20 x 10^{24}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 H_{2}O, 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 10^{23}molecules) (3 atoms/molecule) = 1.81 x 10^{24}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 C_{6}H_{12}O_{6}

(b) 2 mol of (NH_{4})_{2}CO_{3}

(c) 4 mol of H_{2}O

(d) 5 mol of CH_{3}COOH

**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

### Avogadro's Number Problems

(c) 4N x 2 = 8N(d) 5N x 4 = 20N

(d) is the answer.

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**Example #12:** How many oxygen atoms are in 27.2 L of N_{2}O_{5} 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 N_{2}O_{5}:

(1.21353 mol N_{2}O_{5}) (5 mol O / 1 mol N_{2}O_{5}) = 6.06765 mol O

3) Use Avogadro's Number:

(6.06765 mol O) (6.022 x 10^{23}atoms O / mole O) = 3.65 x 10^{24}atoms O (to three sig figs)

**Example #13:** How many carbon atoms are in 0.850 mol of acetaminophen, C_{8}H_{9}NO_{2}?

**Solution:**

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

(0.850 mol C_{8}H_{9}NO_{2}) (8 mol C / mol C_{8}H_{9}NO_{2}) = 6.80 mol C

2) Use Avogadro's Number:

(6.80 mol C) (6.022 x 10^{23}atoms C / mole C) = 4.09 x 10^{24}atoms C (to three sig figs)

**Example #14:** How many atoms are in a 0.460 g sample of elemental phosphorus?

**Solution:**

_{4}. (Not P!!)

0.460 g / 123.896 g/mol = 0.00371279 mol

(6.022 x 10^{23} molecules/mol) (0.00371279 mol) = 2.23584 x 10^{21} molecules of P_{4}

(2.23584 x 10^{21} molecules) (4 atoms/molecule) = 8.94 x 10^{21} atoms (to three sig figs)

Set up using dimensional analysis style:

1 mol | 6.022 x 10^{23} molecules | 4 atoms | ||||

0.460 g x | –––––––– | x | –––––––––––––––––– | x | ––––––––– | = 8.94 x 10^{21} atoms |

123.896 g | 1 mol | 1 molecule |

**Example #15:** Which contains the most atoms?

(a) 3.5 molecules of H_{2}O

(b) 3.5 x 10^{22}molecules of N_{2}

(c) 3.5 moles of CO

(d) 3.5 g of water

**Solution:**

Choice (a): You can't have half of a molecule, so this answer should not be considered. Also, compare it to (b). Since (a) is much less than (b), (a) cannot ever be the answer to the most number of atoms.

Choice (b): this is a viable contender for the correct answer. Since there are two atoms per molecule, we have 7.0 x 10^{22} atoms. We continue to analyze the answer choices.

Choice (c): Use Avogadro's number (3.5 x 10^{23} mol¯^{1}) and compare it to choice (b). You should be able to see, even without the 3.5 moles, choice (c) is already larger than choice (b). Especially when you consider that N_{2} and CO both have 2 atoms per molecule.

Choice (d): 3.5 g of water is significantly less that the 3.5 moles of choice (c). 3.5 / 18.0 equals a bit less that 0.2 moles of water.

**Bonus Example:** A sample of C_{3}H_{8} has 2.96 x 10^{24} H atoms.

(a) How many carbon atoms does the sample contain?

(b) What is the total mass of the sample?

**Solution to (a):**

### Determining Avogadro's Number

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

3 | y | |

––––––– | = | –––––––––––––––– |

8 | 2.96 x 10^{24} H atoms |

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

y = 1.11 x 10^{24}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 10^{24}

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 C_{3}H_{8} present.

2.96 x 10^{24}/ 8 = 3.70 x 10^{23}molecules of C_{3}H_{8}

2) Divide by Avogadro's Number:

3.70 x 10^{23}/ 6.022 x 10^{23}mol¯^{1}= 0.614414 mol <--- I'll keep some guard digits

3) Use the molar mass of C_{3}H_{8}:

0.614414 mol times 44.0962 g/mol = 27.1 g (to three sig figs)

#### Introduction

Atoms and molecules are incredible tiny and weigh hardly anything, so scientists usually count them in terms of moles, which is 6.022140857 x 10^{23} particles. Why? For the same reason that we measure distance in terms of miles and donuts in terms of dozens: when you are counting to big numbers, it is easier to use big units. When eating donuts, it makes more sense to count in dozens than attempt to count individual donuts, and it is simpler to tell someone that you live 5 miles down the road than 26,400 feet.

Avogadro’s number is named to honor Amedeo Avogadro who pioneered some of the molecular theory that led to the discovery of Avogadro’s number. In this lab, you will estimate the number of molecules in a monolayer of stearic acid in order to calculate Avogadro’s number.

#### Background

To estimate Avogadro’s number, you must count the number of molecules. Most of the time, chemists simply use the mass to count molecules because molar mass relates mass and number of molecules:

mass of carbon (g) / molar mass of carbon (g/mole) = number of moles of carbon

number of moles of carbon (moles) x 6.022 x 10^{23} atoms per mole (atoms/mole) = number of atoms of carbon

However, this approach assumes you know Avogadro’s number, so we have to get a little more creative.

When measuring lots of little things, it helps to have a lot of them piled up.

Remember that molecules are physical things that take up space. One molecule is a very little thing that takes up just a little space (microscopic), but if you have a lot of them all lined up, they take up enough space for you to measure (macroscopic). When the dimensions of the stearic acid molecule are known, we can effectively count stearic acid molecules by measuring a volume of stearic acid.

Stearic acid is a non-polar hydrocarbon chain that has a polar carboxylic acid end. When you add it to water, each molecule aligns with the polar end pointing towards the water and the non-polar portion pointing up, and the molecules form a monolayer on top of the water. You can picture each molecule like a tall, skinny rectangle with dimensions 1:5.44, and the monolayer can be approximated as a cylinder. By measuring the volume and surface area of the stearic acid layer, you will be able to calculate the dimensions of the individual molecules via geometry, which is all you need to calculate the volume of the individual molecule. Comparing the volume of the monolayer to the volume of an individual molecule gives you the number of molecules in the monolayer. Since the monolayer has a known mass, and stearic acid has a known molar mass, you can calculate Avogadro’s number. Step-by-step instructions for completing the calculations are on the worksheet.

#### Procedure

Note: this should all take place in a hood to protect you from fumes.

##### Calibration of a pipet

I. Wash a 10 mL beaker (or the smallest beaker you have).

- Wash with soap and water.
- Rinse with ~1 mL of ammonia solution three times. Put the rinsate in the ammonia waste container.
- Rinse with DIH2O three times.
- Rinse with ~1 mL of acetone. Put the rinsate in the acetone waste container. Wait for the beaker to dry (a minute or two).
- Rinse the beaker with ~1/2 mL of hexane
*[CAUTION!]*three times. Put the rinsate in the hexane waste container.

II. Wash a 10 mL graduated cylinder.

- Wash with soap and water.
- Rinse with ~1 mL of acetone. Wait for it to dry.

III. Calibrate the pipet

- Put approximately 3 mL of hexane into the clean beaker.
- Use the pipet and the hexane in the beaker to fill the graduated cylinder up to exactly 1.0 mL. Count the number of drops it takes to fill it to 1 mL. Record the number of drops. Tips to ensure consistent drop size:
- Have one designated dropper. Preferably whoever has steadier hands.
- Be sure to hold the pipet straight up and down.
- Make sure no drops stick to the side of the graduated cylinder.
- Don’t let the dropper touch the sides of the cylinder.
- Work slowly and be patient.

- Pour the hexane out of the graduated cylinder and into the hexane waste container. Wait for the graduated cylinder to dry. Blowing nitrogen on the glassware will help it dry faster.
- Repeat the calibration procedure again. Record the number of drops in 1 mL.
- Repeat again if the first two calibration measurements are not within 10% of one another (example: 20 and 22 drops would be acceptable, but 20 and 25 drops would warrant another calibration).

##### Make a stearic acid monolayer

I. Prepare a large watch glass

- Measure and record the diameter of the watch glass with your ruler.
- Wash the watch glass with soap and water.
- Rinse with ammonia solution. Put rinsate in the ammonia waste container.
- Rinse thoroughly with DIH2O. Wait for it to dry.
- Once clean, be sure to avoid getting fingerprints on it. Handle wearing gloves, and hold it on the edges.
- Place the watch glass on a 400 mL beaker, which will simply hold it steady for you. Make sure the watch glass is parallel to the bench top.

II. Form the monolayer

### How Big Is Avogadro's Number

- Using your wash bottle, fill the watch glass to the brim with DIH2O.
- Pour about 3 mL of the stearic acid solution into the clean 10 mL beaker.
- Fill the pipet with the stearic acid solution. Holding it straight up and down, add one drop of stearic acid solution to the water-filled watch glass. If the watch glass is sufficiently clean, the drop should disappear quickly.
- Add the stearic acid solution drop wise until the last drop, which will remain a lens and not disappear. Record the number of drops you used. You will know you are close when you see a circular pattern forming. If you see a second lens forming, you added too much stearic acid and no longer have a monolayer.

Procedure adapted from http://chemskills.com/?q=avogadro