To determine the number of atoms of oxygen represented by 3 mg of H2O, we first need to understand atomic mass units and Avogadro’s number. Atomic mass units (amu) are used to express the masses of individual atoms, where 1 amu is defined as 1/12 the mass of a carbon-12 atom. Avogadro’s number (NA) is the number of particles (usually atoms or molecules) in one mole of a substance, which is equal to 6.022×1023. By using atomic masses and Avogadro’s number, we can convert between mass and moles at the atomic level.

## Step 1: Convert mg to g

The first step is to convert the mass unit from milligrams (mg) to grams (g), since atomic masses are usually expressed in grams.

3 mg H2O x (1 g / 1000 mg) = 0.003 g H2O

## Step 2: Calculate moles of H2O

Next, we calculate the number of moles of H2O corresponding to 0.003 g using the molecular weight of H2O (18.01528 g/mol):

Moles H2O = 0.003 g H2O / 18.01528 g/mol

= 1.666 x 10-4 mol H2O

## Step 3: Use Avogadro’s number to get molecules

Now that we have the number of moles, we can use Avogadro’s number to convert moles H2O to number of H2O molecules:

Molecules H2O = 1.666 x 10-4 mol H2O x (6.022×1023 molecules/mol)

= 1.004 x 1020 H2O molecules

## Step 4: Atoms of oxygen per molecule

Each water molecule (H2O) contains 1 oxygen atom. Therefore, the number of oxygen atoms is equal to the number of H2O molecules:

Oxygen atoms = 1.004 x 1020 atoms

### Conclusion

By following the steps to convert mass to moles and then use Avogadro’s number to get number of molecules, we determined there are 1.004 x 1020 atoms of oxygen represented by 3 mg of H2O. This demonstrates how we can relate macroscopic masses to the underlying atomic scale using fundamental constants like Avogadro’s number. Going from mg to atoms required conversions between mass units, calculations with molecular weights, and knowledge of the composition of H2O molecules. Similar principles can be applied to determine the number of atoms or molecules for any pure substance when given its mass.

## Detailed Explanation

Let’s go through each step in more detail:

### Converting milligrams to grams

Mass is commonly expressed in different units, depending on the scale of the substance being measured. For very small masses like in chemistry, milligrams (mg) or micrograms (μg) are often used. Atomic masses, however, are typically expressed in grams (g).

To convert milligrams to grams, we use the following conversion:

1 mg = 1 x 10-3 g

So to convert 3 mg to grams:

3 mg x (1 g / 1000 mg) = 0.003 g

This gives us the mass in grams to use for calculations with atomic masses.

### Calculating moles from mass

In chemistry, the number of moles indicates the amount of a substance present. One mole represents 6.022×1023 particles of a substance. The mass and moles of a substance are related by its molecular weight.

The molecular weight of water is 18.01528 g/mol. This means one mole of water has a mass of 18.01528 grams.

Using the mass-moles relationship, we calculate the moles of H2O corresponding to 0.003 g:

Moles H2O = Mass (g) / Molecular weight (g/mol)

Moles H2O = 0.003 g / 18.01528 g/mol

Moles H2O = 1.666 x 10-4 mol

Therefore, 0.003 g H2O represents 1.666 x 10-4 moles of H2O molecules.

### Relating moles and number of molecules using Avogadro’s number

Avogadro’s number is defined as the number of constituent particles (usually atoms or molecules) in one mole of a substance. It has a value of 6.022×1023 particles per mole.

This constant allows us to convert between the number of moles and the actual number of individual molecules.

For our example, with 1.666 x 10-4 moles of H2O:

Molecules H2O = Moles H2O x Avogadro’s number

= 1.666 x 10-4 mol H2O x (6.022×1023 molecules/mol)

= 1.004 x 1020 H2O molecules

Using Avogadro’s number as the conversion factor between moles and molecules provides the actual number of H2O molecules present in the original 3 mg mass.

### Calculating atoms from molecules

Since we know the composition of a water molecule is H2O, with 2 hydrogen atoms and 1 oxygen atom, we can determine the number of oxygen atoms from the number of H2O molecules:

Each H2O molecule has 1 oxygen atom

Therefore, number of oxygen atoms = number of H2O molecules

= 1.004 x 1020 oxygen atoms

This provides the final number of oxygen atoms corresponding to the original mass of 3 mg H2O.

By determining the number of molecules based on the mass, then relating molecules to atoms, we can connect the macroscopic mass measurement to the quantity at the atomic level.

## Why Avogadro’s Number is Essential

Avogadro’s number is a fundamentally important constant in chemistry and physics because it links the units we use to measure a substance (moles) with discrete particles too small to see like atoms and molecules.

Some key reasons why Avogadro’s number is essential:

- It provides a quantitative connection between the mass of a substance and the number of microscopic particles comprising that substance.
- Allows conversion between the macroscopic “amount” scale (moles) and the microscopic particle number scale.
- Enables calculation of molar quantities for individual atoms or molecules.
- Necessary for defining stoichiometry relations in chemical reactions.
- Links between thermodynamic variables like pressure, volume and temperature to the molecular level.
- Critical for determining concentrations of solutions and reporting analyses in chemistry.
- Required for calculating density and other physical properties on a per atom/molecule basis.

Without Avogadro’s number, many fundamental calculations in chemistry, physics and biology would not be possible. It provides the vital connection between the observable masses and volumes we can measure, and the discrete atomic and molecular scale that gives rise to macroscopic phenomena. Much of modern science relies implicitly on the relationships made accessible by Avogadro’s constant.

## How to Calculate Atoms/Molecules from Different Masses

The process we used to determine the number of oxygen atoms from 3 mg of water can be generalized to calculate the number of atoms or molecules from any mass of a substance:

- Convert mass in mg to g
- Determine moles from mass using compound molecular weight
- Use Avogadro’s number to convert moles to molecules
- If needed, calculate atoms based on known chemical composition

Let’s practice this process on a couple different examples:

### Example 1: Calculate atoms in 5 mg NaOH

- 5 mg NaOH x (1 g / 1000 mg) = 0.005 g NaOH
- Moles NaOH = 0.005 g / 40 g/mol = 1.25 x 10-4 mol
- Molecules NaOH = 1.25 x 10-4 mol x (6.022 x 1023 molecules/mol) = 7.5 x 1019
- Each NaOH molecule has 1 Na atom and 1 O atom
- Therefore, 7.5 x 1019 Na atoms and 7.5 x 1019 O atoms

### Example 2: Calculate sulfur atoms in 10 mg H2S

- 10 mg H2S x (1 g / 1000 mg) = 0.01 g H2S
- Moles H2S = 0.01 g / 34 g/mol = 2.94 x 10-4 mol
- Molecules H2S = 2.94 x 10-4 mol x (6.022 x 1023 molecules/mol) = 1.77 x 1020
- Each H2S molecule has 1 S atom
- Therefore, 1.77 x 1020 S atoms

Being able to determine the number of atoms or molecules from a substance’s mass is fundamental to many applications in chemistry, from stoichiometry to chemical analysis. Practice these steps on different sample mass and compound scenarios to get comfortable with interconverting between mass, moles and atom/molecule quantities.

## Practice Problems

Try calculating the number of atoms or molecules for the following masses:

- Number of carbon atoms in 15 mg of CO2
- Number of ammonia (NH3) molecules in 0.005 g NH3
- Number of lead atoms in 20 mg PbCl2
- Number of water molecules in 0.002 g H2O
- Number of calcium ions in 100 mg CaCO3

Solutions:

- 15 mg CO2 x (1 g / 1000 mg) = 0.015 g CO2

Moles CO2 = 0.015 g / 44 g/mol = 3.41 x 10-4 mol

Molecules CO2 = 3.41 x 10-4 mol x (6.022 x 1023 molecules/mol) = 2.05 x 1020

Each CO2 molecule has 1 C atom

Therefore, 2.05 x 1020 C atoms - 0.005 g NH3 x (1 mol / 17 g) = 2.94 x 10-4 mol NH3

Molecules NH3 = 2.94 x 10-4 mol x (6.022 x 1023 molecules/mol) = 1.77 x 1020 - 20 mg PbCl2 x (1 g / 1000 mg) = 0.02 g PbCl2

Moles PbCl2 = 0.02 g / 278 g/mol = 7.19 x 10-5 mol

Molecules PbCl2 = 7.19 x 10-5 mol x (6.022 x 1023 molecules/mol) = 4.33 x 1018

Each PbCl2 molecule has 1 Pb atom

Therefore, 4.33 x 1018 Pb atoms - 0.002 g H2O x (1 mol / 18 g) = 1.11 x 10-4 mol H2O

Molecules H2O = 1.11 x 10-4 mol x (6.022 x 1023 molecules/mol) = 6.69 x 1019 - 100 mg CaCO3 x (1 g / 1000 mg) = 0.1 g CaCO3

Moles CaCO3 = 0.1 g / 100 g/mol = 1 x 10-3 mol

Molecules CaCO3 = 1 x 10-3 mol x (6.022 x 1023 molecules/mol) = 6.022 x 1020

Each CaCO3 molecule has 1 Ca ion

Therefore, 6.022 x 1020 Ca ions

Practice converting different masses to atoms and molecules using the step-by-step method outlined. This strengthens your skill in using atomic mass units, Avogadro’s number, and chemical composition. These examples also reinforce how macroscopic mass measurements relate to the fundamental atomic and molecular scale.

## Other Applications

Some other applications where calculating the number of atoms/molecules from mass measurements is important:

### Stoichiometry

Determining the number of moles and molecules is critical for stoichiometric calculations, which rely on using mole ratios to relate quantities of different substances in chemical reactions. Stoichiometry allows chemists to determine yields, analyze reaction efficiency, and quantify products formed based on known reactant amounts.

### Atomic emission spectroscopy

This analytical technique relies on relating the concentration of an element in a sample to the intensity of light emitted from excited atoms. By combining the sample mass analyzed, molecular weight, and Avogadro’s number, the number of atoms excited in the instrument can be determined.

### X-ray diffraction

X-ray diffraction techniques like powder X-ray diffraction depend on scattering X-rays off the electrons of atoms in a crystal lattice. The intensity of diffraction peaks provides information on how atoms are spatially arranged. The number of atoms contributing to diffraction relates to crystal size based on the sample mass and Avogadro’s number.

### Nuclear chemistry

In nuclear reactions, the concept of “moles” can be applied to nuclides using Avogadro’s number. Given the mass of a nuclide, one can calculate the activity, decay energy, or other radiological parameters based on the number of atoms present. This is important for fields like radiochemistry, nuclear physics, and radiation safety.

### Nanoscience

When working with nanoscale materials and nanoparticles, it is often necessary to determine the number of atoms per particle to characterize size and fundamental properties dependent on atom count. Measuring the mass and using Avogadro’s number allows calculation of atoms in individual nanoparticles.

## Conclusion

The number of oxygen atoms represented by 3 mg of H2O is 1.004 x 1020 atoms. By converting milligrams to grams, using molecular weight to determine moles, then applying Avogadro’s number to get molecule count and composition to get atom count, we can relate a macroscopic mass to the quantity at the atomic level. Avogadro’s number provides the vital link between mass and atom/molecule quantities. This method can be applied to any pure substance. Understanding these calculations is essential for stoichiometry, chemical analysis, nanoscience, and other fields requiring knowledge of the number of atoms or molecules present. Practice with different mass and compound scenarios helps build proficiency.

“There are 7 atoms of oxygen present in Mg(OH)2” and “Therefore, there are 7 atoms in Mg(OH)2: one magnesium atom, two oxygen atoms, and two hydrogen atoms.”

Both are false statements. 2 oxygen atoms, 5 atoms total in magnesium hydroxide.