Have you ever wondered how many molecules or atoms are in a drop of water? This is a common chemistry problem students solve because it deals with several important concepts, including chemical formulas, unit conversions, volume, mass, density, using the periodic table, the mole, and Avogadro’s number. Here is a step-by-step guide for the calculation.

- Use the chemical formula of water to find the mass of one mole of water.
- Use the density of water to find the mass per unit volume.
- Multiply mass per unit volume by the volume of a water drop to find the mass per drop.
- Convert from grams to moles to get moles of water per drop.
- Use Avogadro’s number to find the number of molecules of water per drop.
- Multiply the number of molecules times the number of atoms per molecule to find the number of atoms in a drop of water.

The average water droplet is 0.05 mL of volume. There are over 1.5 sextillion (10^{21}) molecules in a drop of water and over 5 sextillion atoms in the drop.

### How Big Is a Drop of Water?

Defining the volume of a drop of water is the first part of the problem. Water droplets vary dramatically in size. The medical and scientific community commonly define the average volume of a drop of water as exactly 0.05 mL, which means there are 20 drops of water per milliliter. This is a good starting point for calculating molecules and atoms in a drop of water.

### Chemical Formula of Water

Use the chemical formula of water to determine the mass of a single molecule. The formula of water is H_{2}O, which means there are two hydrogen atoms and one oxygen atom in each water molecule. Each water molecules contains a total of 3 atoms.

### Molar Mass of Water

Find the molar mass of water. Do this by looking up the atomic masses of hydrogen and oxygen given on the periodic table. Then add the mass of hydrogen atoms and oxygen atoms in a molecule of water. The mass of hydrogen is 1.008 g/mol and the mass of oxygen is 16.00 g/mol. The mass of one mole of water is:

mass water = 2 x mass hydrogen + mass oxygen

mass water = 2 x 1.008 + 16.00

mass water = 18.016 g/mol

The mass of one mole of water is 18.016 grams.

### Density of Water

Use the density of water to find the mass of water per unit volume. The value for density of water depends on its temperature (cold water is more dense than warm water), but a typical value is 1.00 grams per milliliter (1 g/mL). One milliliter of water has a mass of 1 gram, but a drop of water is a smaller volume than a milliliter. Multiply the mass per unit volume by the volume of a drop to find the mass of a drop:

mass of water drop = 1 g/mL x 0.05 mL

mass of water drop = 0.05 grams

Next find the number of moles of water in a drop. This is a basic grams-to-moles unit conversion using the molar mass of water:

moles in water drop = 0.05 grams x (1 mole / 18.016 grams)

moles in water drop = 0.002775 moles

One drop of water is 0.002775 moles of water.

### Using Avogadro’s Number

Avogadro’s number is the number of molecules in a mole. There are 6.022 x 10^{23} molecules of water per mole of water. Use Avogadro’s number to find the number of molecules in a drop of water. Multiply Avogadro’s number by the number of moles in a drop of water:

molecules in a drop of water = (6.022 x 10^{23} molecules/mole) x 0.002275 moles

molecules in a drop of water = 1.67 x 10^{21} water molecules **There are 1.67 x 10 ^{21} or 1.67 sextillion water molecules in a drop of water.**

There are three atoms per water molecule. Find the number of atoms in a drop of water by multiplying this number of molecules by 3.

atoms in a drop of water = 3 atoms/molecule x 1.67 x 10

^{21}molecules

atoms in a drop of water = 5.01 x 10

^{21}atoms

**There are 5.01 x 10**

^{21}or about 5 sextillion atoms in a drop of water.### Drops of Water in the Ocean

A variant of this problem is to find whether there are more atoms in a drop of water than there are drops of water in the ocean. There are different estimates for the volume of the Earth’s oceans, ranging between 1.3 billion km^{3} and 1.5 km^{3}. The United States Geological Survey uses a value of 1.338 billion km^{3}, which is 1.338 x 10^{21} liters of seawater.

To answer the question, divide this number by the volume of a drop to get the number of drops of water in the sea.

number of water drops in the ocean = 1.338 x 10^{21} liters total volume / 5.0 x 10^{-5} liters per drop

number of drops of water in the ocean = 2.676 x 10^{26} drops

Since 10^{26} is larger than 10^{21} **there are more drops of water in the ocean than there are molecules or atoms in a drop of water**. The size of the drop makes a difference, but there are between 1,000 and 100,000 more drops of water in the ocean than atoms in a drop of water.

### Reference

- Gleick, P.H. (August 28, 2006). “Where is Earth’s Water.”
*Earth’s Water Distribution*. U.S. Geological Survey.