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How Many Molecules Are in a Drop of Water?

To accurately estimate the number of molecules in a drop of water you must know the volume of a raindrop or water droplet, the density of water, the molecular weight of a water molecule, and the number of moles in one gram of water. (In chemistry, 1 mole = 6.0221415×10²³ molecules, where 6.0221415×10²³ is the constant known as Avogadro's Number.)

Here's how you can put all of this information together to calculate the number of molecules in a drop of water, step by step.

1. The Volume of a Drop

A drop of water falling through the air has a spherical shape between 2mm and 5mm across. As the diameter approaches 5mm, the drop becomes unstable and splits into several smaller drops. For the sake of example, let's assume that one drop is 3mm in diameter.

The volume of a 3mm drop of water is (4/3)πr³, where r is the radius. In this case, r = 1.5mm. Thus, the volume of the drop is

(4/3)π1.5³ = 14.137167 mm³, or 0.014137167 cm³, or 0.014137167 mL.

2. The Density of Water and the Weight of a Drop

Water at room temperature (about 21°C or 69.8°F) has a density of 0.997992 g/mL. Cold water is more dense, and warm water is less dense. The least dense form of water is ice. In this example, let's assume the water is about 21°C. Since the volume of the drop is 0.014137167 mL, its weight is

(0.997992 g/ML)(0.014137167 mL) = 0.01410878 g.

3. The Molar Mass (Molecular Weight) of Water and Avogadro's Number

A water molecule is comprised of one oxygen atom and two hydrogen atoms. The atomic mass of oxygen is 15.9994 amu, and that of hydrogen is 1.00794 amu. Thus, one molecule of H₂O is 18.01528 amu. This means that 18.01528 grams of water contains 6.0221415×10²³ molecules, or equivalently, 6.0221415×10²³ water molecules weigh 18.01528 grams.

4. The Number of Molecules in a Drop of Water

Since we know that one drop of water weighs 0.01410878 grams, and there are 6.0221415×10²³ water molecules in 18.01528 grams of water, we can now compute the number of molecules N in a single drop of water. The equation we must solve is

N molecules / 6.0221415×10²³ molecules = 0.01410878 g / 18.01528 g

This gives us

N = 4.716278×10²⁰ molecules.

Note: The most important factor is the size of the drop, which varies according to the situation. While the 3mm diameter estimate is appropriate for an average raindrop, a drop of water hanging from your faucet may be much larger.

The temperature of the water has some effect on the final answer. If you want more precision, here is a list of densities for various water temperatures:

-8°C: 0.99865 g/mL
-4°C: 0.999417 g/mL
0°C: 0.9998425 g/mL
0.1°C: 0.999847 g/mL
3.98°C: 0.999975 g/mL (densest)
10°C: 0.9997026 g/mL
15°C: 0.999099 g/mL
20°C: 0.998203 g/mL
30°C: 0.995646 g/mL
50°C: 0.9880393 g/mL
100°C: 0.9583665 g/mL

Finally, water purity is another factor that must be taken into consideration. The example above assumes the drop is pure water, but the presence of pollutants and tiny gas bubbles will change the final answer.