Why Do Helium Filled Balloons Float?

Helium is an element, atomic number 2, atomic weight 4.003. It is lighter than air, and is used to fill balloons because they will rise, whether party balloons, weather balloons, gas balloons (for travel) or blimps. This is due to helium's lower-than-air density.

Ideal Gas Law

More than a century ago, the basic gas laws were derived. These laws relate pressure, mass and volume. The Ideal Gas Law reads,

pV = nRT

This law says pressure times volume of a gas V, is equal to the "moles" of the gas n, times a constant R, times its temperature T. What are moles? Moles equal the weight of a gas divided by its molecular weight. The gas law becomes,

pV = (w/mw) x (RT)
Pressure times Volume = Weight divided by Molecular Weight times constant R times temperature T

Now let's rearrange this formula so that only the weight and the volume are on the left side of the equation and everything else is on the right side. We get the result,

w/V = (p/RT) x mw

Comparison to Water

Consider a swimming pool of water. If a block of lead is held just under the water's surface and is let go, it will sink, quickly. If the same size piece of wood is put under the water's surface and let go, it will rise. Now if the same size piece of Styrofoam is put in the water, it will suddenly jump to the surface. Why? Because it is much lighter. Being more precise, it is because the weight of a given volume of lead and of Styrofoam are so much heavier and lighter than the same amount of water. The weight per given volume is called "density." Lead sinks because the density of lead is greater than that of water. Styrofoam floats because the density of Styrofoam is less than that of water. Mathematically,

D = w/V
Density = Weight divided by Volume

Modifying the Gas Law

From this, we can modify the Ideal Gas Law to read,

D = (p/RT) x mw

Calculation Using Modification

For simplicity's sake, take the molecular weight of helium to be 4.0 even. Air has a molecular weight of nearly 29 if dry, and a bit less if wet. Let's round off to 28. Then, inside a balloon,

For helium,

D = 4 (p/RT)

For air,

D = 28 (p/RT)

If a balloon is filled with air and another balloon is filled with helium, the pressure is the same inside both. The temperature is the same. The constant is the same. If we divide the two equations, one by the other, we get

4 (p/RT) divided by 28 (p/RT) = 1/7

Evaluation

The equation says helium's density is about 1/7th that of air. The Ideal Gas Law value is very close. If we let go of a party balloon, it jumps out of our hand.

References

• University of Chicago: Kavli Institute for Cosmological Physics

Resources

• Elmhurst College: Density Application With Gases