Trick Question About a Balloon

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Empty BalloonsDo your ears perk up when you are asked a “trick question”? Well I have one about a balloon for you. Which weighs more, an empty balloon or a helium-filled balloon? When I first thought of this one, it took me a minute to realize the answer myself.

Stop for a moment. Close your eyes and think this through. Have you arrived at an answer? Are you sure you’re right? We’ll provide some specifics in giving our answer – an example.

Providing Specifics

A typical empty balloon weighs between 1 and 5 grams. Let’s say it is 3 grams. You could easily fit it within a 1″x1″x1″ cubic block. OK. What size will our balloon be after it becomes a helium-filled balloon? Let’s pick a believable size, say 9″ around. How much helium does that require?

Since a balloon blows up round (See: “Why Balloons Blow Up Round”), we can approximately use the equation of the volume of a sphere* to determine the volume of the balloon. Thus…

V = 4/3 π·r³
V = (4/3)(3.14)(4.5 inches)³
V = 381 cubic inches

where the volume is V and r is the radius or half the diameter of the filled balloon.

How Much “Gas” in the “Tank”?

How much helium gas does this amount to in grams? Well, the equation involved is…

Weight of Helium = Volume of Helium x Density of Helium
Weight = 381 cubic inches x 0.00293 grams/cubic inch
Weight = 1.12 grams

Thus the balloon, when empty, weighs 3 grams, but the balloon, filled, weighs 4.12 grams. The helium filled balloon is heavier than when it was empty. Yet, it floats in the air for the same reason a steel ship does not sink in water. The helium-filled balloon’s average density, rubber plus gas, is lighter than that of the air it displaces.

Going Up!

Let’s check the math on that. The skin of the filled and stretched balloon is nearly zero in thickness, so the volume of the filled balloon is almost the same as the volume of the helium by itself. Thus the average density of the helium-filled balloon in its entirety is very close to…

(3 grams + 1.12 grams)/381 cubic inches = 0.01081 grams/cubic inch

The filled balloon is very, very light! The air above the balloon is much heavier than that, so it sinks, while the helium filled balloon rapidly rises!

Note: You might also enjoy “Which Weighs More – Wet Air or Dry Air?”

* There are additional factors that make this approximation less than ideal, but these do not interfere with our correct conclusion.

References:

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