In 2009 Pixar Animation studios released the movie “Up”.
On it we see a house floating in the air, pulled up by many helium balloons.
What volume of helium is needed to to make a house airborne?
(We assume a single balloon, to make it easy to answer)
You can assume that the balloon has a volume of one cubic meter (1m3). This will give you how much weight can be lifted by such balloon.
You can assume that air and helium are ideal gases.
The ideal gas law says that \(PV=nRT\) where
The house is pulled down by its weight
The ballon is pushed down by its weight
The push-up force is given by Archimedes principle
“The force up is equal to the weight of air displaced by the balloon”
A quick Google Search suggests that such house weight between 120 and 180 tonnes
(a metric ton is 1000 Kg)
It is important that you check the source of these estimations
Use only well recognized sources
Include explicit references to all used sources
We can assume that the balloon has a volume of one cubic meter (1m3)
To know the weight of 1m3 of helium, we need the number of mols
Then we will multiply it by the atomic weight of helium
The ideal gas law says \(PV=nRT\), therefore \[n=\frac{PV}{RT}\]
It is reasonable to assume that \[\begin{aligned} P &= 1 \text{ atm} = 101325 \text{ Pa}\\ T &= 27 \text{°C} = 300\text{ K} \end{aligned}\] If you like, you can use intervals instead
According to WikiPedia, the International Union of Pure and Applied Chemistry (IUPAC) says that the atomic weight of helium is \[A_r°(He) = 4.002602±0.000002\text{ dalton}\]
This is also the weight in grams of a mol of helium
IUPAC also publishes abridged values, rounded to five significant figures.
\[A_{r, \text{abridged}}°(He) = 4.0026\text{ dalton}\]
The uncertainty is small, we can take it as a precise value
The number of moles of air is the same as for helium
Air is a mix of gases. According to WikiPedia:
“By number of molecules, dry air contains
Element | Range |
---|---|
N | [14.00643, 14.00728] |
O | [15.99903, 15.99977] |
Ar | [39.792, 39.963] |
C | [12.0096, 12.0116] |
Keep in mind that in air the molecules are N2, O2, Ar, and CO2
Since the number of moles is the same, the only factor determining the buoyancy is \[\text{Molecular weight of Helium} - \text{Molecular weight of Air}\]
From there the rest is just calculation