Helium balloons deflate–even when they are made from metal foil. How does this happen exactly? Is there any chance that quantum tunneling plays a role here?
The thought is that the helium atoms inside the balloon would have higher energy and smaller mass on average than the air molecules outside of the balloon so that they have higher chance of being found on the other side of the barrier (foil).
I don't expect that this is the primary cause of deflation, but I wonder if this could be part of it. Any quantitative insight into this?
Best Answer
Just about every substance contains very small pores that will allow for the passage of atoms and molecules.
Quantum tunneling need not play a role here, since in quantum tunneling, the movement of particles through a barrier should not have a classical physics explanation, though here, there is$^1$.
The lighter "smaller" helium gas molecules will leave the balloon at a faster rate than if you had the balloon filled with heavier "larger" molecules like nitrogen and oxygen (air).
Quantum tunneling is a quantum mechanical, and not a classical phenomenon, where the wavefunction of an object can spread across a potential barrier. Basically, the objects wavefunction may spread on either side of a barrier, so that there is a probability that the object can appear on the other side of it.
$^1$ I guess one could say that quantum effects happen everywhere, and so why not here? If we were to continuously throw a baseball at wall and watch it bounce off over and over and over again, maybe in a time equal to the entire history of the universe, there is a finite probability that the baseball will pass through the wall? Why not apply the same logic here? The probability is given by $$P\approx e^{-2kL}$$ where $L$ is the barrier width, and $$k=\sqrt\frac{{2m(V-E)}}{h}$$ Doing some very "back of the hand" calculations, for a balloon wall width of about $10^{-4}m$, and the mass of a helium molecule of about $6.64 \times 10^{−27}kg$ and estimating a rough value for the potential energy$^2$, gives a probability of about $$P\approx 3.38\times 10^{-10}$$ that a helium molecule will tunnel$^3$.
$^2$To get the kinetic energy of the helium molecules, one would use the fact that $$\langle E\rangle =\frac{3}{2}kT$$ considering translational degrees of freedom. $T=293^o$ is room temperature in Kelvin, and $k=1.38\times 10^{-23}JK^{-1}$ is Boltzmann's constant.
$^3$This is a very rough calculation and tells us that about one in every $\approx 3\times 10^{10}$ helium molecules will quantum tunnel if there are no pores. But again, this is not exact, and should only be used to point out that the probability for tunneling to occur in this instance is small, so that we can effectively use classical arguments, and conclude that the molecules pass through because they can fit through the pores, and not because of tunneling.