# Modell for the buoyancy of a hot air balloon

I was just working on a problem concerning a ballon for my introdouction to thermodynamics class. The helium ballon is emerged in air and surrounded by earths gravitational field, which shall be considered as homogenous. Furthermore the atmosphere shall be considered as isothermic. The aim is to calculate how much load can be put on the ballon and how far up it can rise until it colapses due to material failure at 120% of it's original volume. For the buoyancy it's asked for a more sophisticated reasoning than just blindly applying Archimedes law.

Obviously the balloon is experiencing a downward force because of it's load $$\mathbf{F}_1=-m_{total}*g*\hat{\mathbf{z}}$$ Since we suppose a temperature $$T\neq0$$ K, I think we also have to consider the air molecules and the helium atoms bouncing against the wall of the ballon. $$\mathbf{F}_2=\oint_{\partial \Omega}(\rho_{air}-\rho_{helium})\mathbf{dA}=\int_\Omega\nabla(\rho_{air}-\rho_{helium})dV$$

Where $$\Omega$$ denotes the balloon.

But this leads me to two questions:

1. How should I calculate $$\rho_{helium}$$? I think I figured out $$\rho_{air}$$, I used the isothermic barometric formula and approximated it for small heights, giving a linear function. This would indeed lead to archimedes law if I just set $$\rho_{helium}=const.$$ But this seems odd to me, especially for very large ballons. Shouldn't we also apply the barometric forumla here? But then again, this would make the second part very hard since I can't just use the relation for a ideal gas to calculate the expansion of the ballon. Because in my understanding it only applys if the pressure is constant throughout the considered volume.
2. And this got me thinking: Suppose we would use the barometric formula to calculate $$\rho_{helium}$$ and there wouldn't be any air, just the garvitational field. Wouldn't this imply that the balloon is falling faster to the ground than a solid of the same mass? For a solid it's obviously $$\mathbf{F}_2=0$$, but for the air ballon this force does make a contribution since $$\rho_{helium}\neq const.$$ The pressure is higher at the bottom of the balloon, than at the ceiling, meaning that the net force is directed downwards. (And yes I know that balloon would collapse pretty fast under these conditions, so suppose it doesn't expand here.)

And well, this seems odd to me. I would be very grateful for any comments.