Given I have two identical balloons on earth, how will the buoyancy compare between the one filled with helium and another filled with hydrogen?
How can I calculate the ratio of buoyancy given two different substances and identical balloons?
I am also interested in the equations relating to this problem.
Balloons are buoyant because the air pushes on them. The air doesn't know what's in the balloon, though. It pushes on everything the same, so the buoyant force is the same on all balloons of the same size.
If the "balloon" is just a lump of air with an imaginary boundary, then the lump won't go anywhere because the air isn't moving on average. So the buoyant force must exactly cancel the gravitational force (the weight). Since the buoyant force is the same on everything, the buoyant force on a balloon is equal to the weight of the air it displaces. In symbols this is
$$F_{buoyant} = \rho g V$$
where $\rho$ is the density of air, $g$ is gravitational acceleration, and $V$ is the balloon's volume.
Hydrogen and helium have less weight than a similar volume of air at the same pressure. That means the buoyant force on them, which is just enough to hold up air, is more than enough to hold up the balloons, and they have to be tethered down.
Assuming they have the same pressure and volume, a hydrogen balloon has less weight than a helium balloon. Things like pressure and volume are roughly decided on a per-molecule basis, at least in gases at low pressure, so at the same pressure and volume hydrogen and helium will have the same number of molecules. Hydrogen is lighter per molecule, so the hydrogen weighs less, and less of the buoyant force is canceled out. That means the net force on a hydrogen balloon is greater. The difference in the net force is small.
Hydrogen is $H_2$, which has atomic mass 2, while air is mostly $N_2$, which has atomic mass 28, so the hydrogen balloon has a net force of about (28-2)/28 = .93 the weight of the air it displaces.
Helium is mostly helium-4, so the net force on a helium balloon is about (28-4)/28 = .86 the weight of the air is displaces.
So net force on the hydrogen balloon is in the neighborhood of 10% more.
Assuming they are filled to the same volume, and the air surrounding them is of the same density, the buoyant force acting on the balloons will be the same. Buoyant force is simply equal to the weight of the amount of surrounding fluid that would occupy the space filled by the balloon, if the balloon were not there. It has nothing to do with the contents or nature of the balloon itself.
However, the net force on the balloon filled with hydrogen would be greater, because helium is more dense than hydrogen. Given that the balloons are filled to the same volume (and the pressure and temperature are the same), the one with helium in it will be heavier; the weight of the helium can cancel out more of the buoyant force than the weight of the hydrogen can.
Mathematically, the buoyant force is given by
$$F_b = \rho_\text{fluid}gV$$
where $\rho_\text{fluid}$ is the density of the surrounding fluid (air in this case), and $V$ is the volume of the balloon. The gravitational force on the balloon itself is given by
$$F_g = -mg - \rho_\text{contents}gV$$
where $m$ is the mass of the balloon itself (not counting the gas inside), and $\rho_\text{contents}$ is the density of the gas filling the balloon. The net force can be calculated as
$$F_\text{net} = (\rho_\text{fluid} - \rho_\text{contents})gV - mg$$
and in this equation, only $\rho_\text{contents}$ depends on the gas filling the balloon.
I've written a blog post about buoyancy in the context of an old Mythbusters episode; some of the information in there might be of interest to you.
To add just a little more to David's excellent post: the densities of interest to you to use in his last equation are...
air (fluid): 1.2 kg/(m^3)
helium (contents): 0.0899 kg/(m^3)
hydrogen (contents): 0.1786 kg/(m^3)
Helium has an atomic mass of about four times that of hydrogen. But it does not form a diatomic molecule like hydrgen (H2) does. So it's density is only twice that of hydrogen, rather than four times.
So the answer to your question is that, at STP, the CONTENTS of a hydrogen filled ballon will give a net bouyant force of about:
(9.8)(1.2kg - 0.0899kg)/(m^3) = 10.0N/(m^3) ---> about 2.44 lb./(m^3)
The CONTENTS of a helium filled ballon will give a net bouyant force of:
(9.8)(1.2kg - 0.1786kg)/(m^3) = 10.0N/(m^3) ---> about 2.25 lb./(m^3)
The hydrogen filled ballon gives about 9% more net bouyant force.
Remember that temperature and pressure both dramatically affect actual bouyant force.
