Lifting force of a helium-filled balloon

We have the calculator which calculates the parameters of a balloon that uses hot air as the lifting gas. In the case of helium gas, the calculation becomes simpler. Again, when it comes to molecular kinetic theory problems, one is asked to find either the mass of helium required to lift a balloon with a given shell mass and mass of a cargo or load, or the mass of a load that can be lifted by a given mass of helium for a given mass of a balloon shell. The calculator below allows you to find any unknown value, including the mass of the shell. It is worth noting that the temperature of the surrounding air and its pressure are usually given in the problems, but they are not really needed for the calculation.

As usual, the theory and calculation formulas are given below the calculator.

Calculating the parameters of a helium-filled balloon

A balloon is lifted by an buoyant force, or Archimedes' force, because Archimedes' principle applies not only to liquids but also to gases. Accordingly, for the balloon to begin to rise, the upward force acting on the balloon must exceed the force of gravity. To solve problems, we usually find a boundary condition (on mass), which allows us to equate the force of gravity and the upward force. That is, a little more mass of helium, or a little less mass of a cargo - and the balloon begins to rise. The equation of equilibrium looks like this:
,
where
M - mass of the shell,
m is the mass of the cargo,
mₐ is the mass of helium in the balloon,
mₑ is the mass of the surrounding air displaced by the balloon,
g is the acceleration of free fall.

Reducing g and transferring the mass of air to the right-hand side, we obtain

Let's deal with the mass of air displaced by the balloon. The key point for helium balloons is that its pressure and temperature are equal to the pressure and temperature of the surrounding air. Therefore, if we consider two equal volumes, each equals to the volume occupied by helium, we can write from the Mendeleev-Clapeyron equation:

Whence, after reducing RT

The molar mass of helium is 4 g/mol, the molar mass of air is 28.98 g/mol (often rounded to 29 in problems). For the equality to hold, the mass of helium must be less than the mass of displaced air. We get the following expression for the mass of air:

Substituting mₑ into the equality above, we get the final formula that relates all the parameters of the balloon:

From this equality we can obtain formulas to calculate the desired unknown.
The mass of the shell:

Mass of the load:

Mass of helium:

It is worth noting that helium-filled balloons are used to ascend to the stratosphere. The air pressure is lower there, so to displace the necessary mass of air needed for the ascent, it is necessary to occupy more volume. Thus, balloons intended for ascent to the stratosphere should be able to expand strongly, i.e. their shell should not resist the change of the balloon volume (this, by the way, is written in the problem conditions). The easiest way to achieve this is to make the shell "with reserve", so such balloons seem almost empty at launch.