I guess I'm asking the wrong question. Let me rephrase with another example.

Assume I have a bottle of liquid hydrogen. It's connected to a non-porous balloon with no elasticity made of, say mylar. The maximum volume of that balloon is 1,000 cubic meters. I open the valve on my hydrogen and the balloon begins to fill. Assuming the balloon never breaks, for every liter of liquid hydrogen I let escape, what is the pressure inside that balloon? How does that pressure change based on temperature? And assuming the maximum volume of that balloon won't increase or decrease, based on altitude will the internal pressure change (I assume it won't)?

Please, I need a simplified mathematical formula I can plug into game code that gives me this pressure. This formula will represent the filling of gas bags in a rigid airship and I basically need to know when to make them rupture based on over/under inflation.

I know this has to do with the ideal gas law. But, for the past 2 days I've been researching and trying to get a working formula with no success.

Thank you.

  • $\begingroup$ More info is required. How rigid are the walls of the gas cell? Does the gas cell expand as the gas pressure increases? At what difference in pressure between the gas cell and the atmosphere does the gas cell rupture? Do you already have equations for pressure vs. height and temperature vs. height for the simulated atmosphere surrounding your airship? This information is needed in order to use the ideal gas law (PV=nRT) to answer your questions. $\endgroup$ Aug 5, 2018 at 15:52
  • $\begingroup$ For simplicity's sake, assume the gas cells do not expand or contract and have a constant max volume. The rupture point can be any arbitrary constant but will change with the size/volume of the gas bags. Atmospheric pressure and temperature based on altitude are already in the game. As I understand it, the pressure in a non-rigid airship would remain mostly constant as the gas bags expand or contract and I could calculate a rupture point based on the strechability of the material. The rigid is what's causing me headaches as the player could put it in a vacuum. $\endgroup$
    – uPrompt
    Aug 5, 2018 at 16:28
  • $\begingroup$ Thinking about it, if this makes the calculations easier, assume the gas bag volume can increase or decrease up to the maximum gas bag volume. The entire airship will need to be able to handle pressures up to 5+ atmospheres and down to a vacuum. If the internal gas pressure is too much higher or too much lower than the external pressure, they need to rupture. $\endgroup$
    – uPrompt
    Aug 5, 2018 at 16:51
  • 1
    $\begingroup$ How can your airship be "rigid" if the gas cells expand? Also, use the ideal gas law to calculate the pressure of the lifting gas inside the airship. $\endgroup$ Aug 5, 2018 at 17:18
  • $\begingroup$ Again, whatever works. Assume it's a balloon inside a box if need be. I've been trying to use the ideal gas law and the results I get don't make sense. I've update the OP to show what I've been trying. $\endgroup$
    – uPrompt
    Aug 5, 2018 at 17:36

1 Answer 1


Despite the title, the balloon in question seems to be unstretchable but flexible, not rigid. Which means that, for an increasing amount of gas, the pressure $P$ inside the balloon equals the external pressure $P_\text{ext}$ up to the point where its volume $V$ reaches the full volume of the balloon, $V_\text{max}$. In particular, if placed in vacuum, any amount of gas will fully inflate the balloon.

In this situation, for $n$ moles of an ideal gas at constant temperature $T$, we have:

\begin{equation} P = \begin{cases} P_\text{ext} &\text{for } V<V_\text{max}\\ nRT/V_\text{max} &\text{for } V\ge V_\text{max}, \end{cases} \end{equation}

where $R$ is the ideal gas constant. In practice, this means:

$$ P = \max(P_\text{ext},\, nRT/V_\text{max}). $$

  • $\begingroup$ I'm glad you understood what I was attempting. You were correct in your assumptions. Perhaps a piston, where the unfilled volume is at atmospheric pressure, would have been a better example. And the max is what I was missing. My problem has been in properly calculating n. And, if I understand this, I'll have to calculate the number of molecules or moles that go into that volume in order to determine the pressure once it exceeds max? $\endgroup$
    – uPrompt
    Aug 6, 2018 at 12:25
  • $\begingroup$ @uPrompt Yes, you'll need the number of moles $n$, which is about half the number of grams of gas (for molecular hydrogen, H$_2$). $\endgroup$
    – stafusa
    Aug 6, 2018 at 12:52

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