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The balloon model is often used to show how ventilation works in the human lungs. You pull on the diaphragm, expanding the volume of the container, and the balloon begins to fill up. My understanding (which may be flawed) is this:

  1. The air in the container is isolated from the air outside the container, so when the container’s volume is increased, the container’s gas pressure decreases in accordance to Boyle’s law.
  2. There becomes an imbalance of forces between the side of the balloon wall that faces the container (i.e. at reduced pressure) and that which is open to the atmosphere (i.e. at atmospheric pressure). This pushes the balloon outward, increasing its volume.
  3. The balloon’s expansion causes the container volume to decrease, increasing the container pressure again. At the same time, the gas pressure inside the balloon decreases as the balloon’s volume increases. This continues until the gas pressures on both sides of the balloon are equal.
  4. The equilibrium pressure is lower than the atmospheric pressure, and soon atmospheric air rushes down a pressure gradient into the balloon.

I confused as to what would happen after this (if these events even occur; if my analysis is wrong, pray tell and explain). Does the pressure in the balloon increase? Does its volume? I’m thinking about the ideal gas law $PV=nRT$, and it’s clear that $RT$ is constant and $n$ is changing. However, which is the depending variable: $P$ or $V$?

(And though this part is probably more suited for the Bio SE, I’d also like to ask if this balloon model, in light of what I wrote above, is an accurate depiction of the lungs’ ventilation mechanism. There is no attachment between the balloon and container wall, yet in the human body, the alveoli are nestled in the lung tissue, the lung tissue is connected to the thoracic cavity via pleural membranes, and the thoracic cavity is bounded by the ribs and diaphragm, which whose muscles accomplish the volume expansion process.)

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The 'balloon in a jar' demonstration just illustrates the principle of lung ventilation, but really shouldn't accepted too closely in modeling the dynamics of breathing. In a real lung the only gas-filled space is the space that communicates with the upper ways, and terminates at the other extreme with sac-like structures called alveoli. The space outside this branching structure of airways that terminate in the alveoli is the pleural space which is filled with fluid that's maintained at a pressure slightly below atmospheric pressure. This keeps the lung expanded even under rest conditions. Without that negative pressure the branching structure would collapse to near-walnut size balls.

The space outside the pleural sack surrounding the lungs and within the bounds of the rib cage and diaphragm below is also filled with fluid and tissues. Inspiratory action, mostly by the diaphragm, and partially by the intercostal muscles of the upper chest can reduce the hydraulic pressure of these fluids and this further reduces the pressure in the pleural space causing the alveoli to expand in size. Collectively the expansion creates a difference in pressure between these inner spaces and the space above the trachea. And this pressure difference is what leads to the air flowing into the lung.

During exhalation the muscles relax, and the expulsion of air becomes mostly passive - during restful breathing. The lungs are highly elastic, but rather than springing back sharply though like in the balloon in a bottle model does, they slowly collapse against the forces of viscosity caused by tissues sliding against one another. The lung is a highly visco-elastic structure.

For excellent mathematical models of the lung that can provide you with greater insight over the balloon in a bottle demo, you can search google for authors such as Jason Bates and Ken Lutchen. I've also written some papers on the subject.

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I think the referenced experiment is interesting but not an exact model of the lungs. But in the "close enough" realm, what's happening is that the movement of the diaphragm (and the ribcage) happens as various muscles act against the external air pressure. Once there's additional volume inside (and assuming no collapsed lung, the interior of your lungs, attached as they are via tubes to the outside air, expand because their internal pressure is close to 1 atmosphere while the chest cavity (as the diaphragm retreats) is at a lower pressure. The lungs are not attached to the muscle structure, which is why the commonest form of lung collapse occurs: air enters the cavity between the lung and the structure, and with no pressure differential the lung has to way to expand.

Try this: exhale, close your nose & mouth, and try to expand your chest. You won't be able to, or not very much. The point is that your body cannot maintain much of a pressure difference, so the $PV=nRT$ equation's operation consists of $\Delta n$ following $\Delta V$ pretty closely ($P$ remaining close to constant).

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