Question

1) Why is a hot air balloon stiff?
2) Is the pressure inside the balloon higher than the pressure outside (atmospheric pressure)?
3) If the pressure inside is higher than the outside, how is it explained by ideal gas law?

Answer 1

To understand the physics of inflated balloons you have to understand curved membranes under tension.

If there's a membrane under tension T (for simplicity assumed to be isotropic), with fluids on both sides, it will be flat (planar) unless there is a pressure difference between the two fluids. If you notice that it's curved with radius of curvature R (again assumed to be isotropic, like a sphere), then there must be a pressure difference given by

Of course, the membrane bulges away from the side where the pressure is higher, out into the side where the pressure is lower.

Now that we have this basic concept, the answers to your questions all come from it because they're all related.

The hot air balloon is stiff because it is under tension. If a membrane is not under tension it could have folds or wrinkles because there is no energy penalty for having extra area. But since the material of the balloon is under tension it's going to be as smooth as possible. It would be flat if not for the pressure difference, which forces it to have a smooth curve.

The pressure inside the balloon is indeed higher than the pressure outside, and the difference is given by the formula above.

This pressure difference is not really "explained" by the ideal gas law, but it is consistent with it. If you start with a deflated balloon on the ground and gradually fill it up with hot air, you're increasing N (because there are more air molecules inside the balloon than when you started) and V, while p and T remain roughly constant (the T inside being higher than the T outside). When V finally gets to the volume that will make the balloon stiff, the pressure begins to go up slightly above the ambient pressure, with the extra pressure being provided by the tension of the balloon. However, the density of the gas inside the balloon is still less than that of the air around it, because although p is greater (tending to increase the density), T is also greater (tending to decrease the density), and the temperature effect dominates.