Confusion regarding buoyancy & lighter-than-air vehicles So school started pretty recently and the first thing we are learning in Aerospace Engineering is buoyancy, however I've kinda reached a block in my understanding on how hot air balloons (and dirigibles) rise. The teacher provided several resources but most of my confusion revolves around this link in the paragraphs under "Using The Ideal Gas Law".
I feel like I'm getting two contradictory explanations on lift so I'll just tell what I'm hearing from the resource and hopefully I can get this confusion cleared up.
1.) Volume must remain constant in the air balloon's envelope, therefore when the air inside gets heated it will expands, however, because it has nowhere to go, it applies an upward force on the balloon causing it to rise
2.) Volume must remain constant in the air balloon's envelope, therefore when air is heated it will expand, this pressure pushes out air from the envelopes bottom opening, causing the balloon to rise from pressure difference
I guess my main confusion is on whether 1.) the air escapes, making the inside lighter or 2.) the air (and volume) remains the same so it has nowhere to go but rise
 A: The resource you linked could perhaps be improved. In particular, I wonder if the content

As the volume of gas increases, it fills the balloon. The hot air balloon then becomes full of hot air. If the air continues to heat up, it will expand the [sic] and flow out of the hole in the bottom of the balloon. [emphasis added]

led to your deduction that

the air inside...applies an upward force on the balloon causing it to rise

In other words, the resource maybe led you to believe that propulsion of hot air from the bottom of the balloon directs the balloon upward in the manner of a rocket. This is not the case.
A better summary might be:

As the air inside the balloon heats up, the air expands and leaks from the bottom of the balloon.
We can also predict and quantify this expansion using the ideal gas law $PV=nRT$. The overall pressure stays about the same (at atmospheric pressure), the balloon volume stays about the same (as the balloon walls don't really stretch), and the temperature increases; thus, the amount of gas $n$ within the balloon must decrease, and this forms the leakage.
Less gas weighs less, and objects rise if their weight is less than the weight of the fluid they displace. Once the total weight of the balloon material plus the payload plus the enclosed low-density hot air decreases below the weight of an identical volume of air at ambient temperature, the balloon begins to gently rise.

Does this help clarify things?
