Hot air balloon - mass Short background
I'm aware that a hot air balloon rises up because of the less density than air.
But I'm not sure about how that lower density is reached, and how it can be sustained over time.
At constant P and mass, we can write:
$$\frac{T}{V}=k$$
Then as T increases, if mass is constant, density decreases.
Question
But doesn't the mass change also?
 A: After the initial heating of the air in the balloon, while the balloon is still on the ground, it stretches a little and takes its shape. 
Further heating does change its volume much, which causes the increase in pressure, but, since the balloon is open at the bottom, high pressure inside pushes the air out, which roughly equalizes the pressure inside to the pressure outside and keeps it more or less constant.
So a more accurate formula for the hot air balloon should be $Tn=k$, where $k=PV/R$, i.e., as the temperature rises, the mass and therefore the density of the air inside the balloon decreases, which creates buoyancy. 
A: 
But doesn't the mass escape from the balloon?

At what point?  During the heating phase, yes.  Expanding warm air will push some other air out of the envelope.  Lower mass remains, which is now supported by buoyancy.

And really the volume increase withouth changing pressure? How would you model the phenomena?

Because the envelope is open to the atmosphere, the pressure can't change (at least not at the surface prior to release).  It's always 1atm inside, minus the pressure gradient that already exists in the atmosphere with height.
