Re: How does a hot air balloon rise.

You are correct that it has to do with the density difference between
the hot air in the balloon and the cooler air surrounding it.

The cool air above the balloon is heavier than the heated air, so that if
it flows down around the balloon and the balloon rises to replace it,  we
have succeeded in letting the force of gravity bring more of the mass of 
the air and the earth closer together. Gravity always likes to do this, and
it is called lowering the total gravitational potential energy. This poten-
tial energy is converted to random motion of the air molecules, i.e. heat.

Now, if you ask the question, "why do masses always attract each other, or,
put another way, why is there such a thing as gravity?" and you have asked
a very, very deep question that is right at the heart of physics. The 
rising of your hot air balloon is an example of one of the fundamental
forces of the cosmos.