Answer 1:
Hot air rises because when you heat air (or
any other gas for that matter), it expands. When
the air expands, it becomes less dense than the
air around it. The less dense hot air then
floats in the more dense cold air much like wood
floats on water because wood is less dense than
water. This floating effect in a less dense
medium is called a buoyant force or a
displacement force.
We can model what I just said with a little
physics. Consider the air to be an ideal gas
(this is a good approximation which neglects the
interaction of air molecules with each other).
The pressure P, volume V, amount of gas N,
and temperature T can all be related by the
ideal gas equation as
P*V=N*R*T
where R is the ideal gas constant (8.314 in
units of J/(mol*K)).
Say you start with an amount of air N at
pressure P, volume V_{1}, and
Temperature T_{1} and heat it in a
balloon (we will neglect the mass and elastic
properties of the balloon for simplicity) to
temperature T_{2}.
The ideal gas equation can be rewritten as
P*V_{1}/(N*T_{1})
=R=P*V_{2}/(N*T_{2})
which with a little algebra can be solved to give
V_{2}=V_{1}*T_{2}/T_{1
}.
From this we see that if you heat a gas such
that
T_{2}>T_{1}
you get
V_{2}>V_{1}
meaning that the gas expands. Now since you have
the same amount of gas, the mass does not change
so the density before heating is
d_{1}=m/V_{1}
and the density after heating is
d_{2}=m/V_{2}.
Now the force of gravity on the hot air
(F_{g}) is
given by the mass of the hot air times the
gravitational constant g (9.8 m/s^{2})
(F_{g}=d_{2}*V_{2}*g).
Here the mass of the hot air is given by the
density of the hot air times the volume of the
hot air. The buoyant force on the balloon is
given by the mass of the air displaced times the
gravitational constant
(F_{b}=d_{1}*V_{2}*g).
Here the mass of the air displace is given by
the density of the surrounding cool air times
the volume of the balloon (hot air). Now the
buoyant force is up and the gravitational force
is down so the net force is given by
F=F_{b}
F_{g}=d_{1}*V_{2}*g
d_{2}*V_{2}*g=m*g*
(V_{2}/V_{1}
V_{2}/V_{2})
which can be rewritten using the relation from
the ideal gas law
(V_{2}=V_{1}*T_{2}/T_{
1}) to give:
F=m*g*(T_{2}/T_{1}1).
This equation shows that as you heat the air in
the balloon T_{2}>T_{1} (i.e.
T_{2}/T_{1}>1) you get a
positive force (F>0) which means that the force
is up and the balloon rises.
In summary, hot air rises because it is less
dense than the surrounding air.
