Mass of air inside air balloon

Question

I have an exercise here with a solution.

The basic outline of the problem is:

Calculate the mass of air inside a hot air balloon, given the mass of the air balloon at rest is $m_b = 500~\mathrm{kg}$.

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The solution goes like this:

The buoyant force of the air and the gravitational pull on the balloon are in equilibrium:

$$F_{B,a} = F_{g,a}$$

From which follows:

$$\rho_a g V = m_b g$$

Rewriting as an expression for volume:

$$V = \frac{m_b}{\rho_a}$$

Using the ideal gas law in terms of density:

$$V = \frac{m_b R T_o}{M_a P}$$

$T_o$ is the temperature outside the air balloon, $M_a$ is the molar mass of dry air.

Rewriting the ideal gas law, in this case in terms of the amount of moles of air inside the balloon:

$$n = \frac{PV}{RT_i} = \frac{m_b}{M_a} \frac{T_o}{T_i}$$

Here $T_i$ is the temperature inside the balloon.

Finally, the mass of the air inside is given by the relation:

$$m_a = m_b \frac{T_o}{T_i}$$

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There's one thing I'm missing here: why does the mass of the air inside the balloon pulling it down not play a part ?

Answer 1

Almost there. Since this looks like a homework question I'll just give a hint. The force balance equation is due to the buoyancy force which is due to the difference in density inside the ballon vs. outside the ballon.

Once you work that out, the weight that you missing will come back into the equation.

Answer 2

The air inside the balloon is less dense than the air outside; this difference is what causes the lift for the balloon. When you heat the air in the balloon, it expands until the balloon is full. At this point the balloon is still on the ground since there is not enough lift.

You need to heat the air more, which expands the air more and causes some of the air to escape from the balloon. Eventually enough air leaves the balloon and it lifts off the ground.

Added:
From Wikipedia: (1) The most-used size is about $2,800 m^3$ (99,000 cu ft), and can carry 3 to 5 people.
(2) At sea level and at 15 °C, air has a density of approximately $1.225 kg/m^3$ ($0.001225 g/cm^3$, $0.0023769 slugs/ft^3$).
So, the air in a $2,800 m^3$ balloon filled by a fan (without heating) would have a mass of nearly 3500 kg.