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 $m_b = 500~\mathrm{kg}$.
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:
$$m_a = m_b \frac{T_o}{T_i}$$
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 ?
Best Answer
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.