ideal gas law/hot air balloon problem (Thermal Physics by D. Schroeder, prob 1.15)

[BoogieBot]

1. The problem statement, all variables and given/known data

Estimate the average temperature of the air inside a hot air balloon. Assume that the total mass of the unfilled balloon and payload is 500kg. what is the mass of the air inside the balloon?

No volume or radius is given, so I estimated V = 2800 m^3 & r = 8.75 m
estimate T = 370 Kelvin
mass of 1 molecule of air calculated to be 29.00 g/mol


2. Relevant equations

PV = NkT
m{air} = # of molecules[N] * mass of 1 molecule {m}
P = mass/volume


3. The attempt at a solution

PV = NkT ---> N = PV/kT

m{air} = (PV/kT)*m

I don't know how to find pressure w/o knowing the mass of air inside the balloon. P = F/A (surface area calculation is trivial so I'm not going to do it here). I assume F is the buoyant force?

In equilibrium, Fbuoy = Fgrav, so: Fbuoy = (m{air} + m{payload})* g

but it looks like I need m{air} to find Fbuoy. What do?

[Andrew Mason]

What is the buoyant force that is required? Write the equation relating balloon volume and mass of the balloon + hot air and the mass of the displaced air (relate the densities of the hot air and displaced air to temperature using the ideal gas law). Hint: the pressure is the same inside as outside the balloon so: nRT/V is the same inside and outside. How is this related to density?

AM