Question

Determine the volume (m^3) necessary to fill with hot air for a hot air balloon to lift a payload of 800 kg. Assume that the temperature inside the balloon is 99.0 C and the temperature outside is 20.0 C. Assume that the pressure (inside and out) is 1.00 atm and that the molar mass of air is 28.9 g/mol.

Answer 1

PV = nRT (1)

Payload = (no-ni)g - wtb (2)

Where: no = (n outside) is the number of moles of the cold displaced air

ni = (n inside) is the number of moles of the hot internal air

g = the number of grams of air per mole

wtb = the weight of the balloon

From equation (1) we can solve for the n’s in terms of pressure, volume and temperature, no=PoVo/RTo and ni=PiVi/RTi (3)

Where To is the absolute cold air temperature and Ti is the absolute hot air temperature. Substituting these expressions into equation (2),

We get Payload = (PoVo/RTo - PiVi/RTi)g - wtb (4)

Since the internal pressure of a hot-air balloon is the same as the outside ambient pressure we let Pi=Po=P.

Likewise, the volume of displaced air outside of the balloon equals the volume of the hot air inside the balloon so, Vi=Vo=V.

Rewriting (4), Payload = (PV/RTo - PV/RTi)g - wtb and factoring out PV/R

Payload = PV/R (1/To - 1/Ti)g - wtb

Moving g to the other side of the equation,

Payload = PVg/R (1/To - 1/Ti) - wtb

Given, g=28.9 gm/mole.

V= (800 + wtb)*0.082/1/0.0289/(1/293-1/372) L

We need to know the weight of the balloon (wtb) to get the exact volume.