What would this curve look like? (hot air balloon problem)

[Terminus]

I'm just wondering what a curve would look like if the x-axis is the temperature inside a hot air balloon and the y-axis is the diameter (in feet) of the balloon. There's a relation in how many pounds a hot air balloon can lift based on this curve and I'm looking to see what a good balance would be between both x and y to be able to lift 400 pounds into the air.

 

[Bob3141592]

I'm just wondering what a curve would look like if the x-axis is the temperature inside a hot air balloon and the y-axis is the diameter (in feet) of the balloon. There's a relation in how many pounds a hot air balloon can lift based on this curve and I'm looking to see what a good balance would be between both x and y to be able to lift 400 pounds into the air.

Let's take a shot at this. I think you only need two formulas.

Assume the balloon takes on a spherical shape, so it's Volume is
$$V = \frac{\pi D^3}{6}$$

And assuming the air inside the balloon acts as an ideal gas
$$P V = n R T$$

Once the balloon is fully expanded, the pressure is constant, so as the temperature (in Kelvins) increases, the number of gas molecules in the balloon decreases proportionally. So the change in density of the heated air is inversely proportional to the differnce in temperature between the ambient air and the air inside the balloon.

I understand the optimum temperature for the air inside a hot air balloon is just under the boiling point of water. Assume the outside air is a chill 5 degrees C.

If you knew the mass of the air contained in the volume of the balloon, that should be enough to calculate the lifting power.

Is that close enough?