You are a maker of hot air balloons and wish to construct a balloon that will lift you and a friend to places unknown. Your standard balloon design is a perfectly spherical, nylon fabric balloon with a small hole on the bottom for the burner that heats the air. Nylon starts to degrade if the air temperature exceeds 120o C, so you don't want to exceed this temperature for your hot air. What is the minimum radius in meters you need for your balloon?
Details and assumptions
You, your friend, the balloon, the burner, and the basket have a total mass of 300 kg.
The ambient pressure is 1 atm=101,325 Pa and the temperature of the surrounding air is 20o C.
Air has a molar mass of μ=29 g/mol.
12.3
You are wrong hgj.
11.8
To determine the minimum radius of your hot air balloon, we'll use the ideal gas law and the principle of buoyancy.
First, let's calculate the molar mass of air in kilograms:
Molar mass (μ) of air = 29 g/mol = 0.029 kg/mol
Next, we need to find the number of moles of air in the balloon. We know the mass, so we can use the formula:
Number of moles (n) = mass (m) / molar mass (μ)
Number of moles (n) = 300 kg / 0.029 kg/mol
Now, let's convert the ambient temperature from Celsius to Kelvin:
Temperature (T) = 20°C + 273.15
Now, let's calculate the minimum volume of the balloon required. We'll use the ideal gas law:
PV = nRT
Where:
P is the pressure (101,325 Pa)
V is the volume of the balloon
n is the number of moles of air
R is the ideal gas constant (8.314 J/(mol·K))
T is the temperature in Kelvin
To solve for V, rearrange the equation:
V = (nRT) / P
Plugging in the values:
V = (n * R * T) / P
V = (0.029 * 8.314 * T) / 101,325
Now, let's calculate the maximum temperature we can use without exceeding the nylon fabric's degradation temperature of 120°C:
Maximum temperature (T_max) = 120°C + 273.15
Lastly, we can calculate the minimum radius of the balloon using the formula for the volume of a sphere:
V = (4/3) * π * r^3
Rearrange the equation to solve for r:
r = ((3 * V) / (4 * π))^(1/3)
Plugging in the values:
r = ((3 * V) / (4 * π))^(1/3)
r = ((3 * ((0.029 * 8.314 * T_max) / 101,325)) / (4 * π))^(1/3)
Now, you can substitute the value for T_max and calculate the minimum radius of the balloon.