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.
1.47
Anonymous, you're wrong, check back next week for solution.
2.36
anonymous- you fool!!
just do the rho g V = m * g
Where P.Mr= rho*RT
Get it?
hEY CROW ! What's Mr ?
molar mass
So finally crow what's the answer ?
To find the minimum radius, we need to calculate the temperature of the hot air inside the balloon. This can be done using the ideal gas law:
PV = nRT
Where:
- P is the pressure (in Pascal)
- V is the volume (in m^3)
- n is the number of moles
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature (in Kelvin)
First, we need to find the number of moles of air in the balloon. We can use the mass and molar mass of air:
n = m / μ
where:
- m is the mass (in kg)
- μ is the molar mass of air (in kg/mol)
Given that the mass of you, your friend, the balloon, the burner, and the basket is 300 kg, and the molar mass of air is 29 g/mol (0.029 kg/mol), we can calculate the number of moles:
n = 300 kg / 0.029 kg/mol
n ≈ 10344.83 mol
Now, let's calculate the volume of the balloon. Since the balloon is assumed to be perfectly spherical, we can use the formula for the volume of a sphere:
V = (4/3)πr^3
where r is the radius of the balloon.
Next, we can rearrange the ideal gas law equation to solve for the temperature:
T = PV / (nR)
Given that the pressure is 1 atm (101,325 Pa) and the temperature of the surrounding air is 20°C (293.15 K), we can substitute these values along with the calculated values for the number of moles and volume into the equation:
T = (1 atm) * (V / (10344.83 mol * 8.314 J/(mol·K)))
T = (101,325 Pa) * (V / (10344.83 mol * 8.314 J/(mol·K))) [since 1 atm = 101325 Pa]
Since we want to find the minimum radius that ensures the temperature doesn't exceed 120°C (393.15 K), we can set up the equation:
393.15 K = (101,325 Pa) * (V / (10344.83 mol * 8.314 J/(mol·K)))
Now, we can substitute the volume equation into the equation above to solve for the minimum radius.