A typical volume of a modern hot air balloon is 2500 cubic metres, and a typical maximum temperature of the hot air is 120 degrees Celsius. Given these figures, and an outside air temperature and density of 15 ∘C and 1.225 kg/m3 respectively, compute the maximum mass (in kilograms) of the balloon, basket and payload
Archimedes says:
volume of fluid displaced * density of fluid * g = upward buoyant force = weight is neutrally buoyant
mass = weight/g = volume*density
2500 * 1.225 = 3063 kg up
now subtract the mass of air in the balloon
2500 * rho
what is rho, the density of air inside at 120 C ?
approximate as perfect gas ?
P V = n R T
P is close to the same in and out or balloon would burst
V the same
so
n T is about constant
density proportional to mass in the balloon volume
n hot T hot = n cold Tcold
n hot /n cold = Tcold/Thot
Tcold = 273 + 15 = 288
T hot = 273 + 120 = 393
so density hot/density cold = 288/393 = .733
.733 * 1.225 = .8977 kg/m^3
so mass of air in balloon = .8977*2500 = 2244 kg
then finally
mass of other stuff = mass of cold air displaced - mass of hot air = 3063 -2244 = 819 kg
Why did the hot air balloon bring a calculator to the party? Because it wanted to calculate its maximum mass, of course!
To determine the maximum mass of the balloon, basket, and payload, we need to take into account the difference in air density between the hot air inside the balloon and the cool air outside.
First, let's find the mass of the air inside the balloon at the maximum temperature. We can use the ideal gas law to do this:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature in Kelvin
Converting the temperature to Kelvin:
T_hot = 120 + 273.15 = 393.15 K
Now, let's find the number of moles of hot air inside the balloon:
n_hot = PV / RT_hot
Given:
P = outside air pressure = ?
V = volume of the balloon = 2500 m^3
R = ideal gas constant = 8.314 J/(mol·K)
Uh-oh! Looks like we're missing the outside air pressure. Do you happen to know that value? If not, we'll need to make an assumption to proceed.
To compute the maximum mass of the balloon, basket, and payload, we need to consider the change in density of the air inside the balloon due to the difference in temperature.
Step 1: Calculate the density of the hot air inside the balloon
We can use the ideal gas law to calculate the density of the hot air inside the balloon:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Assuming the pressure inside the balloon is the same as the outside pressure, we can rearrange the equation to solve for the number of moles:
n = PV / RT
Given:
Volume (V) = 2500 m^3
Temperature (T) = 120 °C + 273.15 = 393.15 K (converting to Kelvin)
Ideal gas constant (R) = 8.314 J/(mol·K)
Plugging in these values, we can calculate the number of moles:
n = (2500 m^3 * 1.225 kg/m^3) / (8.314 J/(mol·K) * 393.15 K)
Step 2: Calculate the mass of the air inside the balloon
Using the molar mass of air, we can convert the number of moles to mass:
Molar mass of air = 28.97 g/mol
Mass of air inside balloon = n * molar mass of air
Step 3: Calculate the mass of the balloon, basket, and payload
Given:
Mass of empty balloon = 100 kg
Total mass of balloon, basket, and payload = mass of air inside balloon + mass of empty balloon
Now we can perform the calculations:
n = (2500 m^3 * 1.225 kg/m^3) / (8.314 J/(mol·K) * 393.15 K)
n ≈ 934.69 mol
Mass of air inside balloon = 934.69 mol * 28.97 g/mol / 1000 (to convert grams to kilograms)
Mass of air inside balloon ≈ 27 kg
Total mass of balloon, basket, and payload = 27 kg + 100 kg
Total mass of balloon, basket, and payload ≈ 127 kg
Therefore, the maximum mass of the balloon, basket, and payload would be approximately 127 kilograms.
To compute the maximum mass of the balloon, basket, and payload, we need to consider the difference in air density between the hot air inside the balloon and the surrounding outside air. The buoyant force that allows the hot air balloon to float is generated due to this difference in air density.
Here's how to calculate the maximum mass:
Step 1: Calculate the density of hot air inside the balloon.
Density of air = Mass / Volume
Given that the volume of the balloon is 2500 cubic meters and the temperature of the hot air is 120 degrees Celsius (or 393 Kelvin), we can use the Ideal Gas Law to calculate the density of the hot air.
Ideal Gas Law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Rearranging the equation, we get: n = PV / RT
Since the number of moles (n) is directly proportional to the mass (m), we can write: m = (PV / RT) * M, where M is the molar mass of air.
The molar mass of air is the sum of the molar masses of nitrogen, oxygen, carbon dioxide, and other trace gases. It is approximately 28.97 grams/mole.
So, substituting the values into the equation, we get:
m = (P * V * M) / (R * T)
Step 2: Calculate the mass of the surrounding air displaced by the balloon.
The displaced air mass can be calculated using the density of air at the outside temperature.
Mass = Density * Volume
Given that the density of air at the outside temperature is 1.225 kg/m^3 and the volume of the balloon is 2500 cubic meters, we can calculate the mass of the displaced air:
Mass_displaced = Density * Volume
Step 3: Calculate the maximum mass of the balloon, basket, and payload.
The maximum mass is equal to the mass of the displaced air plus the mass of the balloon, basket, and payload:
Max_Mass = Mass_displaced + Mass_balloon + Mass_basket + Mass_payload
Since the balloon, basket, and payload are composed of materials with known densities, we can calculate their individual masses by multiplying their volumes by their respective densities.
Finally, add the mass of the balloon, basket, and payload to the mass of the displaced air.
Remember to convert units if necessary.
Following these steps, you can calculate the maximum mass of the balloon, basket, and payload based on the given information.