To calculate the payload of the balloon, we first need to determine the buoyant force acting on it, which is equal to the weight of the air displaced by the balloon. Then, we will find the weight of the helium inside the balloon and finally calculate the payload by subtracting the mass of the balloon from the buoyant force. Let’s break this down step by step.
Step 1: Calculate the Volume of the Balloon
The volume \( V \) of a sphere (which is the shape of the balloon) can be calculated using the formula:
V = (4/3)πr³
Given that the radius \( r \) is 10 m:
V = (4/3)π(10)³ = (4/3)π(1000) ≈ 4188.79 m³
Step 2: Determine the Mass of Displaced Air
Using the density of air, we can find the mass of the air displaced by the balloon:
Mass of displaced air = Volume × Density of air
Given that the density of air is 1.2 kg/m³:
Mass of displaced air = 4188.79 m³ × 1.2 kg/m³ ≈ 5026.55 kg
Step 3: Calculate the Mass of Helium in the Balloon
To find the mass of helium, we first need to calculate the number of moles of helium using the Ideal Gas Law:
PV = nRT
Where:
- P = pressure in bar = 1.66 bar
- V = volume in dm³ = 4188790 dm³ (since 1 m³ = 1000 dm³)
- R = gas constant = 0.083 bar dm³ K⁻¹ mol⁻¹
- T = temperature in Kelvin = 27°C + 273.15 = 300.15 K
Rearranging the Ideal Gas Law to solve for \( n \) (number of moles):
n = PV / RT
Substituting the values:
n = (1.66 bar × 4188790 dm³) / (0.083 bar dm³ K⁻¹ mol⁻¹ × 300.15 K) ≈ 26681.34 moles
Step 4: Calculate the Mass of Helium
The molar mass of helium is approximately 4 g/mol, which is 0.004 kg/mol. Thus, the mass of helium can be calculated as:
Mass of helium = n × Molar mass
Mass of helium = 26681.34 moles × 0.004 kg/mol ≈ 106.73 kg
Step 5: Calculate the Payload
Now, we can find the payload by subtracting the mass of the balloon from the mass of the displaced air:
Payload = Mass of displaced air - (Mass of balloon + Mass of helium)
Payload = 5026.55 kg - (100 kg + 106.73 kg) ≈ 4820.82 kg
Final Result
The payload of the balloon, when filled with helium at the given conditions, is approximately 4820.82 kg. This means that the balloon can lift this amount of weight in addition to its own mass and the mass of the helium inside it.
