A plastic bag is filled with nitrogen at atmospheric pressure and 22.0 oC. Assume that the mass of the bag is negligible and take the temperature of the outside air to be 0 oC. With the help of this "balloon" we want to lift a 50.0 kg girl off her feet.
How large a volume of nitrogen (in m3) is required?
If hot air is used instead of nitrogen, what is the required volume of the balloon if the air inside can be maintained at 37.0 oC ?
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To answer these questions, we need to use the ideal gas law and the concept of buoyancy.
1. For the nitrogen-filled plastic bag:
Firstly, we need to determine the pressure difference between the inside and outside of the balloon. At atmospheric pressure, this pressure difference is equal to the difference in weight between the girl and the volume of gas inside the balloon.
Step 1: Determine the weight of the girl:
Weight = mass * gravitational acceleration
Weight = 50.0 kg * 9.8 m/s^2
Weight = 490 N (Newtons)
Step 2: Determine the pressure difference:
Pressure difference = Weight / Area
Here, we assume the area of the bottom of the bag is the same as the top and equal to the area of the girl's feet. Since the mass of the bag is negligible, its weight is not considered.
Step 3: Use the ideal gas law to determine the volume:
The ideal gas law equation is: 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 in Kelvin.
We can rearrange the equation to solve for volume:
V = (nRT) / P
Since we have the pressure, temperature, and want to find the volume of nitrogen, we can plug the values in and solve for n (number of moles).
Step 4: Determine the molar mass of nitrogen:
The molar mass of nitrogen is approximately 28.0134 g/mol.
Step 5: Calculate the number of moles:
n = mass / molar mass
The mass in grams is assumed to be the mass of air needed to lift the girl.
Step 6: Calculate the volume using the ideal gas equation with the number of moles we just determined.
2. For the hot air-filled plastic bag:
Repeat the above steps, but this time use the temperature of 37.0°C (310.15 K) for the air inside the bag.
Please note that in practice, lifting a person using a balloon filled with gas has many safety considerations and additional factors need to be taken into account.