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 ?
To answer these questions, we can use the ideal gas law equation: 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.
Let's solve the first part of the question using nitrogen gas:
Step 1: Convert the temperatures to Kelvin.
22.0 oC + 273.15 = 295.15 K (temperature inside the bag)
0 oC + 273.15 = 273.15 K (temperature outside the bag)
Step 2: Calculate the pressure inside the bag.
Since the bag is filled with nitrogen at atmospheric pressure, the pressure inside the bag is also the same as atmospheric pressure, which is typically around 1 atm.
Step 3: Calculate the number of moles.
Since the mass of the bag is negligible, the mass of nitrogen gas inside the bag is also negligible. Therefore, we can assume that all the air inside the bag is nitrogen gas. The number of moles can be calculated using the molar mass of nitrogen, which is approximately 28.0134 g/mol:
Number of moles = mass / molar mass = 50.0 kg / 28.0134 g/mol = 1783.43 mol
Step 4: Calculate the required volume.
Using the ideal gas law equation, we can rearrange the equation to solve for the volume:
V = (nRT) / P
Plugging in the values:
V = (1783.43 mol * 0.0821 L·atm/(mol·K) * 295.15 K) / 1 atm
V ≈ 47,764.09 L or 47.76 m^3
Therefore, approximately 47.76 m^3 of nitrogen gas is required to lift the girl off her feet.
Now let's solve the second part of the question using hot air:
Step 1: Convert the temperature to Kelvin.
37.0 oC + 273.15 = 310.15 K (temperature inside the bag)
Step 2: Calculate the pressure inside the bag.
Since the bag is filled with hot air and not nitrogen, the pressure inside the bag will not be atmospheric pressure. Let's assume it is still approximately 1 atm.
Step 3: Calculate the number of moles.
Since the mass of the bag is negligible, we need to find the number of moles of air that has an equivalent mass to the girl's mass. The molar mass of air is approximately 28.97 g/mol.
Number of moles = mass / molar mass = 50.0 kg / 28.97 g/mol ≈ 1725.76 mol
Step 4: Calculate the required volume.
Using the ideal gas law equation, we can rearrange the equation to solve for the volume:
V = (nRT) / P
Plugging in the values:
V = (1725.76 mol * 0.0821 L·atm/(mol·K) * 310.15 K) / 1 atm
V ≈ 42,682.79 L or 42.68 m^3
Therefore, approximately 42.68 m^3 of hot air is required to lift the girl off her feet when the temperature is maintained at 37.0 oC.