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 find the volume of nitrogen or hot air required in the balloon to lift the girl off her feet, we can apply the ideal gas law, which states:
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 start with the first scenario using nitrogen.
1. Convert the temperatures to Kelvin:
Temperature in Kelvin = Temperature in Celsius + 273.15
For the nitrogen scenario:
Temperature of nitrogen inside the balloon, T1 = 22.0 °C + 273.15 = 295.15 K
Temperature of outside air, T2 = 0 °C + 273.15 = 273.15 K
2. Calculate the pressure difference:
Since the balloon is at atmospheric pressure initially, the pressure difference is given by:
ΔP = P1 - P2 = P1 - 1 atm (since air pressure outside is considered atmospheric pressure)
3. Calculate the mass of air needed:
Mass = girl's weight = 50.0 kg
4. Calculate the number of moles of nitrogen:
n = Mass / Molar mass of nitrogen gas
The molar mass of nitrogen (N2) gas is approximately 28.0134 g/mol.
5. Plug in the values into the ideal gas law:
PV = nRT
(Volume of the balloon) * (Pressure difference) = (Number of moles of nitrogen) * (Ideal gas constant) * (Temperature difference)
V1 * ΔP = n * R * (T1 - T2)
Rearrange the equation to solve for V1 (the volume of nitrogen):
V1 = (n * R * (T1 - T2)) / ΔP
Plug in the values, where R is the ideal gas constant (8.314 J/(mol·K)), T1 = 295.15 K, T2 = 273.15 K, and ΔP = (1 atm - 1 atm) = 0 (since the initial and final pressures are the same):
V1 = (n * R * (T1 - T2)) / ΔP = (50.0 kg / 28.0134 g/mol) * 8.314 J/(mol·K) * (295.15 K - 273.15 K) / 0
Since dividing by zero is undefined, we cannot proceed with this calculation for nitrogen. This means that a balloon filled solely with nitrogen gas cannot lift the girl off her feet.
Now, let's move on to the second scenario using hot air.
1. Convert the temperature to Kelvin:
Temperature of the air inside the balloon, T = 37.0 °C + 273.15 = 310.15 K
2. Calculate the pressure difference:
Again, since the balloon is at atmospheric pressure initially, the pressure difference remains the same: ΔP = P1 - P2 = P1 - 1 atm
3. Calculate the number of moles of air (hot air is primarily composed of nitrogen and oxygen):
n = Mass / Molar mass of air
The molar mass of air is approximately 28.97 g/mol.
4. Plug in the values into the ideal gas law:
V2 = (n * R * (T - T2)) / ΔP
Where T2 is the temperature of the outside air (0 °C = 273.15 K).
Substituting the values, we have:
V2 = (50.0 kg / 28.97 g/mol) * 8.314 J/(mol·K) * (310.15 K - 273.15 K) / (P1 - 1 atm)
Calculating the value of P1 - 1 atm might require additional information, such as the air pressure at the location where this scenario is taking place. Once you have the value of P1 - 1 atm, substitute it into the equation above to get the required volume V2.