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 this question, we need to use the ideal gas law equation: PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Let's start with the first scenario where a plastic bag is filled with nitrogen at atmospheric pressure (1 atm) and 22.0 °C.
Step 1: Convert temperature to Kelvin
To convert from Celsius to Kelvin, we add 273.15. So, 22.0 °C + 273.15 = 295.15 K.
Step 2: Determine the number of moles of nitrogen required to lift the girl
To calculate the number of moles, we need to use the molar mass of nitrogen, which is approximately 28.0 g/mol. However, the problem does not provide the force required to lift the girl, so we will assume it is the same force as her weight due to the atmospheric pressure. The weight of the girl is 50.0 kg × 9.8 m/s^2 = 490 N.
Step 3: Convert the force to pressure
Pressure is force per unit area. Since the bag is assumed to have a negligible mass, the entire force will be distributed over the surface area of the bag when it is lifted off the girl's feet. So, we divide the force by the surface area of the bag that comes in contact with the girl's feet.
Step 4: Calculate the volume using the ideal gas law
Rearrange the ideal gas law equation to solve for volume (V):
V = (nRT) / P
Substitute the values into the equation:
V = ((490 N / A) × 295.15 K) / (1 atm)
Now, we need to calculate the surface area in order to proceed.
If we assume the girl is standing on two feet, each having an area of 50 cm^2 (0.005 m^2), then the total surface area is = 2 * 0.005 m^2 = 0.01 m^2.
Substitute this value for A in the equation:
V = ((490 N / 0.01 m^2) × 295.15 K) / (1 atm)
Now, calculate the volume.
Now, let's move on to the second scenario where hot air is used instead of nitrogen and the temperature inside the balloon can be maintained at 37.0 °C.
Step 1: Convert temperature to Kelvin
37.0 °C + 273.15 = 310.15 K.
Step 2: Determine the number of moles of air required to lift the girl
Use the same approach as in the first scenario to calculate the number of moles of air required.
Step 3: Calculate the volume using the ideal gas law
Substitute the values into the ideal gas law equation:
V = ((490 N / A) × 310.15 K) / (1 atm)
Substitute the appropriate surface area (A) calculated earlier.
Now, you can calculate the required volume of the balloon in both scenarios.