To an order of magnitude, how many helium-filled toy balloons would be required to lift you? Because helium is an irreplaceable resource, develop a theoretical answer rather than an experimental answer. Assume that you have a mass of approximately 70 kg, the radius of a balloon is 12.5 cm, the helium is at STP and it is immersed in air at 0°C and 1 atm.
__________My work now___ I assumed that the balloon was a sphere, so 4/3*pi*(r^3) = 0.008181231 m^3. I used 0.125 m for the radius. Also the weight of the person is equal to m*g=686 Newtons.
Thus, the force needed to lift up the person is 686 Newtons. I found the density of air at STP to be 1.2754 kg/m^3. So the buoyant force is density of air * g * Volume of air displaced which is the same as the volume of one balloon. Thus, the buoyant force for one balloon is is 1.2754 x 9.8 x 0.008181231 m^3.
The weight of the person was 686 newtons, so 686/force of one balloon = # of balloons, which is 6710 balloons. This is INCORRECT. Also, I thought the question stated "what order of magnitude" so I guessed 7,000 balloons, but my online homework still said, "response differs from the correct answer by more than 10%".
Please help explain if you can how to solve this problem.
To solve this problem, we need to take into account a few additional factors. First, let's correct the calculation of the buoyant force for one balloon.
The buoyant force is given by the equation: Buoyant force = (density of air) x (acceleration due to gravity) x (volume of displaced air)
Using the given values:
Density of air at STP = 1.2754 kg/m^3
Acceleration due to gravity = 9.8 m/s^2
Volume of one balloon = (4/3) x π x (radius^3) = (4/3) x π x (0.125 m)^3
Calculating the buoyant force for one balloon:
Buoyant force = 1.2754 kg/m^3 x 9.8 m/s^2 x (4/3) x π x (0.125 m)^3
Next, let's calculate the total buoyant force required to lift a person with a mass of 70 kg. Since the weight of the person is equal to the mass multiplied by the gravitational acceleration, we can get the total force needed to lift them up in Newtons.
Total force needed to lift the person = mass x acceleration due to gravity = 70 kg x 9.8 m/s^2
Now, we can find the number of balloons required by dividing the total force needed to lift the person by the force exerted by one balloon:
Number of balloons = Total force needed to lift the person / Buoyant force for one balloon
Substituting the respective values:
Number of balloons = (70 kg x 9.8 m/s^2) / (Buoyant force for one balloon)
This calculation will give us the number of balloons required to theoretically lift the person. Keep in mind that this answer does not take into account any additional factors such as the weight of the balloons themselves, variations in atmospheric conditions, or the possibility of the person being lifted safely with balloons.
To determine the number of helium-filled balloons required to lift you, we can use Archimedes' principle. According to Archimedes' principle, an object immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.
First, let's calculate the volume of helium needed to lift your weight. You correctly calculated the volume of one balloon as 0.008181231 m^3. However, we need to consider the volume of the helium inside the balloon, not the entire balloon itself.
Assuming helium behaves as an ideal gas at STP, we can use the ideal gas law to find the volume of helium required. The ideal gas law states:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
At STP (Standard Temperature and Pressure), the pressure is 1 atm, the temperature is 0°C (which is equivalent to 273.15 K), and R is the ideal gas constant (approximately 0.0821 L.atm/mol.K).
We can rearrange the ideal gas law to solve for the volume V:
V = nRT / P
To find the number of moles of helium required, we can use the molar mass of helium, which is approximately 4 g/mol.
n = (mass of person / molar mass of helium)
n = (70 kg / 0.004 kg/mol)
n = 17500 moles
Now, let's substitute the values into the equation for volume:
V = (17500 mol) * (0.0821 L.atm/mol.K) * (273.15 K) / (1 atm)
V ≈ 3888 L
Since we previously calculated that the volume of one balloon is approximately 0.008181231 m^3, we need to convert the volume of helium to the equivalent number of balloons:
Number of balloons = (3888 L / 0.008181231 m^3)
Number of balloons ≈ 475,027
Therefore, to an order of magnitude, we would need around 500,000 helium-filled balloons to lift you, assuming each balloon can lift the weight of the person.