The buoyant force on a balloon is equal to the mass of air it displaces. The gravitational force on the balloon is equal to the sum of the masses of the balloon, the gas it contains, and the balloonist. If the balloon and balloonist together weight 175 kg, what would the diameter of a spherical hydrogen-filled balloon have to be in meters if the rig is to get off the ground at 22 degrees Celsius and 752 mmHg? (Take MM air= 29.0 g/mol)
HELP! We're currently studying gases but I don't even know where to start with this question.
To determine the diameter of the hydrogen-filled balloon, we need to find the volume of the balloon using the ideal gas law equation.
The ideal gas law equation is as follows:
PV = nRT
Where:
P = pressure (in Pascals)
V = volume (in cubic meters)
n = number of moles
R = ideal gas constant (8.314 J/K⋅mol)
T = temperature (in Kelvin)
First, let's convert the given pressure and temperature to the appropriate units:
Temperature:
22 degrees Celsius = 22 + 273 = 295 Kelvin
Pressure:
752 mmHg = 752 × 133.322 Pa = 100314.544 Pa
Next, we need to find the number of moles of hydrogen gas. To do that, we can use the ideal gas law with the given pressure and temperature, assuming the volume is 1 mole:
PV = nRT
Rearranging the equation to solve for n:
n = PV / RT
n = (100314.544) / (8.314 × 295)
Now, we need to find the mass of air that the balloon displaces, which will be equal to the buoyant force acting on the balloon. The buoyant force is given by the formula:
Buoyant Force = Mass of Air Displaced × Acceleration Due to Gravity
Since the buoyant force is equal to the gravitational force on the balloon, which is equal to the total weight of the balloon and balloonist (175 kg) multiplied by the acceleration due to gravity (9.8 m/s^2), we can write:
Buoyant Force = (175 kg) × (9.8 m/s^2)
The mass of air displaced can be calculated using the ideal gas law equation with the number of moles of hydrogen gas:
Mass of Air Displaced = n × MM air
Where:
MM air = molar mass of air
The molar mass of air is given as 29.0 g/mol. So,
Mass of Air Displaced = n × 29.0 g/mol
Now, we can use the formula for the volume of a sphere to find the diameter of the balloon. The volume of a sphere is given by:
Volume = (4/3) × π × (radius)^3
Since we're given the diameter, we can use the formula to find the radius of the balloon:
Radius = Diameter / 2
Substituting the values into the equation:
Volume = (4/3) × π × [(Diameter / 2)^3]
Finally, we can rearrange the equation to solve for the diameter:
Diameter = 2 * ([(Volume × 3) / (4π)])^(1/3)