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
Don't worry, I'm here to help you! To solve this problem, we can break it down into three steps:
Step 1: Calculate the volume of the spherical hydrogen-filled balloon.
Step 2: Convert the volume to mass using the molar mass.
Step 3: Equate the buoyant force and the gravitational force to find the diameter of the balloon.
Let's begin with step 1:
Step 1: Calculate the volume of the spherical hydrogen-filled balloon.
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
where V is the volume and r is the radius of the sphere.
Since we want to find the diameter of the balloon, which is equal to 2 times the radius, we can rewrite the formula as:
V = (4/3) * π * (d/2)^3
where V is the volume and d is the diameter of the balloon.
Step 2: Convert the volume to mass using the molar mass.
The ideal gas law can be used to convert the volume of a gas to its mass. The formula is:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
In this case, we are given the pressure (752 mmHg), temperature (22 degrees Celsius), and the molar mass of air (29.0 g/mol).
Using the ideal gas law, we can solve for the number of moles, n:
n = (PV) / (RT)
where P is the pressure in atm, V is the volume in liters, R is the ideal gas constant (0.0821 L*atm/mol*K), and T is the temperature in Kelvin. To convert Celsius to Kelvin, add 273 to the Celsius temperature.
Once we have the number of moles, we can calculate the mass of the gas using the molar mass:
Mass of gas = n * molar mass
Step 3: Equate the buoyant force and the gravitational force to find the diameter of the balloon.
The buoyant force on the balloon is equal to the weight of the air it displaces. This can be calculated using Archimedes' principle:
Buoyant force = Density of air * Volume of balloon * Gravity
The gravitational force on the balloon is equal to the sum of the masses of the balloon, the gas it contains, and the balloonist.
Gravitational force = Mass of balloon + Mass of gas + Mass of balloonist
We can now equate the two forces:
Buoyant force = Gravitational force
By substituting the known values and solving for the diameter, we can find the answer to the question.
Remember to convert units appropriately and double-check your calculations.