1)We consider a hot-air balloon of mass 251 kg (basket and envelope). The spherical envelope of the balloon has a diameter of 19 m when fully inflated. To what temperature (in °C) must the enclosed air be heated for the balloon to carry five standard men? Assume the surrounding air is at 20°C and is treated as an ideal gas. Use 29 g/mol for the molar mass of air. (Assume the mass of each standard man is 70 kg.)
so ive been looking through my notes and such for a long time and i cant find out how to do this!! i dont need an answer persay, just maybe a little bit of help on how to start this question off right. I know for sure it has something to do with bouyancy. its just our prof hasnt explained it very well lately. thanks!
total weight = (251+350)(9.81)
+ rhogas 9.81(4/3) pi (9.5)^3
where rhogas is the mass per unit volume of the gas in the balloon
weight must = weight of air displaced = rhoout g (4/3)pi(9.5)^3
rho gas = kg/mol * n = .029 n
where n is mols in PV =nRT
T = 273 + T that we want
rho air = same but T = 273+20 = 293
assume P = 1 atm inside and out
To solve this problem, you need to consider the principles of buoyancy and ideal gas laws.
First, let's start with the principle of buoyancy. According to Archimedes' principle, the buoyant force experienced by an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In the case of a hot-air balloon, the buoyant force allows the balloon to float because the hot air inside is less dense than the surrounding cool air.
Now let's try to break down the problem step by step:
1. Determine the net force required to lift the balloon and the men:
The net force is the difference between the weight of the balloon and the men and the buoyant force acting upward. For the balloon to carry the weight of the men, the net force must be positive.
2. Calculate the weight of the balloon and the men:
The weight is calculated by multiplying the mass by the acceleration due to gravity (9.8 m/s^2). The combined weight of the balloon and the men is given as 251 kg (mass) in the question.
3. Calculate the buoyant force:
The buoyant force can be calculated using the formula F_b = ρ_fluid * V * g, where ρ_fluid is the density of the fluid, V is the volume displaced by the balloon, and g is the acceleration due to gravity.
To find the volume displaced by the balloon, you need to know the density of air when it is heated to a certain temperature. This is where the ideal gas law comes into play.
4. Use the ideal gas law:
The ideal gas law states that 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 Kelvin.
In this problem, you have the molar mass of air and need to determine the volume of air displaced by the balloon. Rearrange the ideal gas law equation to solve for V/V_molar (the volume divided by the molar volume).
5. Calculate the volume of air displaced by the balloon:
Multiply V_molar by the number of moles of air, which can be found by dividing the mass of air by the molar mass of air.
6. Substitute the values into the buoyant force equation:
Plug in the calculated volume from step 5, ρ_fluid (density of air at 20°C), and g into the buoyant force equation.
7. Solve for the net force:
Subtract the weight of the balloon and the men (step 2) from the buoyant force (step 6). The net force should be positive to lift the balloon and the men.
8. Rearrange the net force equation to solve for the temperature of the air inside the balloon:
The net force is equal to the weight of the balloon and the men. Thus, you can equate the weight to the buoyant force and solve for T.
These steps should guide you in finding the required temperature in °C for the enclosed air to lift the balloon and carry the five men. Remember to convert all units to the appropriate form and use absolute temperature (Kelvin) in the calculations using the ideal gas law.