Pay load is defined as the difference between the mass of displaced air and the mass of the ballon Calculate the pay-load when a balloon of radius `10m` mass `100 kg` is filled with helium at 1.66 bar at `27^(@)C` (Density of air ` = 1.2 kg m^(-3)` and `R = 0.083` nar `dm^(-3) K^(-1) mo1^(-1)`) .

Explanation

Step 1. Calculation of the mass of displaced air
Radius of balloon `(r) = 10 m`
Volume of balloon `(V) = 4//3 pir^(3) = 4//3 xx 22//7 xx (10 m)^(3) = 4190.5 dm^(3)`
Mass of the displaced air `= "Volume of air (balloon)" xx "Density of air"`
`= (4190.5 m^(-3)) xx (1.2 kg m^(-3)) = 5028.6 kg`
Step II. Calculation of the mass of the filled balloon.
Moles of He present `(n) = (PV)/(RT) = ((1.66 "bar" xx 4190.5 xx 10^(3) dm^(3)))/((0.083 "bar" dm^(3) K^(-1) mol^(-1)) xx (300 K))`
`= 279.37 xx 10^(3)` mol.
Mass of He present `= "Moles of He" xx "Molar mass of He"`
`= (279.37 xx 10^(3) mol) xx (4 g mol^(-1))`
`= 1117.48 xx 10^(3) g = 1117.48 kg`
Mass of filled balloon `= 100 + 1117.48 = 1217.48 kg`
Step III. Calculation of the pay load
Pay load = Mass of displaced air - Mass of filled balloon
`= 5028.6 - 1217.48 = 3811.12 kg`.