Pay load is defined as the difference between the mass of displaced air and the mass of the balloon. Calculate the pay load when a balloon of radius 10 m, 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 bar `dm^(3)K^(-1)mol^(-1)`).

Explanation

Calculation of balloon.s volume :
Radius of balloon r = 10 m
Volume of balloon `= (4)/(3)pi r^(3)=(4)/(3)((22)/(7))(10 m)^(3)`
`= 4190.48 m^(3)`
Calculation of weight of displaced air :
`therefore` Volume of He gas in balloon = Volume of balloon
`= 4190.48 m^(3)`
`~~4190.5 m^(3) = V`
`therefore (("The weight of air"),("displaced by balloon"))=(("Volume of"),("balloon (V)"))xx(("Air of "),("density"))`
`=(4190.5 m^(3))(1.2 kg m^(-3))`
`= 5028.6` kg air
Calculation of weight (w) of He gas in balloon :
where, Volume of Balloon (V) `= 4190.5 m^(3)=4190.5xx10^(3)`
Pressure of `H_(2)` gas in `dm^(3)` in balloon (p) = 1.66 bar
Temperature (T) of He in Balloon,
`(T)=(27+273)=300K`
Moles of He in Balloon `= ("Weight (w)")/("Molecular Mass (M)")`
Gas Constant (R ) = 0.083 bar `dm^(3)K^(-1)mol^(-1)`
Molecular mass (M) of He gas = 4.0 gm `mol^(-1)`
`= 4.0xx10^(-3)kg mol^(-1)`
`pV = nRT=(wRT)/(M)`
`therefore w=(pVM)/(RT)`
`therefore w=(("1.66 bar")(4190.5xx10^(3)dm^(3))(4.0xx10^(-3)kg mol^(-1)))/(("0.083 bar dm"^(3)K^(-1)mol^(-1))(300K))`
`= 1117.47 kg ~~ 1117.5 kg`
Total calculation of balloon and He :
Total weight = (weight of balloon) + (weight of He)
`= 100 kg + 1117.5 kg`
`= 1217.5 kg`
Calculation of pay load :
Pay load `= (("Mass of air"),("displaced"))-(("Total weight of"),("He nad balloon"))`
`= 5028.6 kg - 1217.5 kg`
= 3811.1 kg