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 $27^{\circ} C$ . (Density of air = $1.2 k g m^{- 3} a n d R = 0.083 b a r d m^{3} K^{- 1} m o l^{- 1}$ ).

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Solution

Given,
Radius of the balloon, r = 10 m
$∴$ Volume of the balloon $= \frac{4}{3} π r^{3}$
$= \frac{4}{3} \times \frac{22}{7} \times 10^{3} = 4190.5 m^{3} (a p p r o x)$
Thus, the volume of the displaced air is $4190.5 m^{3}$ .
Given,
Density of air $= 1.2 k g m^{- 3}$
Then, mass of displaced air $= 4190.5 \times 1.2 k g = 5028.6 k g$
Now, mass of helium (m) inside the balloon is given by,
$m = \frac{M p V}{R T}$
Here,
$M = 4 \times 10^{- 3} k g m o l^{- 1}$
p=1.66 bar
V= Volume of the balloon
$= 4190.5 m^{3} R = 0.083 b a r d m^{3} K^{- 1} m o l^{- 1} T = 27^{\circ} = 300 K$
Then, $m = \frac{4 \times 1.66 \times 4190.5 \times 10^{3}}{0.083 \times 300} = 1117.5 k g (a p p r o x)$
Now, total mass of the balloon filled with helium = (100 + 1117.5) kg
= 1217.5 kg
Hence, pay load = (5028.6 – 1217.5) kg= 3811.1 kg
Hence, the pay load of the balloon is 3811.1 kg.

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