Question

Payload is defined as the difference between the mass of displaced air and the mass of the balloon. Calculate the payload, when a balloon of radius 10 m of mass 100 kg is filled with helium at 1.66 bar at 27 ${}^{\circ }C$. (Density of air = 1.2 kg m${}^3$ and R = 0.083 bar dm $3$ K${}^{-1}$ mol${}^{-1}$)

Answer 1

The volume of the balloon is $V=\frac{4}{3}\pi r^3$.
The radius of balloon is 10 m.
Hence, the volume of the balloon is $V=\frac{4}{3}\times 3.1416\times (10{)}^3=4186.7m^3$.
The mass of displaced air is obtained from the product of volume and density. It is $4186.7\times 1.2=5024.04kg$.

The number of moles of gas present are $n=\frac{PV}{RT}$ $=\frac{1.666\times 4186.7\times {10}^3}{0.083\times 300}=279.11\times {10}^3$.

Note: Here, the unit of volume is changed from $m^3$ to $d{m}^3$.

$1m^3=1000d{m}^3$.

Mass of helium present is obtained by multiplying the number of moles with molar mass. It is $279.11\times {10}^3\times 4=1116.44\times {10}^{3}g=1116.4$ kg.
The mass of filled balloon is the sum of the mass of the empty ballon and the mass of He. It is $100+1116.4=1216.4$ kg.
Pay load $=$ mass of displaced air $-$ mass of balloon $=5024.04-1216.44=3807.6$ kg