question_answer 21)

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 and mass 100 kg is filled with helium at 1.66 bar at\[27{{\,}^{{}^\circ }}C\]. (Density of air \[=1.2kg\,{{m}^{-3}}\]and \[R=0.083\,bar\,d{{m}^{3}}{{K}^{-1}}mo{{l}^{-1}})\]

Answer:

Volume of balloon = \[\frac{4}{3}\pi {{r}^{3}}\] \[=\frac{4}{3}\times \frac{22}{7}\times {{(10)}^{3}}\] \[=4190.5{{m}^{3}}\] Mass of displaced air = V x d \[=4190.5\times 1.2=5028.6kg\] Mass of helium in the balloon can be calculated as \[PV=\frac{w}{M}\times RT\] \[w=\frac{PMV}{RT}=\frac{1.66\times 4\times 4190.5}{0.083\times 300}\] \[=1117.48\times {{10}^{3}}g=1117.48\,kg\] Mass of filled balloon = 1117.48 +100 = 1217.48 kg Payload = Mass of displaced air - Mass of filled balloon \[=5028.6-1217.48=3811.12kg\]