question_answer 54)

                  A hot air balloon is a sphere of radius 8 m. The air inside is at a temperature of \[{{60}^{o}}C\]. How large a mass can the balloon lift when the outside temperature is \[{{20}^{o}}C\]? (Assume air is an ideal gas\[R=8.314\,J\,mol{{e}^{-1}}\,{{K}^{-1}},\] 1 atm. \[=1.013\times {{10}^{5}}\,Pa;\] the membrane tension is \[5\,N\,\,{{m}^{-1}}\].)

Answer:

                  Excess pressure inside balloon,                 \[{{P}_{i}}-\,{{P}_{o}}\,\,\frac{2\sigma }{r}\]                                          ... (i)                 Here \[\sigma \,=\] surface tension                 For ideal gas, \[PV=\mu \,RT\]                 \[\therefore \] \[\mu =\,\frac{PV}{R{{T}_{i}}}\] \[\text{=}\,\frac{\text{Mass}\,\,\text{of}\,\,\text{air}\,\,\text{inside}\,\,\text{balloon}\,\,\text{(}{{\text{M}}_{\text{i}}}\text{)}}{\text{Molar}\,\,\text{mass}\,\,\text{of}\,\,\text{air}\,\text{(}{{\text{M}}_{\text{A}}}\text{)}}\]                 are the number of moles inside balloon Similarly, number of moles outside the balloon                 \[{{\mu }_{0}}=\,\frac{{{P}_{0}}V}{R{{T}_{0}}}\] \[\text{=}\,\frac{\text{mass}\,\,\text{of}\,\,\text{air}\,\,\text{outside}\,\,\text{balloon}\,\text{(}{{\text{M}}_{\text{0}}}\text{)}}{\text{molar}\,\,\text{mass}\,\text{of}\,\text{air}\,\,\text{(}{{\text{M}}_{\text{A}}}\text{)}}\]                 Molar mass of air, \[{{M}_{A}}=28.84\,g\]                 \[=28.84\times {{10}^{-3}}\,kg\]                 Weight raised by balloon \[=({{M}_{o}}-{{M}_{i}})g\]     ?. (ii)                 Now \[{{M}_{o}}=\frac{{{P}_{o}}V{{M}_{A}}}{R{{T}_{o}}}\] and \[{{M}_{i}}=\frac{{{P}_{i}}V\,{{M}_{A}}}{R{{T}_{i}}}\]                 \[\therefore \] Weight raised, \[W=\,\frac{V{{M}_{A}}}{R}\,\left( \frac{{{P}_{0}}}{{{T}_{0}}}-\ \frac{{{P}_{i}}}{{{T}_{i}}} \right)g\,\] ...(iii)                 From eqn. (i),  \[{{P}_{i}}=\,{{P}_{0}}+\,\frac{2\sigma }{r}\]                 \[\therefore \]\[W=\,\frac{V{{M}_{A}}}{R}\,\,\left[ \frac{{{P}_{o}}}{{{T}_{o}}}-\frac{{{P}_{o}}}{{{T}_{i}}}\,-\frac{2\sigma }{r{{T}_{i}}} \right]\,g\]                 or mass raised, m                 \[=\,\frac{V{{M}_{A}}}{R}\,\left[ {{P}_{o}}\,\left( \frac{1}{{{T}_{o}}}-\frac{1}{{{T}_{i}}} \right)-\frac{2\sigma }{r{{T}_{i}}} \right]\]                 \[=\,\frac{\frac{4}{3}\,\pi {{r}^{3}}\,{{M}_{A}}}{R}\,\left[ {{P}_{o}}\left( \frac{1}{{{T}_{o}}}-\frac{1}{{{T}_{o}}} \right)-\frac{2\sigma }{r{{T}_{i}}} \right]\]                 or \[m=\,\frac{4\times \,3.14\,\times \,{{(8)}^{3}}\times \,28.84\,\times \,{{10}^{-3}}}{3\times \,8.314}\]                 \[\left[ 1.013\times \,{{10}^{5}}\left( \frac{1}{293}-\frac{1}{373} \right)\,-\frac{2\times \,5}{8\times \,333} \right]\]                 \[=304.42\,kg\]