A hot air balloon is a sphere of radius 8 m. The sir inside is at a temperature of 60oC. How large a mass can the balloon lift when the outside temperature is 20oC?
Pressure inside (Pi) balloon is larger than outer pressure (Po) of atmosphere. ∴Pi−Po=R2σ, where σ = surfaace tension in membrane of balloon R =radius of balloon. Gas or air inside is perfect (considered) ∴PiV=nRTi V = volume of balloon ni = no. of moles of gas in balloon R = gas constant Ti = temperature of balloon ni=RTiPiV=molecular mass(MA)mass of air balloon(Mi) ni=MAMi=RTiPiV similarly, no=RToPoV By principle of floatation W+Mig=Mog W= weight lifted by balloon W=Mog−Mig W=(Mo−Mi)g where no = no. of molecules of air displaced by balloon V = Volume of air displaced by balloon equal to volume of balloon If Mo mass of air is displaced by balloon MA = Molecular mass inside or outside balloon ni=MAMi=RTiPiV similarly, no=RToPoV By principle of floatation W+Mig=Mog W= weight lifted by balloon W=Mog−Mig W=(Mo−Mi)g where no = no. of molecules of air displaced by balloon V = Volume of air displaced by balloon equal to volume of balloon If Mo mass of air is displaced by balloon MA = Molecular mass inside or outside balloon ni=MAMi=RTiPiV similarly, no=RToPoV By principle of floatation W+Mig=Mog W= weight lifted by balloon W=Mog−Mig W=(Mo−Mi)g where no = no. of molecules of air displaced by balloon V = Volume of air displaced by balloon equal to volume of balloon If Mo mass of air is displaced by balloon MA = Molecular mass inside or outside balloon ∴no=MAMo or no=MAMo=RToPoV⟹Mo=RToPoVMA Also, Mi=RTiPiVMA ∴W=[RToPoVMA−RTiPiVMA]g ∴W=RVMA[ToPo−TiPi]g MA=21%O2+79%N2 MA=21×332+0.79×28=4[0.21×8+0.79×7] MA=4[1.68+5.53]=4[7.21]=28.84g MA=0.02884kg, Pt=Po+R2σ W=8.31434π×8×8×8×0.02884[(273+20)1.013×106−273+60Pt]g Pi=Po+Pdue to S.T. of membrane=Po+R2σ Pi=[1.013×105+82×5]=1.013×105 ∴W=3×8.3144×3.14×8×8×8×0.02884[2931.013×105− −3331.013×105]g W=3×8.3144×3.14×8×8×8×0.02884[2931−3331]g W=3×8.3144×3.14×8×8×8×0.02884[293×333333−293]g W=3×8.314×293×3334×3.14×8×8×8×0.02884 W=3044.2N
