# A balloon of diameter 20 metre weighs 100kg . Calculate its pay-load if it is filled with He at 1.0atm and $27^{\large\circ}C$ . Density of air is 1.2 $kgm^{-3}$ . [$R = 0.082dm^3\;atm\; K^{-1}\;mol^{-1}$]

$\begin{array}{1,1}(a)\;424.67\times10^4g\\(b)\;4.2467\times10^4g\\(c)\;42.467\times10^4g\\(d)\;0.424\times10^4g \end {array}$

Weight of balloon = 100 kg or $10\times10^4g$
Volume of balloon = $\large\frac{4}{3}\pi r^3$
$\;\;\;\;\;\;\;\;\;= \large\frac{4}{3}\times\large\frac{22}{7}\times(\large\frac{20}{2}\times100)^3$
$\;\;\;\;\;\;\;\;\;=4190\times10^6cm^3$
$\;\;\;\;\;\;\;\;\;=4190\times10^3\;litre$
Weight of gas (He) in balloon = $\large\frac{PV_m}{RT}$
(Since PV = $\large\frac{w}{m}RT$)
$=\large\frac{1\times4190\times10^3\times4}{0.082\times300}$
$=68.17\times10^4g$
$\therefore$ Total weight of gas and balloon = $68.13\times10^4+10\times10^4 = 78.13\times10^4$
Weight of air displaced = $\large\frac{1.2\times4190\times10^6}{10^3}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= 502.8\times10^4g$
$\therefore$ Pay load = weight of air displaced - (weight of balloon + weight of gas)
$\therefore$ Pay load = $502.8\times10^4 - 78.13\times10^4$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=424.67\times10^4g$