Question

A man weighs 72.1Kg and want to fly in the sky with the aid of balloons itself weighing 20kg and each containing 50 moles of ${{H}_{2}}$ gas at 0.05 atm and ${{27}^{\circ }}c$ .If the density of air at the given condition is 1.25g/litre, how many such types of balloons he is needed to fly in the sky.

Answer 1

Hint: Firstly, write the given which may help us figure out the exact need easily. ${{H}_{2}}$ is an ideal gas so can make use of an ideal gas equation to find volume and other such quantities. Always make sure to write units.

Complete step by step solution:
Given,
Number of moles of ${{H}_{2}}$= 50 moles
The pressure of ${{H}_{2}}$ = 0.05 atm
The density of air = 1.25gm/L
The weight of the man = 72.15 Kg
Mass of ${{H}_{2}}$ in a balloon is 2 $\times $50 = 100g
We know ${{H}_{2}}$is an ideal gas. Ideal gas is that gas which obeys charle's law, Boyle's law and avagadro's law.
Using ideal gas equation we can find the volume,
The ideal gas equation: PV = nRT
$V=\dfrac{~nRT}{P}$
\[V=\dfrac{50\times 0.0821\times 300}{0.05}\]
V = 29630L
Thus, the volume of air that is displaced because of one balloon is 29630L.
To calculate the mass of air in a balloon that is been displaced,
The formula to calculate density:
\[Density=\dfrac{mass}{Volume}\]
Using this we can find out mass as the value of density is given and the volume is been calculated:
Mass = Density $\times $ Volume
$\implies$ 1.25 $\times $ 29630
$\therefore$ 30787g
Thus, the mass of air displaced is 30787g.
The mass or volume displaced means the mass or volume that has been lifted.
The mass lifted = mass of balloons + mass of ${{H}_{2}}$+ mass lift for payload
 30787 = 20 + 100 + mass lift for payload
Mass lift of payload = 30567g
The weight of the man = 72.15 Kg
= 72150g
Number of balloon required = $\dfrac{72150}{30567}$
$\implies 2.6$
$\therefore 3$ (approximately)

Thus, the number of balloons required is 3.

Note: Analyzing this question is a bit difficult, so it is very important to write the given part with their units. If we don't trend put up the units, somewhere we may go wrong that leads us to the wrong answer. The number balloon is 3, practically we cannot take 2.6 balloons so approximately 3 balloons are taken.