Question

My weight is 80 kg. I want to fly in the sky with the help of balloons each containing 50 moles of ${H_2}$ gas at 0.05atm pressure and ${27^0}C$ temperature. If the density of air at the condition is $1.25gm{L^{ - 1}}$. How many such types of balloons do I need to attach with myself to fly in the sky? [mass of balloon can be neglected]
(a) 2
(b) 3
(c) 7
(d) 9

Answer 1

A gas balloon is a balloon that rises and floats in the air because it is filled with a gas lighter than air (such as helium or hydrogen). When not in flight, it is tethered to prevent it from flying away and is sealed at the bottom to prevent the escape of gas.

Formula used: $PV = nRT$
mass lift generated = mass of balloon +mass of ${H_2}$(m) + mass lift for payload (${m_1}$).

Complete step-by-step solution:
Given data which can be filter out from the question is:
For one balloon: moles of gas n = 50, pressure (P) = 0.05 atm
Therefore the mass of ${H_2}$ in one balloon will be twice the number of moles (m) = $50 \times 2 = 100g$ $m = 100g$
Temperature = ${27^0}C = 300K$
The mass lifted by one balloon, say ${m_1}$ can be calculated.
Using the ideal gas equation $PV = nRT$, we find out the volume of air displaced by one balloon, say it to be V. $PV = nRT$ substituting the values of n =50mol, P = 0.05atm, R =0.0821Latm/mol/K, and T = 300k we get:
$V = \dfrac{{50 \times 0.0821 \times 300}}{{0.05}} = 24630L$
Now we find out the mass displaced by one balloon, Mass = mass lift.
According to the formula $Density = \dfrac{{mass}}{{volume}}$
$mass = volume \times Density$, substituting V = 24630 L and density of air = 1.25 gm/L.
$masslift = 1.25 \times 24630 = 30787g$
We know that:
The mass lift generated = mass of balloon +mass of ${H_2}$(m) + mass lift for payload (${m_1}$).
From the above substituting all the mass in this equation we get:
$30787 = 0 + m + {m_1} \\\ 30787 = 100 + {m_1} \\\$
${m_1} = 30787 - 100 \\\ {m_1} = 30687g \\\$
For lifting the weight of 80 Kg = 80000g. Number of balloons required will be: weight/mass lifted by one balloon that is it should be greater than
Number of balloons required = $\dfrac{{80000}}{{{m_1}}} = \dfrac{{80000}}{{30687}} = 2.6$
So the least integer which is greater than 2.6 is 3. Hence the number of balloons required to lift the 80 kg weight in the air is 3.

Hence the correct option is (b).

Note: The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function.