Question

Pay load is defined as the difference between the mass of displaced air and the mass of the balloon. Calculate the pay load when a balloon of radius 10 m, mass 100 kg is filled with helium at 1.66 bar at 27°C. (Density of air = 1.2 kg $m^{-3}$ and R = 0.083 bar $dm^{3}$$K^{-1}$ $mol^{-1}$).

Answer 1

Payload of the balloon = mass of the displaced air - mass of the balloon

Radius of the balloon, r = 10 m

Mass of the balloon, m = 100 kg

Therefore, volume of the balloon =$\frac { 4 }{ 3 } \pi { r3 }$

=$\frac { 4 }{ 3 } \times \frac { 22 }{ 7 } \times { (10) }^{ 3 }=4190.5{ m }^{ 3 }$

Now the volume of the displaced air =$\quad 4190.5\quad { m }^{ 3 }$

From the given,

Density of air = $1.2 kg{ m }^{ -3 }$

Therefore, mass of the displaced air = $\quad 4190.5\quad \times \quad 1.2\quad =\quad 5028.6\quad kg$

Let w be the mass of helium gas filled into the balloon,

Then, pv = $\quad \left( \frac { w }{ m }  \right)$ RT

Or

$w=\frac { PVM }{ RT } =\quad \frac { 1.66\quad \times \quad 4190.5\quad \times \quad { 10 }^{ 3 }\quad \times \quad 4) }{ (0.083\times 300) } \quad =\quad 1117\quad kg$

Total mass of the balloon filled with He = 1117 + 100 = 1217 kg

Therefore payload of the balloon = 5028.6 - 1217 = 3811.6 kg

Hence, the payload of the balloon is 3811.6 kg

Answer 2

Answer:

The volume of the balloon is V=  

3

4

​  

πr  

3

.

The radius of balloon is 10 m.

Hence, the volume of the balloon is V=  

3

4

​  

×3.1416×(10)  

3

=4186.7m  

3

.

The mass of displaced air is obtained from the product of volume and density. It is 4186.7×1.2=5024.04kg.

The number of moles of gas present are n=  

RT

PV

​  

 =  

0.083×300

1.666×4186.7×10  

3

 

​  

=279.11×10  

3

.

Note: Here, the unit of volume is changed from m  

3

 to dm  

3

.  

1m  

3

=1000dm  

3

.

Mass of helium present is obtained by multiplying the number of moles with molar mass. It is 279.11×10  

3

×4=1116.44×10  

3

g=1116.4 kg.

The mass of filled balloon is the sum of the mass of the empty ballon and the mass of He. It is  100+1116.4=1216.4 kg.

Pay load = mass of displaced air − mass of balloon =5024.04−1216.44=3807.6 kg

Explanation: