Question

Weather balloons are high altitude balloons that are capable of reaching an altitude of 40 km (or 25 miles). They are usually filled with hydrogen or helium gas. One such balloon is filled with gas that is initially at a temperature of 300 K and at atmospheric pressure. This particular balloon can expand to a maximum diameter of 32 m when it reaches an altitude where the pressure and temperature are 0.028 atm and 190 K respectively. Determine the diameter of the balloon at lift off

Answer 1

`D_(1)=11.32 m`

Step-by-step explanation:

We will need to use the ideal gas equation. The equation is given by:

`PV=nRT`

• P is the pressure
• V is the volume
• n is the amount of molecules
• R is the ideal gas constant

As we have the same amount of molecules in the initial and final steps, therefore we can do this:

`(P_(1)V_(1))/(T_(1))=(P_(2)V_(2))/(T_(2))` (1)

- P(1) is the atmospheric pressure (P(1) = 1 atm) and P(2) is 0.028 atm

- T(1) is 300 K and T(2) is 190 K

- V(1) is the volume of the balloon in the first step, we can consider a spherical geometry so:

`V_(1)=(4)/(3)\pi ((D_(1))/(2))^(3)` (2)

`V_(2)=(4)/(3)\pi ((D_(2))/(2))^(3)` (3)

- D(2) = 32 m

So
`V_(2)=17157.3 m^(3)`

Let's solve the equation (1) for V(1)

`V_(1)=(T_(1)P_(2)V_(2))/(P_(1)T_(2))`

`V_(1)=(300*0.028*17157.3)/(1*190)`

`V_(1)=758.53 m^(3)`

And using the equation (2) we can find D.

`D_(1)=2((3)/(4)V_(1))^(1/3)`

`D_(1)=2((3)/(4\pi)*758.53)^(1/3)`

`D_(1)=11.32 m`

I hope it helps you!