Question

A weather balloon is designed to expand to a maximum radius of 24 m at its working altitude, where the air pressure is 0.030 atm and the temperature is 200 K. If the balloon is filled at atmospheric pressure and 349 K, what is its radius at liftoff

Answer 1

Radius at liftoff 8.98 m

Step-by-step explanation:

At the working altitude;

maximum radius = 24 m

air pressure = 0.030 atm

air temperature = 200 K

At liftoff;

temperature = 349 K

pressure = 1 atm

radius = ?

First, we assume balloon is spherical in nature,

and that the working gas obeys the gas laws.

from the radius, we can find the volume of the balloon at working atmosphere.

Volume of a sphere =
`(4)/(3) \pi r^(3)`

volume of balloon =
`(4)/(3)` x 3.142 x
`24^(3)` = 57913.34 m^3

using the gas equation,

`(P1V1)/(T1)` =
`(P2V2)/(T2)`

The subscript 1 indicates the properties of the gas at working altitude, and the subscript 2 indicates properties of the gas at liftoff.

imputing values, we have

`(0.03*57913.34)/(200)` =
`(1*V2)/(349)`

0.03 x 57913.34 x 349 = 200V2

V2 = 606352.67/200 = 3031.76 m^3 this is the volume occupied by the gas in the balloon at liftoff.

from the formula volume of a sphere,

V =
`(4)/(3) \pi r^(3)` =
`(4)/(3)` x 3.142 x
`r^(3)` = 3031.76

4.19
`r^(3)` = 3031.76

`r^(3)` = 3031.76/4.19

radius r of the balloon on liftoff =
`\sqrt[3]{723.57}` = 8.98 m