Question

A weather balloon is designed to expand to a maximum radius of 22 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 280 K, what is its radius at lift-off?

Answer 1

R_2 = 7.647 m

Step-by-step explanation:

HI!

Let us consider that the aballon is filled with a gas that follows the ideal gas equation. Since the amount (number of moles) of the gas is constant we should have:

`PV/T = constant`

Therefore, we can have the following relationship between two differnet states given their Volume, Pressure and Tempreature:

`(P_1 V_1)/(T_1) = (P_2 V_2)/(T_2)`

Since the volume of a sphere is

`V = \frac{4 \text{$\pi $R}^3}{3}`

The relathionship will be:

`(P_1 R_1^3)/(T_1) = (P_2 R_2^3)/(T_2)`

Solving for R_2:

`R_2 = R_1 \sqrt[3]{(P_1 T_2)/(P_2 T_1)}`

Where the index 2 is the state at the lift-off and the index 1 denotes the state at its maximum radius. Replacing all the values given we find that:

R_2 = 7.647 m