Question

A weather balloon is designed to expand to a maximum radius of 21 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 323 K, what is its radius at liftoff?

Answer 1

Answer:

7.65 m

Explanation:

$P_1$ = Initial pressure = 0.03 atm

$P_2$ = Final pressure = 1 atm

$r_1$ = Inital radius = 21 m

$V_1$ = Intial volume of gas = $\frac{4}{3}\pi r_1^3$

$V_2$ = Final volume of gas = $\frac{4}{3}\pi r_2^3$

$T_1$ = Initial temperature = 200 K

$T_2$ = Final temperature = 323 K

From ideal gas law we have

$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\\\Rightarrow \frac{P_1\frac{4}{3}\pi r_1^3}{T_1}=\frac{P_2\frac{4}{3}\pi r_2^3}{T_2}\\\Rightarrow \frac{P_1 r_1^3}{T_1}=\frac{P_2r_2^3}{T_2}\\\Rightarrow r_2=\frac{P_1r_1^3T_2}{T_1P_2}\\\Rightarrow r_2=\left(\frac{0.03\times 21^3\times 323}{200\times 1}\right)^{\frac{1}{3}}\\\Rightarrow r_2=7.65\ m$

The radius at liftoff is 7.65 m