Answer:
[tex]D_{1}=11.32 m[/tex]
Explanation:
We will need to use the ideal gas equation. The equation is given by:
[tex]PV=nRT[/tex]
As we have the same amount of molecules in the initial and final steps, therefore we can do this:
[tex]\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}[/tex] (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:
[tex]V_{1}=\frac{4}{3}\pi (\frac{D_{1}}{2})^{3}[/tex] (2)
[tex]V_{2}=\frac{4}{3}\pi (\frac{D_{2}}{2})^{3}[/tex] (3)
- D(2) = 32 m
So [tex]V_{2}=17157.3 m^{3}[/tex]
Let's solve the equation (1) for V(1)
[tex]V_{1}=\frac{T_{1}P_{2}V_{2}}{P_{1}T_{2}}[/tex]
[tex]V_{1}=\frac{300*0.028*17157.3}{1*190}[/tex]
[tex]V_{1}=758.53 m^{3}[/tex]
And using the equation (2) we can find D.
[tex]D_{1}=2(\frac{3}{4}V_{1})^{1/3}[/tex]
[tex]D_{1}=2(\frac{3}{4\pi}*758.53)^{1/3}[/tex]
[tex]D_{1}=11.32 m[/tex]
I hope it helps you!
