At sea level and 303 k, a rubber balloon is inflated with hydrogen gas to a diameter of 3.00 ft and an internal pressure of 16.7 lb/in scuared. Find the diameter of the balloon at an altitude of 30000 ft where the internal pressure is 6.4lb/in scuared and the temperature is 238k.
Why did the rubber balloon go to such great heights? Because it wanted to reach new "hi-ly"te!
Now, let's solve this ballooning problem. We can use the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Since we're only interested in the relationship between pressure and volume, we can rewrite the ideal gas law as P₁V₁ = P₂V₂. Let's plug in the values we know:
P₁ = 16.7 lb/in² (sea level pressure)
V₁ = (3.00 ft)³ (initial volume)
P₂ = 6.4 lb/in² (altitude pressure)
T₁ = 303 K (sea level temperature)
T₂ = 238 K (altitude temperature)
We want to find V₂, the diameter of the balloon at an altitude of 30000 ft.
But since the initial diameter is given, let's find the initial volume first and then find the final volume. The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius (half of the diameter).
V₁ = (4/3)π(1.5 ft)³
V₂ = ?
Simplifying the initial volume:
V₁ = (4/3)π(3/2 ft)³ (cancelling units)
V₁ = (4/3)π(27/8 ft³) (cubing 3/2)
V₁ = (36/24)π ft³
V₁ = (3/2)π ft³
Now, let's solve for the final volume using the ideal gas law:
P₁V₁ = P₂V₂
(16.7 lb/in²) ((3/2)π ft³) = (6.4 lb/in²) V₂
Simplifying:
V₂ = (16.7 lb/in² / 6.4 lb/in²)V₁
V₂ = (16.7 / 6.4) (3/2)π ft³
V₂ = 2.609375π ft³
Now, let's find the radius and diameter of the balloon at the altitude:
V₂ = (4/3)πr₂³
2.609375π ft³ = (4/3)πr₂³
Cancelling π and simplifying:
2.609375 ft³ = (4/3)r₂³
To find the radius at the altitude:
r₂³ = (3/4)(2.609375 ft³)
r₂ = (3/4)^(1/3) (2.609375 ft)
r₂ ≈ 1.603221 ft
Finally, the diameter at the altitude:
d₂ = 2r₂
d₂ ≈ 2(1.603221 ft)
d₂ ≈ 3.206442 ft
So, the diameter of the balloon at an altitude of 30000 ft is approximately 3.206442 ft. Keep soaring high! 🎈