Suppose you have a spherical balloon filled with air at room temperature and 1.0 atm pressure; its radius is 17 cm . You take the balloon in an airplane, where the pressure is 0.89 atm.
If the temperature is unchanged, what's the balloon's new radius?
p1/p2=V2/V2
but volume is proportional to radius cubed, so
p1/p2=r2^3/r1^3
1/.89 * 17^3cm=R2^3
r2=17 cubrt(1/.89)
To find the new radius of the balloon, we can use Boyle's Law, which states that the pressure and the volume of a gas are inversely proportional, assuming constant temperature.
First, let's determine the initial volume of the balloon using the formula for the volume of a sphere:
V = (4/3)πr³
where V is the volume and r is the initial radius.
Substituting the given value of the radius (17 cm), we have:
V = (4/3)π(17 cm)³
Now, let's use Boyle's Law equation to find the new volume of the balloon when the pressure is 0.89 atm. Boyle's Law can be expressed as:
P₁V₁ = P₂V₂
where P₁ and V₁ are the initial pressure and volume, and P₂ and V₂ are the final pressure and volume.
Rearranging the equation to solve for V₂, we get:
V₂ = (P₁V₁) / P₂
Substituting the given values, we have:
V₂ = (1.0 atm) * [(4/3)π(17 cm)³] / (0.89 atm)
Now, we can use the formula for the volume of a sphere to find the new radius of the balloon:
V₂ = (4/3)πr₂³
Substituting the calculated value of V₂, we can solve for r₂:
(4/3)πr₂³ = (1.0 atm) * [(4/3)π(17 cm)³] / (0.89 atm)
Simplifying the equation, we find:
r₂³ = [(1.0 atm) * (4/3)π(17 cm)³] / (0.89 atm) / (4/3)π
Canceling out units and simplifying further:
r₂³ = (1.0 * 17³) / 0.89
r₂³ ≈ 17.22
Taking the cube root of both sides, we find:
r₂ ≈ ∛(17.22)
Therefore, the new radius of the balloon in the airplane at a pressure of 0.89 atm is approximately 12.50 cm.