When a soap bubble expands in a vacuum bell jar, the pressure inside the jar decreases. To find the new pressure inside the bell jar after the bubble expands to a radius of 1.0 cm, we can use the principle of Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume at constant temperature.
Calculating the Initial Volume of the Soap Bubble
First, let's calculate the initial volume of the soap bubble when its radius is 1.0 mm.
Given that the radius of the soap bubble, r = 1.0 mm = 0.001 m
The initial volume, V₁, of the soap bubble can be calculated using the formula for the volume of a sphere: V = (4/3)πr³
Substitute the radius into the formula to find the initial volume:
V₁ = (4/3)π(0.001)³
Calculating the Final Volume of the Soap Bubble
Next, let's calculate the final volume of the soap bubble when its radius is 1.0 cm.
Given that the radius of the soap bubble, r = 1.0 cm = 0.01 m
The final volume, V₂, of the soap bubble can be calculated using the same formula for the volume of a sphere: V = (4/3)πr³
Substitute the radius into the formula to find the final volume:
V₂ = (4/3)π(0.01)³
Applying Boyle's Law
As per Boyle's Law, p₁V₁ = p₂V₂, where p₁ and V₁ are the initial pressure and volume, and p₂ and V₂ are the final pressure and volume, respectively.
Now, we have the initial and final volumes of the soap bubble. We also know the initial pressure inside the jar is 100 kPa.
Substitute the initial volume, final volume, and initial pressure into Boyle's Law to find the final pressure:
100 kPa * V₁ = p₂ * V₂
Solve for p₂ to find the new pressure inside the bell jar after the soap bubble expands.
This calculation will give you the new pressure inside the bell jar after the soap bubble expands to a radius of 1.0 cm.
