To determine the radius of the soap bubble when the external pressure drops to zero, we can apply the principles of fluid mechanics and the physics of bubbles. The key concept here is the balance of pressures acting on the bubble, which includes the internal pressure due to the surface tension of the soap film and the external pressure acting on the bubble.
Understanding Pressure in a Soap Bubble
A soap bubble has two surfaces, and the pressure inside the bubble is greater than the pressure outside due to the surface tension of the soap film. The relationship between the internal pressure (P_inside) and the external pressure (P_outside) can be expressed using the formula:
P_inside = P_outside + (4 * γ / r)
Where:
- P_inside is the pressure inside the bubble.
- P_outside is the pressure outside the bubble.
- γ is the surface tension of the soap film.
- r is the radius of the bubble.
Initial Conditions
Initially, the bubble has a radius of r₀ = 2.0 mm (or 0.002 m) and is floating in a bell jar at atmospheric pressure (P = 1 atm). The internal pressure can be calculated as follows:
Converting atmospheric pressure to SI units:
P = 1 atm = 101325 Pa
Now, substituting the values into the pressure equation:
P_inside = 101325 Pa + (4 * 2.50 × 10-2 N/m / 0.002 m)
Calculating the term involving surface tension:
4 * 2.50 × 10-2 N/m / 0.002 m = 50 N/m²
Thus, the internal pressure becomes:
P_inside = 101325 Pa + 50 Pa = 101375 Pa
Final Conditions When External Pressure is Zero
As the vacuum pump is turned on, the external pressure (P_outside) decreases to zero. We need to find the new radius (r) of the bubble when:
P_outside = 0 Pa
Using the pressure equation again:
P_inside = 0 Pa + (4 * γ / r)
Setting P_inside equal to its previous value:
101375 Pa = (4 * 2.50 × 10-2 N/m) / r
Rearranging to solve for r:
r = (4 * 2.50 × 10-2 N/m) / 101375 Pa
Calculating the radius:
r = (0.1 N/m) / 101375 Pa ≈ 9.86 × 10-7 m
Converting this back to millimeters:
r ≈ 0.000986 m = 0.986 mm
Final Result
When the external pressure drops to zero, the radius of the soap bubble decreases to approximately 0.986 mm. This illustrates how the balance of pressures and the effects of surface tension govern the behavior of bubbles in varying pressure environments.
