Soap Bubble Pressure Difference and Air Flow

To determine the behavior of the system just after opening the valve, we analyze the excess pressure inside the soap bubbles at both ends of the tube.

1. Excess Pressure in a Soap Bubble

The excess pressure inside a spherical soap bubble with two surfaces in contact with air is given by Young-Laplace equation:
$\Delta P = P_{\text{in}} - P_0 = \frac{4T}{R}$
where:

• $T$ is the surface tension of the soap solution,
• $R$ is the radius of curvature of the spherical bubble cap,
• $P_0$ is the atmospheric pressure.

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2. Analysis of Curvature and Pressure at Each End

For a bubble formed at the circular opening of a tube with radius $r$:

• The minimum possible radius of curvature occurs when the bubble is a hemisphere, where $R = r$.
• For a sub-hemispherical cap, the radius of curvature $R$ is strictly greater than the tube radius ($R > r$).

At End 1:

The bubble is hemispherical, so its radius of curvature is:
$R_1 = r$
Thus, the internal pressure at End 1 is:
$P_1 = P_0 + \frac{4T}{r}$

At End 2:

The bubble is sub-hemispherical, so its radius of curvature is:
$R_2 > r$
Thus, the internal pressure at End 2 is:
$P_2 = P_0 + \frac{4T}{R_2}$

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3. Comparison of Internal Pressures

Since $R_2 > r$, we have:
$\frac{4T}{r} > \frac{4T}{R_2} \implies P_1 > P_2$

The pressure inside End 1 is greater than the pressure inside End 2.

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4. Air Flow and Volume Change

• Air flows naturally from the region of higher pressure to the region of lower pressure. Thus, air flows from End 1 towards End 2.
• As air leaves End 1, the volume of the soap bubble at End 1 decreases (it flattens further).

Hence, the correct option is (B).