Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m^

Q: Why is the excess pressure inside a soap bubble $\frac{4S}{r}$ and not $\frac{2S}{r}$?

A soap bubble is a thin liquid film with air both inside and outside. This means it has two free liquid-gas interfaces. Each interface contributes a force due to surface tension, resulting in double the excess pressure of a single-surface liquid drop, which is $\frac{2S}{r}$.

Q: Why do we assume the temperature of both bubbles is the same?

Since both bubbles are kept in the same closed chamber for a sufficiently long time, they reach thermal equilibrium with the chamber environment. Therefore, their temperatures equalize to the chamber's temperature.

Q: What would happen if the chamber pressure was zero (vacuum)?

If the chamber pressure $P_0 = 0$, the absolute pressure inside the bubbles would just be their excess pressure, i.e., $P_A = \frac{4S}{r_A}$ and $P_B = \frac{4S}{r_B}$. The ratio of moles would then simplify to $\frac{n_B}{n_A} = \frac{r_B^2}{r_A^2}$.

Q: Can we use the formula $V = \frac{4}{3}\pi r^3$ even though the bubble is distorted by gravity?

The problem explicitly states to neglect the effect of gravity. In the absence of gravity, surface tension forces dominate completely, pulling the bubble into a mathematically perfect sphere to minimize surface area.

Question

Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure $8	ext{ Nm}^{-2}$. The radii of bubbles A and B are $2	ext{ cm}$ and $4	ext{ cm}$, respectively. Surface tension of the soap-water used to make bubbles is $0.04	ext{ Nm}^{-1}$. Find the ratio $\frac{n_B}{n_A}$, where $n_A$ and $n_B$ are the number of moles of air in bubbles A and B, respectively. [Neglect the effect of gravity]

Sigma Percentile · Accepted Answer

### The Magic of Soap Bubbles

Have you ever wondered why soap bubbles are always perfectly spherical? Or why a smaller bubble feels 'tighter' and more pressurized than a larger one? Today, we are going to dive deep into a classic JEE Advanced problem from 2009 that beautifully connects the mechanics of surface tension with the elegant laws of thermodynamics.

Imagine a closed chamber filled with air at a steady pressure of $8	ext{ Nm}^{-2}$. Inside this chamber float two soap bubbles of different sizes: Bubble A with a radius of $2	ext{ cm}$ and Bubble B with a radius of $4	ext{ cm}$. Our mission is to find the ratio of the number of moles of air trapped inside these two bubbles, $\frac{n_B}{n_A}$.

### The Physics of Surface Tension

To understand what is happening inside these bubbles, we must first understand the force that creates them: **surface tension**.

At the microscopic level, molecules in a liquid experience attractive intermolecular forces from their neighbors. A molecule deep inside the liquid is pulled equally in all directions. However, a molecule on the surface has no liquid neighbors above it, resulting in a net inward cohesive force. This inward pull makes the liquid surface behave like a stretched elastic membrane, always trying to minimize its surface area.

When you blow a soap bubble, you trap a pocket of air inside a thin spherical shell of soapy water. This shell has **two free surfaces**: an inner surface in contact with the trapped air, and an outer surface in contact with the surrounding atmosphere. Both of these surfaces are trying to shrink, which compresses the air inside.

### Deriving Excess Pressure

Because the surface tension forces are pulling inward, the pressure inside the bubble must be higher than the pressure outside to prevent the bubble from collapsing. This difference in pressure is known as the **excess pressure** ($P_{	ext{excess}}$).

Let us derive this using a simple force balance. Imagine cutting a soap bubble of radius $r$ in half. The surface tension force acts along the circumference of the cut, pulling the two halves together. Since there are two surfaces (inner and outer), the total surface tension force is:

$$F_{	ext{ST}} = 2 	imes (2\pi r 	imes S) = 4\pi r S$$

This force is balanced by the excess pressure acting over the circular cross-sectional area of the cut:

$$F_{	ext{pressure}} = P_{	ext{excess}} 	imes \pi r^2$$

Equating these two forces for equilibrium:

$$P_{	ext{excess}} 	imes \pi r^2 = 4\pi r S \implies P_{	ext{excess}} = \frac{4S}{r}$$

This is a fundamental result in fluid mechanics! The excess pressure is inversely proportional to the radius of the bubble. This means **smaller bubbles have higher internal pressure** than larger bubbles.

### Calculating the Absolute Pressures

Now let's apply this to our two bubbles. The absolute pressure inside any bubble is the sum of the external chamber pressure ($P_0$) and the excess pressure:

$$P_{	ext{inside}} = P_0 + \frac{4S}{r}$$

Let's substitute the given values:
- Chamber pressure, $P_0 = 8	ext{ Nm}^{-2}$
- Surface tension, $S = 0.04	ext{ Nm}^{-1}$

For **Bubble A** ($r_A = 2	ext{ cm} = 0.02	ext{ m}$):

$$P_A = 8 + \frac{4 	imes 0.04}{0.02} = 8 + \frac{0.16}{0.02} = 8 + 8 = 16	ext{ Nm}^{-2}$$

For **Bubble B** ($r_B = 4	ext{ cm} = 0.04	ext{ m}$):

$$P_B = 8 + \frac{4 	imes 0.04}{0.04} = 8 + \frac{0.16}{0.04} = 8 + 4 = 12	ext{ Nm}^{-2}$$

Notice how the pressure inside the smaller Bubble A ($16	ext{ Nm}^{-2}$) is significantly higher than the pressure inside the larger Bubble B ($12	ext{ Nm}^{-2}$).

### Connecting to Thermodynamics

Since both bubbles are in the same closed chamber, they are in thermal equilibrium with their surroundings. This means their temperatures are equal ($T_A = T_B = T$).

We can now use the **Ideal Gas Law** to find the number of moles of air inside each bubble:

$$PV = nRT \implies n = \frac{PV}{RT}$$

For Bubble A and Bubble B, the number of moles are:

$$n_A = \frac{P_A V_A}{RT} \quad 	ext{and} \quad n_B = \frac{P_B V_B}{RT}$$

Taking the ratio of the two equations:

$$\frac{n_B}{n_A} = \frac{P_B V_B}{P_A V_A}$$

Since the bubbles are spherical, their volumes are given by $V = \frac{4}{3}\pi r^3$. Substituting this into our ratio:

$$\frac{n_B}{n_A} = \frac{P_B \left(\frac{4}{3}\pi r_B^3ight)}{P_A \left(\frac{4}{3}\pi r_A^3ight)} = \frac{P_B}{P_A} \left(\frac{r_B}{r_A}ight)^3$$

### The Final Computation

Let's plug in our calculated pressures and the given radii into this elegant formula:

$$\frac{n_B}{n_A} = \frac{12}{16} 	imes \left(\frac{4	ext{ cm}}{2	ext{ cm}}ight)^3$$

Simplifying the terms:
- The pressure ratio is $\frac{12}{16} = \frac{3}{4}$
- The radius ratio is $\frac{4}{2} = 2$

Now, substitute these back:

$$\frac{n_B}{n_A} = \frac{3}{4} 	imes (2)^3 = \frac{3}{4} 	imes 8 = 6$$

And there we have it! The ratio of the number of moles of air in Bubble B to Bubble A is exactly **6**.

### A Beautiful Paradox: The Rich Get Richer

What would happen if we connected these two bubbles with a thin tube?

Instinct might tell you that air would flow from the larger bubble to the smaller bubble until they are equal in size. But physics tells a different story!

Since the smaller bubble has a higher internal pressure ($16	ext{ Nm}^{-2}$) than the larger bubble ($12	ext{ Nm}^{-2}$), air will actually flow from the **smaller bubble to the larger bubble**. As air leaves the smaller bubble, its radius decreases, which makes its internal pressure go even higher! Meanwhile, the larger bubble grows, and its pressure decreases further. This runaway process continues until the smaller bubble completely collapses, emptying all its air into the larger one.

This phenomenon is often called the 'two-bubble paradox' or the 'rich-get-richer' effect in physics, and it is a beautiful demonstration of how nature sometimes behaves in ways that defy our everyday intuition.

Enter Numerical Value:

Visualized Solution

Visualizing the Closed Chamber and Soap Bubbles

Understanding Excess Pressure

Calculating Pressure inside Bubble $A$

Evaluating $P_{A}$

Calculating Pressure inside Bubble $B$

Evaluating $P_{B}$

Applying the Ideal Gas Law

Setting up the Ratio $\frac{n _{B}}{n _{A}}$

Calculating the Final Ratio

Exploring Further Scenarios

The Sigma Insight: Surface Tension and Capillary Action

The Magic of Soap Bubbles

The Physics of Surface Tension

Deriving Excess Pressure

Calculating the Absolute Pressures

Connecting to Thermodynamics

The Final Computation

A Beautiful Paradox

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