The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. What is the ratio between the volume of the first and the second bubble?

Options

A. `9:1`
B. `1:3`
C. `1:9`
D. `3:1`

Explanation

To solve the problem, we need to determine the ratio between the volumes of two soap bubbles given that the excess pressure inside the first bubble is three times that of the second bubble.

### Step-by-Step Solution:

1. **Understanding Excess Pressure in Soap Bubbles**:
The excess pressure \( P \) inside a soap bubble is given by the formula:
\[ P = \frac{4S}{R} \]
where \( S \) is the surface tension of the soap solution and \( R \) is the radius of the bubble.

2. **Setting Up the Ratios**:
Let \( P_1 \) be the excess pressure in the first bubble and \( P_2 \) be the excess pressure in the second bubble. According to the problem:
\[ P_1 = 3P_2 \]

3. **Expressing Excess Pressure**:
From the formula for excess pressure, we can express the pressures in terms of their respective radii:
\[ P_1 = \frac{4S}{R_1} \quad \text{and} \quad P_2 = \frac{4S}{R_2} \]

4. **Substituting into the Pressure Ratio**:
Substituting these expressions into the ratio given in the problem:
\[ \frac{4S}{R_1} = 3 \cdot \frac{4S}{R_2} \]
Simplifying this, we can cancel \( 4S \) from both sides:
\[ \frac{1}{R_1} = \frac{3}{R_2} \]

5. **Finding the Relationship Between Radii**:
Rearranging gives us:
\[ R_2 = 3R_1 \]

6. **Volume of the Bubbles**:
The volume \( V \) of a sphere (bubble) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the volumes of the two bubbles can be expressed as:
\[ V_1 = \frac{4}{3} \pi R_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi R_2^3 \]

7. **Finding the Volume Ratio**:
The ratio of the volumes \( \frac{V_1}{V_2} \) can be calculated as:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi R_1^3}{\frac{4}{3} \pi R_2^3} = \frac{R_1^3}{R_2^3} \]

8. **Substituting for \( R_2 \)**:
Since \( R_2 = 3R_1 \):
\[ \frac{V_1}{V_2} = \frac{R_1^3}{(3R_1)^3} = \frac{R_1^3}{27R_1^3} = \frac{1}{27} \]

9. **Final Ratio**:
Thus, the ratio of the volumes of the first bubble to the second bubble is:
\[ V_1 : V_2 = 1 : 27 \]

### Conclusion:
The ratio between the volume of the first bubble and the second bubble is \( 1 : 27 \).