Pressures inside two soap bubbles are 1.01 and 1.02 atmospheres. Ratio between their volumes is

Options

A. `102:101`
B. `(102)^(3):(101)^(3)`
C. `8:1`
D. `2:1`

Answer

To solve the problem of finding the ratio of the volumes of two soap bubbles given their internal pressures, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Pressures:**
- Let the pressures inside the two soap bubbles be:
- \( P_1 = 1.01 \) atm
- \( P_2 = 1.02 \) atm

2. **Calculate the Pressure Differences:**
- The pressure difference (\( \Delta P \)) for each bubble is given by:
- \( \Delta P_1 = P_1 - P_{outside} = 1.01 - 1 = 0.01 \) atm
- \( \Delta P_2 = P_2 - P_{outside} = 1.02 - 1 = 0.02 \) atm

3. **Relate Pressure Difference to Radius:**
- For soap bubbles, the relationship between the pressure difference and the radius is given by:
\[ \Delta P = \frac{4S}{R} \]
- Here, \( S \) is the surface tension, and \( R \) is the radius of the bubble.

4. **Set Up the Ratios:**
- From the above relationship, we can write:
\[ \Delta P_1 = \frac{4S}{R_1} \quad \text{and} \quad \Delta P_2 = \frac{4S}{R_2} \]
- Dividing the two equations gives:
\[ \frac{\Delta P_1}{\Delta P_2} = \frac{R_2}{R_1} \]

5. **Substituting the Values:**
- Substitute the values of \( \Delta P_1 \) and \( \Delta P_2 \):
\[ \frac{0.01}{0.02} = \frac{R_2}{R_1} \]
- Simplifying this gives:
\[ \frac{1}{2} = \frac{R_2}{R_1} \quad \Rightarrow \quad \frac{R_1}{R_2} = 2 \]

6. **Finding the Volume Ratio:**
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
- Therefore, the ratio of the volumes \( V_1 \) and \( V_2 \) is:
\[ \frac{V_1}{V_2} = \frac{R_1^3}{R_2^3} \]
- Substituting \( \frac{R_1}{R_2} = 2 \):
\[ \frac{V_1}{V_2} = \left(\frac{R_1}{R_2}\right)^3 = 2^3 = 8 \]

7. **Final Ratio:**
- Thus, the ratio of the volumes of the two soap bubbles is:
\[ \frac{V_1}{V_2} = 8:1 \]

### Conclusion:
The ratio of the volumes of the two soap bubbles is \( 8:1 \).