The pressure inside two soap bubbles is 1.01 and 1.02 atmosphere. The ration of their respective 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 pressure inside the first soap bubble, \( P_1 = 1.01 \) atm.
- Let the pressure inside the second soap bubble, \( P_2 = 1.02 \) atm.

2. **Calculate the Excess Pressures**:
- The excess pressure inside a soap bubble is defined as the difference between the internal pressure and the atmospheric pressure (1 atm).
- For the first bubble:
\[ \Delta P_1 = P_1 - P_{\text{atm}} = 1.01 - 1 = 0.01 \text{ atm} \]
- For the second bubble:
\[ \Delta P_2 = P_2 - P_{\text{atm}} = 1.02 - 1 = 0.02 \text{ atm} \]

3. **Relate Excess Pressure to Radius**:
- The formula for excess pressure in a soap bubble is given by:
\[ \Delta P = \frac{4\sigma}{r} \]
- From this, we can see that the excess pressure is inversely proportional to the radius of the bubble:
\[ \Delta P_1 \propto \frac{1}{r_1} \quad \text{and} \quad \Delta P_2 \propto \frac{1}{r_2} \]
- Therefore, we can write:
\[ \frac{\Delta P_1}{\Delta P_2} = \frac{r_2}{r_1} \]

4. **Substitute the Values of Excess Pressures**:
- Substitute the values of \( \Delta P_1 \) and \( \Delta P_2 \):
\[ \frac{0.01}{0.02} = \frac{r_2}{r_1} \]
- Simplifying this gives:
\[ \frac{1}{2} = \frac{r_2}{r_1} \implies r_2 = \frac{1}{2} r_1 \]

5. **Calculate the Volume Ratio**:
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Thus, the volumes of the two bubbles are:
\[ V_1 = \frac{4}{3} \pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi r_2^3 \]
- The ratio of the volumes is:
\[ \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]
- Since \( r_2 = \frac{1}{2} r_1 \):
\[ \frac{V_1}{V_2} = \frac{r_1^3}{\left(\frac{1}{2} r_1\right)^3} = \frac{r_1^3}{\frac{1}{8} r_1^3} = 8 \]

6. **Final Ratio**:
- Therefore, the ratio of the volumes \( V_1 : V_2 = 8 : 1 \).

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