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

Options

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

Explanation

To find the ratio of the volumes of two soap bubbles with given internal pressures, we can follow these steps:

### Step 1: Understand the relationship between pressure and radius
The excess pressure inside a soap bubble is given by the formula:
\[ \Delta P = \frac{4S}{r} \]
where \( S \) is the surface tension and \( r \) is the radius of the bubble.

### Step 2: Calculate the excess pressures
Given the internal pressures of the two soap bubbles:
- For Bubble 1: \( P_1 = 1.01 \) atm
- For Bubble 2: \( P_2 = 1.02 \) atm

The external pressure (atmospheric pressure) is approximately \( 1 \) atm. Therefore, the excess pressures can be calculated as follows:
- Excess pressure in Bubble 1:
\[ \Delta P_1 = P_1 - P_{outside} = 1.01 - 1 = 0.01 \text{ atm} \]
- Excess pressure in Bubble 2:
\[ \Delta P_2 = P_2 - P_{outside} = 1.02 - 1 = 0.02 \text{ atm} \]

### Step 3: Relate the pressures to the radii
Using the relationship from Step 1, we can set up the ratios of the pressures:
\[ \frac{\Delta P_1}{\Delta P_2} = \frac{4S/r_1}{4S/r_2} = \frac{r_2}{r_1} \]
Substituting the values of excess pressures:
\[ \frac{0.01}{0.02} = \frac{r_2}{r_1} \]
This simplifies to:
\[ \frac{1}{2} = \frac{r_2}{r_1} \]
Thus, we find:
\[ r_1 = 2r_2 \]

### Step 4: Calculate the volume ratio
The volume \( V \) of a sphere (bubble) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the volume ratio \( \frac{V_1}{V_2} \) can be expressed as:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3} \]
Substituting \( r_1 = 2r_2 \):
\[ \frac{V_1}{V_2} = \frac{(2r_2)^3}{r_2^3} = \frac{8r_2^3}{r_2^3} = 8 \]

### Final Answer
Thus, the ratio of the volumes of the two soap bubbles is:
\[ V_1 : V_2 = 8 : 1 \]