Two spherical soap bubbles of radii `r_(1) and r_(2)` in vaccum coalesce under isothermal conditions. The resulting bubble has a radius R such that

Options

A. `(r_(1) + r_(2))//2`
B. `R = [r_(1)r_(2)]//r_(1) + r+(2)`
C. `R = sqrt((r_(1)^(2) + r_(2)^(2)))`
D. `R = r_(1) + r_(2)`

Explanation

To solve the problem of two spherical soap bubbles coalescing under isothermal conditions, we will derive the formula for the radius \( R \) of the resulting bubble from the radii \( r_1 \) and \( r_2 \) of the original bubbles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two soap bubbles with radii \( r_1 \) and \( r_2 \).
- When these bubbles coalesce, they form a new bubble with radius \( R \).
- The process occurs under isothermal conditions, meaning the temperature remains constant.

2. **Pressure Inside the Bubbles**:
- The pressure 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.
- Therefore, the pressures inside the two bubbles are:
\[ P_1 = \frac{4S}{r_1} \quad \text{and} \quad P_2 = \frac{4S}{r_2} \]

3. **Volume of the Bubbles**:
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
- Hence, the volumes of the two bubbles are:
\[ V_1 = \frac{4}{3} \pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi r_2^3 \]

4. **Using the Isothermal Condition**:
- Under isothermal conditions, the total pressure-volume product before coalescence equals that after coalescence:
\[ P_1 V_1 + P_2 V_2 = P R^3 \]
- Substituting the expressions for pressure and volume:
\[ \left(\frac{4S}{r_1}\right) \left(\frac{4}{3} \pi r_1^3\right) + \left(\frac{4S}{r_2}\right) \left(\frac{4}{3} \pi r_2^3\right) = \left(\frac{4S}{R}\right) \left(\frac{4}{3} \pi R^3\right) \]

5. **Simplifying the Equation**:
- Cancel out common factors (\( 4S \) and \( \frac{4}{3} \pi \)):
\[ \frac{r_1^2}{R} + \frac{r_2^2}{R} = R^2 \]
- Rearranging gives:
\[ R^2 = \frac{r_1^2 + r_2^2}{R} \]
- Multiplying through by \( R \):
\[ R^3 = r_1^2 + r_2^2 \]

6. **Finding the Radius of the New Bubble**:
- Therefore, the radius \( R \) of the new bubble is:
\[ R = \sqrt{r_1^2 + r_2^2} \]

### Final Result:
The radius \( R \) of the resulting bubble after coalescence is given by:
\[ R = \sqrt{r_1^2 + r_2^2} \]