Two spherical soap bubbles of radii `r_1` and `r_2` in vacuume collapse under isothermal condition. The resulting bubble has radius `R` such that

Options

A. `R=r_1+r_2`
B. `R=(r_1+r_2)^((1)/(2))`
C. `R=sqrt(r_1^2+r_2^2)`
D. `R=[(r_2r_2(r_1+r_2)]`

Explanation

To solve the problem of two spherical soap bubbles collapsing under isothermal conditions to form a new bubble, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
We have two soap bubbles with radii \( r_1 \) and \( r_2 \) in a vacuum. The pressure inside each bubble is influenced by the surface tension of the soap film.

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 and \( r \) is the radius of the bubble. Therefore, the pressures for 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:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the volumes of the two bubbles are:
\[ V_1 = \frac{4}{3} \pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi r_2^3 \]

4. **Applying the Isothermal Condition**:
Under isothermal conditions, the product of pressure and volume for the two initial bubbles equals the product of pressure and volume for the final bubble:
\[ P_1 V_1 + P_2 V_2 = P_f V_f \]
where \( P_f \) is the pressure of the final bubble and \( V_f \) is its volume.

5. **Volume of the Final Bubble**:
The volume of the resulting bubble with radius \( R \) is:
\[ V_f = \frac{4}{3} \pi R^3 \]

6. **Substituting the Pressures and Volumes**:
Substitute the expressions for pressures and volumes into the equation:
\[ \left(\frac{4S}{r_1}\right) \left(\frac{4}{3} \pi r_1^3\right) + \left(\frac{4S}{r_2}\right) \left(\frac{4}{3} \pi r_2^3\right) = \left(\frac{4S}{R}\right) \left(\frac{4}{3} \pi R^3\right) \]

7. **Simplifying the Equation**:
Cancel out common terms (like \( \frac{4}{3} \pi \) and \( 4S \)):
\[ \frac{r_1^3}{r_1} + \frac{r_2^3}{r_2} = \frac{R^3}{R} \]
This simplifies to:
\[ r_1^2 + r_2^2 = R^2 \]

8. **Final Expression for R**:
Taking the square root of both sides gives:
\[ R = \sqrt{r_1^2 + r_2^2} \]

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