Under isothermal condition two soap bubbles of radii `r_(1)` and `r_(2)` coalesce to form a single bubble of radius r. The external pressure is `p_(0)`. Find the surface tension of the soap in terms of the given parameters.

Explanation

To find the surface tension of the soap in terms of the given parameters when two soap bubbles of radii \( r_1 \) and \( r_2 \) coalesce to form a single bubble of radius \( r \) under isothermal conditions, we can follow these steps:

### Step 1: Understand the excess pressure in soap bubbles
The excess pressure inside a soap bubble is given by the formula:
\[ P_i = P_0 + \frac{4T}{R} \]
where \( P_i \) is the internal pressure of the bubble, \( P_0 \) is the external pressure, \( T \) is the surface tension, and \( R \) is the radius of the bubble.

### Step 2: Write the equations for the two initial bubbles
For the two bubbles with radii \( r_1 \) and \( r_2 \), the internal pressures can be expressed as:
\[ P_1 = P_0 + \frac{4T}{r_1} \]
\[ P_2 = P_0 + \frac{4T}{r_2} \]

### Step 3: Write the equation for the final bubble
After coalescing, the new bubble has a radius \( r \) and its internal pressure is:
\[ P = P_0 + \frac{4T}{r} \]

### Step 4: Apply the principle of conservation of volume
The total volume of the two smaller bubbles must equal the volume of the larger bubble:
\[ \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 = \frac{4}{3} \pi r^3 \]
This simplifies to:
\[ r_1^3 + r_2^3 = r^3 \]

### Step 5: Set up the equation for pressures
Using the pressures from the bubbles, we can write:
\[ P_1 V_1 + P_2 V_2 = P V \]
Substituting the volumes and pressures:
\[ \left( P_0 + \frac{4T}{r_1} \right) \frac{4}{3} \pi r_1^3 + \left( P_0 + \frac{4T}{r_2} \right) \frac{4}{3} \pi r_2^3 = \left( P_0 + \frac{4T}{r} \right) \frac{4}{3} \pi r^3 \]

### Step 6: Simplify the equation
Eliminating \( \frac{4}{3} \pi \) from both sides:
\[ \left( P_0 + \frac{4T}{r_1} \right) r_1^3 + \left( P_0 + \frac{4T}{r_2} \right) r_2^3 = \left( P_0 + \frac{4T}{r} \right) r^3 \]

### Step 7: Expand and rearrange the equation
Expanding the left-hand side:
\[ P_0 r_1^3 + \frac{4T}{r_1} r_1^3 + P_0 r_2^3 + \frac{4T}{r_2} r_2^3 = P_0 r^3 + \frac{4T}{r} r^3 \]
Combining like terms gives:
\[ P_0 (r_1^3 + r_2^3) + 4T \left( \frac{r_1^3}{r_1} + \frac{r_2^3}{r_2} \right) = P_0 r^3 + 4T \frac{r^3}{r} \]

### Step 8: Solve for T
Rearranging to isolate \( T \):
\[ 4T \left( r_1^2 + r_2^2 \right) = P_0 (r^3 - r_1^3 - r_2^3) \]
Thus,
\[ T = \frac{P_0 (r^3 - r_1^3 - r_2^3)}{4 (r_1^2 + r_2^2)} \]

### Final Result
The surface tension \( T \) of the soap in terms of the given parameters is:
\[ T = \frac{P_0 (r^3 - r_1^3 - r_2^3)}{4 (r_1^2 + r_2^2)} \]