Two identical soap bubbles each of radius r and of the same surface tension `T` combine to form a new soap bubble of radius `R`. The two bubbles contain air at the same temperature. If the atmospheric pressure is `p_(0)` then find the surface tension T of the soap solution in terms of `p_(0)`, r and `R`. Assume process is isothermal.

Explanation

To solve the problem of finding the surface tension \( T \) of the soap solution in terms of \( p_0 \), \( r \), and \( R \), we will follow these steps:

### Step 1: Understand the Pressure Inside the Soap Bubbles
For a soap bubble of radius \( r \), the pressure inside the bubble \( P_1 \) can be expressed as:
\[ P_1 = p_0 + \frac{4T}{r} \]
where \( p_0 \) is the atmospheric pressure and \( T \) is the surface tension.

### Step 2: Pressure Inside the New Bubble
When the two identical soap bubbles combine to form a larger bubble of radius \( R \), the pressure inside the new bubble \( P_2 \) is given by:
\[ P_2 = p_0 + \frac{4T}{R} \]

### Step 3: Equate the Pressures
Since the two original bubbles are identical and combine to form the larger bubble, we can set \( P_1 = P_2 \):
\[ p_0 + \frac{4T}{r} = p_0 + \frac{4T}{R} \]

### Step 4: Simplify the Equation
By eliminating \( p_0 \) from both sides, we get:
\[ \frac{4T}{r} = \frac{4T}{R} \]
This equation indicates that the pressures inside the bubbles are equal when they combine.

### Step 5: Volume Conservation
The volume of the two smaller bubbles combined should equal the volume of the larger bubble. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, for two smaller bubbles:
\[ V_1 = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
And for the larger bubble:
\[ V_2 = \frac{4}{3} \pi R^3 \]
Setting these equal gives:
\[ \frac{8}{3} \pi r^3 = \frac{4}{3} \pi R^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides results in:
\[ 2r^3 = R^3 \]

### Step 6: Relate the Surface Tension to the Pressures
From the pressure equations we derived, we can express \( T \) in terms of \( p_0 \), \( r \), and \( R \). Rearranging the earlier equation gives:
\[ \frac{4T}{r} - \frac{4T}{R} = 0 \]
This leads to:
\[ 4T \left( \frac{1}{r} - \frac{1}{R} \right) = 0 \]
However, we need to consider the conservation of energy and the fact that the total pressure must balance out.

### Step 7: Final Expression for Surface Tension
Using the volume relationship and substituting back into the pressure equations, we can derive:
\[ 2 \left( p_0 + \frac{4T}{r} \right) \cdot \frac{8}{3} \pi r^3 = \left( p_0 + \frac{4T}{R} \right) \cdot \frac{4}{3} \pi R^3 \]
This simplifies to:
\[ 2p_0 \cdot \frac{8}{3} r^3 + \frac{8T}{r} = p_0 \cdot \frac{4}{3} R^3 + \frac{4T}{R} \]

Rearranging gives us:
\[ 8T \cdot r^2 - 4T \cdot R = p_0 \cdot \left( R^3 - 2r^3 \right) \]

Finally, solving for \( T \):
\[ T = \frac{p_0 \cdot (R^3 - 2r^3)}{8r^2 - 4R} \]

### Conclusion
The surface tension \( T \) of the soap solution in terms of \( p_0 \), \( r \), and \( R \) is:
\[ T = \frac{p_0 (R^3 - 2r^3)}{8r^2 - 4R} \]