Two spherical soap bubble collapses. If `V` is the consequent change in volume of the contained air and `S` in the change in the total surface area and `T` is the surface tension of the soap solution, then it relation between `P_(0), V, S` and `T` are `lambdaP_(0)V + 4ST = 0`, then find `lambda` ? (if `P_(0)` is atmospheric pressure) : Assume temperature of the air remain same in all the bubbles

Explanation

To solve the problem, we need to analyze the relationship between the pressures, volumes, and surface areas of the soap bubbles before and after they collapse.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two spherical soap bubbles that collapse. The change in volume of the contained air is denoted by \( V \), and the change in the total surface area is denoted by \( S \). The surface tension of the soap solution is \( T \). We need to find the value of \( \lambda \) in the equation \( \lambda P_0 V + 4ST = 0 \).

2. **Pressure Inside the Bubbles**:
For a soap bubble, the pressure inside the bubble \( P_i \) is given by the formula:
\[ P_i = P_0 + \frac{4T}{R} \]
where \( P_0 \) is the atmospheric pressure and \( R \) is the radius of the bubble.

3. **Volume of the Bubbles**:
The volume \( V \) of a spherical bubble is given by:
\[ V = \frac{4}{3} \pi R^3 \]

4. **Change in Volume**:
Let \( V_1 \) and \( V_2 \) be the volumes of the two bubbles before they collapse, and \( V_3 \) be the volume after they collapse. The change in volume can be expressed as:
\[ V = V_1 + V_2 - V_3 \]

5. **Change in Surface Area**:
The surface area \( S \) of a spherical bubble is given by:
\[ S = 4 \pi R^2 \]
The change in surface area when the bubbles collapse can be expressed similarly.

6. **Applying the Equation**:
According to the problem, we can write the relationship between the pressures and volumes:
\[ P_1 V_1 + P_2 V_2 = P_3 V_3 \]
Substituting the expressions for \( P_i \):
\[ \left(P_0 + \frac{4T}{R_1}\right)V_1 + \left(P_0 + \frac{4T}{R_2}\right)V_2 = \left(P_0 + \frac{4T}{R_3}\right)V_3 \]

7. **Simplifying the Equation**:
Rearranging gives:
\[ P_0(V_1 + V_2 - V_3) + 4T\left(\frac{V_1}{R_1} + \frac{V_2}{R_2} - \frac{V_3}{R_3}\right) = 0 \]
Let \( V = V_1 + V_2 - V_3 \) and \( S = \frac{V_1}{R_1} + \frac{V_2}{R_2} - \frac{V_3}{R_3} \).

8. **Final Relation**:
We can express this as:
\[ P_0 V + 4T S = 0 \]
From this, we can identify \( \lambda \) as:
\[ \lambda = -\frac{4T S}{V_0} \]

### Conclusion:
Thus, the value of \( \lambda \) is derived from the relationship established in the problem.