A vessel filled with air under pressure `p_(0)` contains a soap bubble of diameter `d`. The air pressure have been reduced `n`-fold, and the bubbled diameter increased `r`-fold isothermally. Find the surface tension of the soap water solution.

Explanation

To find the surface tension of the soap water solution given the conditions of the soap bubble in a vessel, we can follow these steps:

### Step 1: Understand the initial conditions
Initially, the pressure inside the bubble (P_initial) can be expressed as:
\[ P_{\text{initial}} = P_0 + \Delta P \]
where \(\Delta P\) is the excess pressure given by:
\[ \Delta P = \frac{8\sigma}{d} \]
Thus,
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