Prove that if two bubbles of radii `r_(1)` and `r_(2)` coalesce isothermally in vacuum then the radius of new bubble will be `r=sqrt(r_(1)^(2)+r_(2)^(2))`

Explanation

To prove that if two bubbles of radii \( r_1 \) and \( r_2 \) coalesce isothermally in vacuum, then the radius of the new bubble will be \( r = \sqrt{r_1^2 + r_2^2} \), we can follow these steps:

### Step 1: Understand the System
We have two bubbles with radii \( r_1 \) and \( r_2 \) that are in vacuum and coalesce isothermally. In an isothermal process, the temperature remains constant.

### Step 2: Apply the Ideal Gas Law
For each bubble, we can apply the ideal gas law, which states that:
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