A cylinder with a movable piston contains air under a pressure `p_(1)` and a soap bubble of radius 'r' . The pressure `p_(2)` to which the air should be compressed by slowly pushing the piston into the cylinder for the soap bubble to reduce its size by half will be (The surface tension is `sigma` and the temperature T is maintained constant).

Options

A. `[8p_(1)+(24 sigma)/(r )]`
B. `[4p_(1)+(24 sigma)/(r )]`
C. `[2p_(1)+(24 sigma)/(r )]`
D. `[2p_(1)+(24 sigma)/(r )]`

Explanation

To solve the problem, we need to analyze the situation step by step, applying the principles of fluid mechanics and the properties of soap bubbles.

### Step 1: Understand the Initial Conditions
Initially, the soap bubble has a radius \( r \) and is in equilibrium with the pressure \( p_1 \) in the cylinder. The pressure inside the soap bubble can be expressed as:
\[ P_{\text{inside}} = p_1 + \frac{4\sigma}{r} \]
where \( \sigma \) is the surface tension of the soap bubble.

**Hint:** Remember that the pressure inside a soap bubble is affected by the surface tension and the radius of the bubble.

### Step 2: Determine the Final Conditions
When the piston is pushed slowly, the radius of the soap bubble reduces to \( \frac{r}{2} \). The new pressure inside the soap bubble becomes:
\[ P'_{\text{inside}} = p_2 + \frac{4\sigma}{\frac{r}{2}} = p_2 + \frac{8\sigma}{r} \]

**Hint:** The pressure inside the bubble changes with the change in radius, so make sure to adjust the formula accordingly.

### Step 3: Apply Boyle's Law
Since the process is isothermal (temperature \( T \) is constant), we can apply Boyle's Law, which states that:
\[ p_1 V_1 = p_2 V_2 \]
The volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \). The initial volume \( V_1 \) is:
\[ V_1 = \frac{4}{3} \pi r^3 \]
The final volume \( V_2 \) when the radius is \( \frac{r}{2} \) is:
\[ V_2 = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3 = \frac{4}{3} \pi \frac{r^3}{8} = \frac{1}{6} \left(\frac{4}{3} \pi r^3\right) = \frac{V_1}{6} \]

**Hint:** When the radius changes, the volume changes significantly due to the cubic relationship.

### Step 4: Set Up the Equation Using Boyle's Law
Substituting the volumes into Boyle's Law gives:
\[ p_1 \left(\frac{4}{3} \pi r^3\right) = p_2 \left(\frac{1}{6} \left(\frac{4}{3} \pi r^3\right)\right) \]
This simplifies to:
\[ p_1 = \frac{p_2}{6} \]
Thus, we can express \( p_2 \) in terms of \( p_1 \):
\[ p_2 = 6p_1 \]

**Hint:** Make sure to isolate \( p_2 \) to find its relationship with \( p_1 \).

### Step 5: Relate the Pressures Inside the Bubble
Now we can equate the two expressions for the pressure inside the bubble:
\[ p_1 + \frac{4\sigma}{r} + \frac{8\sigma}{r} = p_2 \]
Substituting \( p_2 = 6p_1 \) into the equation gives:
\[ p_1 + \frac{12\sigma}{r} = 6p_1 \]
Rearranging this leads to:
\[ 5p_1 = \frac{12\sigma}{r} \]
Thus, we can express \( p_1 \):
\[ p_1 = \frac{12\sigma}{5r} \]

### Step 6: Find \( p_2 \)
Finally, substituting \( p_1 \) back into the equation for \( p_2 \):
\[ p_2 = 6p_1 = 6 \times \frac{12\sigma}{5r} = \frac{72\sigma}{5r} \]

**Hint:** Always check your final expression to ensure it is in terms of the quantities given in the problem.

### Conclusion
The pressure \( p_2 \) to which the air should be compressed for the soap bubble to reduce its size by half is:
\[ p_2 = \frac{72\sigma}{5r} \]