To solve the problem step by step, we will use the concepts of pressure in a soap bubble and Boyle's law.
### Step 1: Understand the Initial Conditions
We have a soap bubble with an initial radius \( r_1 = 2.4 \times 10^{-4} \, \text{m} \) and initial pressure \( P_1 = 10^5 \, \text{N/m}^2 \). The surface tension of the soap film is given as \( \sigma = 0.08 \, \text{N/m} \).
### Step 2: Calculate the Initial Pressure Inside the Bubble
The pressure inside the soap bubble can be calculated using the formula:
\[
P_{inside} = P_{outside} + \frac{4\sigma}{r}
\]
Here, \( P_{outside} \) is the atmospheric pressure (initial pressure in the air cylinder), and \( r \) is the radius of the bubble.
Substituting the values:
\[
P_{inside} = 10^5 + \frac{4 \times 0.08}{2.4 \times 10^{-4}}
\]
Calculating the term:
\[
\frac{4 \times 0.08}{2.4 \times 10^{-4}} = \frac{0.32}{2.4 \times 10^{-4}} \approx 1333.33 \, \text{N/m}^2
\]
Thus,
\[
P_{inside} = 10^5 + 1333.33 \approx 101333.33 \, \text{N/m}^2
\]
### Step 3: Determine the Final Radius
The radius of the bubble is halved, so the final radius \( r_2 \) is:
\[
r_2 = \frac{r_1}{2} = \frac{2.4 \times 10^{-4}}{2} = 1.2 \times 10^{-4} \, \text{m}
\]
### Step 4: Calculate the Final Pressure Inside the Bubble
Using the same formula for the final pressure inside the bubble:
\[
P_{inside, final} = P_{outside, final} + \frac{4\sigma}{r_2}
\]
Let \( P_{outside, final} \) be the final pressure in the air cylinder, which we need to find.
Substituting the values:
\[
P_{inside, final} = P_{outside, final} + \frac{4 \times 0.08}{1.2 \times 10^{-4}}
\]
Calculating the term:
\[
\frac{4 \times 0.08}{1.2 \times 10^{-4}} = \frac{0.32}{1.2 \times 10^{-4}} \approx 2666.67 \, \text{N/m}^2
\]
Thus,
\[
P_{inside, final} = P_{outside, final} + 2666.67
\]
### Step 5: Apply Boyle's Law
Since the process is isothermal, we can apply Boyle's law:
\[
P_1 V_1 = P_2 V_2
\]
Where \( V_1 \) is the initial volume and \( V_2 \) is the final volume. The volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \).
The initial volume \( V_1 \) is:
\[
V_1 = \frac{4}{3} \pi (2.4 \times 10^{-4})^3
\]
The final volume \( V_2 \) is:
\[
V_2 = \frac{4}{3} \pi (1.2 \times 10^{-4})^3 = \frac{1}{8} V_1
\]
Substituting into Boyle's law:
\[
P_1 V_1 = P_2 \left(\frac{1}{8} V_1\right)
\]
This simplifies to:
\[
P_1 = \frac{P_2}{8}
\]
Thus,
\[
P_2 = 8 P_1
\]
### Step 6: Substitute the Values
Using \( P_1 = P_{inside} \):
\[
P_{outside, final} + 2666.67 = 8 \times 101333.33
\]
Calculating \( 8 \times 101333.33 \):
\[
8 \times 101333.33 = 810666.67 \, \text{N/m}^2
\]
So,
\[
P_{outside, final} + 2666.67 = 810666.67
\]
Thus,
\[
P_{outside, final} = 810666.67 - 2666.67 = 808000 \, \text{N/m}^2
\]
### Step 7: Express the Final Pressure
The final pressure can be expressed as:
\[
P_{outside, final} = 8.08 \times 10^5 \, \text{N/m}^2
\]
Thus, comparing with the given form \( 8.08 \times 10^n \), we find \( n = 5 \).
### Final Answer
The value of \( n \) is:
\[
\boxed{5}
\]
