At constant temperature if an air bubble present at the bottom of a lake at 8 atm pressure and with radius 0.1 cm rises to the surface then its new radius will become

Options

A. 0.4 cm
B. 0.3 cm
C. 0.2 cm
D. 0.1 cm

Answer

To solve the problem of finding the new radius of an air bubble as it rises from the bottom of a lake to the surface, we can use Boyle's Law, which states that at constant temperature, the pressure and volume of a gas are inversely proportional.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Initial pressure \( P_1 = 8 \, \text{atm} \)
- Initial radius \( r_1 = 0.1 \, \text{cm} \)

2. **Calculate Initial Volume:**
- The volume \( V_1 \) of a sphere is given by the formula:
\[ V_1 = \frac{4}{3} \pi r_1^3 \]
- Substituting the value of \( r_1 \):
\[ V_1 = \frac{4}{3} \pi (0.1)^3 = \frac{4}{3} \pi (0.001) = \frac{4}{3000} \pi \, \text{cm}^3 \]

3. **Identify Final Conditions:**
- Final pressure \( P_2 = 1 \, \text{atm} \) (at the surface of the lake)

4. **Apply Boyle's Law:**
- According to Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
- Rearranging gives:
\[ V_2 = \frac{P_1}{P_2} V_1 \]
- Substituting the known values:
\[ V_2 = \frac{8 \, \text{atm}}{1 \, \text{atm}} V_1 = 8 V_1 \]

5. **Calculate New Volume:**
- Since \( V_1 = \frac{4}{3} \pi (0.1)^3 \):
\[ V_2 = 8 \times \frac{4}{3} \pi (0.1)^3 = \frac{32}{3} \pi (0.001) = \frac{32}{3000} \pi \, \text{cm}^3 \]

6. **Relate Volume to Radius:**
- The volume of the bubble at the new radius \( r_2 \) is:
\[ V_2 = \frac{4}{3} \pi r_2^3 \]
- Setting the two volume equations equal gives:
\[ \frac{4}{3} \pi r_2^3 = \frac{32}{3000} \pi \]

7. **Cancel \( \pi \) and Solve for \( r_2^3 \):**
- Cancel \( \frac{4}{3} \) and \( \pi \):
\[ r_2^3 = \frac{32}{3000} \times \frac{3}{4} = \frac{32 \times 3}{12000} = \frac{96}{12000} = \frac{8}{1000} = 0.008 \, \text{cm}^3 \]

8. **Calculate \( r_2 \):**
- Taking the cube root:
\[ r_2 = \sqrt[3]{0.008} = 0.2 \, \text{cm} \]

### Final Answer:
The new radius of the bubble when it reaches the surface is \( r_2 = 0.2 \, \text{cm} \).