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 that rises from the bottom of a lake to the surface, we can use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional when temperature is held constant.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Initial pressure (P1) = 8 atm (at the bottom of the lake)
- Initial radius (r1) = 0.1 cm

2. **Calculate Initial Volume (V1)**:
- The volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- Therefore, the initial volume (V1) can be calculated as:
\[ V1 = \frac{4}{3} \pi (0.1)^3 = \frac{4}{3} \pi (0.001) = \frac{4\pi}{3000} \text{ cm}^3 \]

3. **Identify Final Conditions**:
- Final pressure (P2) = 1 atm (at the surface of the lake)

4. **Apply Boyle's Law**:
- According to Boyle's Law:
\[ P1 \cdot V1 = P2 \cdot V2 \]
- Rearranging gives:
\[ V2 = \frac{P1 \cdot V1}{P2} \]

5. **Substitute Values**:
- Substitute P1, V1, and P2 into the equation:
\[ V2 = \frac{8 \cdot V1}{1} = 8 \cdot V1 \]

6. **Calculate New Volume (V2)**:
- Since we know that V1 = \(\frac{4}{3} \pi (0.1)^3\):
\[ V2 = 8 \cdot \left(\frac{4}{3} \pi (0.1)^3\right) = \frac{32}{3} \pi (0.1)^3 \]

7. **Relate Volume to Radius**:
- The volume of the new bubble (V2) can also be expressed in terms of the new radius (r2):
\[ V2 = \frac{4}{3} \pi r2^3 \]

8. **Set the Two Volume Equations Equal**:
- Set the two expressions for V2 equal:
\[ \frac{32}{3} \pi (0.1)^3 = \frac{4}{3} \pi r2^3 \]

9. **Cancel Common Terms**:
- Cancel \(\frac{4}{3} \pi\) from both sides:
\[ 32 \cdot (0.1)^3 = r2^3 \]

10. **Calculate r2**:
- Calculate \(r2^3\):
\[ r2^3 = 32 \cdot 0.001 = 0.032 \]
- Now take the cube root to find r2:
\[ r2 = \sqrt[3]{0.032} \approx 0.2 \text{ cm} \]

### Final Answer:
The new radius of the air bubble when it rises to the surface is approximately **0.2 cm**.