The diameter of a gas bubble formed at the bottom of a pond is `d = 4 cm`. When the bubble rises to the surface, its diameter tension of water `= T = 0.07 Nm^(-1)`

Explanation

To solve the problem of the gas bubble rising in a pond, we will follow these steps:

### Step 1: Understand the Given Information
- Diameter of the gas bubble at the bottom of the pond, \( d = 4 \, \text{cm} = 0.04 \, \text{m} \)
- Surface tension of water, \( T = 0.07 \, \text{N/m} \)

### Step 2: Calculate the Volume of the Bubble
The volume \( V \) of a sphere (which we can approximate the bubble to be) is given by the formula:
\[ V = \frac{\pi d^3}{6} \]
Substituting the diameter:
\[ V = \frac{\pi (0.04)^3}{6} = \frac{\pi (0.000064)}{6} \approx 3.33 \times 10^{-5} \, \text{m}^3 \]

### Step 3: Calculate the Surface Area of the Bubble
The surface area \( A \) of a sphere is given by:
\[ A = \pi d^2 \]
Substituting the diameter:
\[ A = \pi (0.04)^2 = \pi (0.0016) \approx 0.0050265 \, \text{m}^2 \]

### Step 4: Calculate the Pressure Difference Due to Surface Tension
The pressure difference \( \Delta P \) across the surface of the bubble due to surface tension is given by:
\[ \Delta P = \frac{4T}{d} \]
Substituting the values:
\[ \Delta P = \frac{4 \times 0.07}{0.04} = \frac{0.28}{0.04} = 7 \, \text{N/m}^2 \]

### Step 5: Calculate the Hydrostatic Pressure at Depth
The hydrostatic pressure \( P \) at a depth \( h \) in a fluid is given by:
\[ P = \rho g h \]
Where:
- \( \rho \) is the density of water \( \approx 1000 \, \text{kg/m}^3 \)
- \( g \) is the acceleration due to gravity \( \approx 9.81 \, \text{m/s}^2 \)

### Step 6: Set Up the Equation for Pressure at the Bottom of the Pond
At the bottom of the pond, the total pressure \( P_b \) is the atmospheric pressure \( P_0 \) plus the hydrostatic pressure:
\[ P_b = P_0 + \rho g h \]
Where \( P_0 \) is the atmospheric pressure \( \approx 10^5 \, \text{Pa} \).

### Step 7: Solve for Height \( h \)
Using the pressure difference calculated earlier:
\[ P_b = P_0 + \Delta P \]
Substituting the values:
\[ P_0 + \rho g h = P_0 + 7 \]
This simplifies to:
\[ \rho g h = 7 \]
Now substituting the values for \( \rho \) and \( g \):
\[ 1000 \times 9.81 \times h = 7 \]
Solving for \( h \):
\[ h = \frac{7}{1000 \times 9.81} \approx 0.000712 \, \text{m} \approx 0.712 \, \text{cm} \]

### Step 8: Conclusion
The height \( h \) at which the bubble rises to the surface is approximately \( 0.712 \, \text{cm} \).