A spherical air bubble is formed in water at a depth of 1.2 m from the surface. The diameter of the bubble is 0.6 mm and surface tension of water is `0.073 Nm^(-1)`. Calculate the pressure inside . Atmospheric pressure `= 10.3m` of water.

Explanation

To calculate the pressure inside a spherical air bubble formed in water at a depth of 1.2 m, we will follow these steps:

### Step 1: Understand the Given Data
- Depth of the bubble (h) = 1.2 m
- Diameter of the bubble = 0.6 mm
- Surface tension of water (T) = 0.073 N/m
- Atmospheric pressure (Patm) = 10.3 m of water

### Step 2: Convert Units
1. Convert the diameter of the bubble to radius in meters:
\[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{0.6 \text{ mm}}{2} = 0.3 \text{ mm} = 0.3 \times 10^{-3} \text{ m} \]

### Step 3: Calculate Atmospheric Pressure
2. Use the formula for atmospheric pressure:
\[ P_{\text{atm}} = \rho g h_1 \]
where:
- \(\rho\) (density of water) = 1000 kg/m³
- \(g\) (acceleration due to gravity) = 10 m/s²
- \(h_1\) = 10.3 m (height of water column)

Substituting the values:
\[ P_{\text{atm}} = 1000 \times 10 \times 10.3 = 103000 \text{ N/m}^2 = 1.03 \times 10^5 \text{ N/m}^2 \]

### Step 4: Calculate Pressure Due to Water Column
3. Calculate the pressure due to the water column at the depth of the bubble:
\[ P_{\text{water}} = \rho g h = 1000 \times 10 \times 1.2 = 12000 \text{ N/m}^2 = 12 \times 10^3 \text{ N/m}^2 \]

### Step 5: Calculate the Pressure Inside the Bubble
4. The pressure inside the bubble can be calculated using the formula:
\[ P_{\text{inside}} = P_{\text{atm}} + P_{\text{water}} + \frac{2T}{r} \]
where \(T\) is the surface tension and \(r\) is the radius of the bubble.

Substituting the values:
\[ P_{\text{inside}} = 1.03 \times 10^5 + 12 \times 10^3 + \frac{2 \times 0.073}{0.3 \times 10^{-3}} \]

Calculate \(\frac{2T}{r}\):
\[ \frac{2 \times 0.073}{0.3 \times 10^{-3}} = \frac{0.146}{0.0003} = 486.67 \text{ N/m}^2 \]

Now substituting this back:
\[ P_{\text{inside}} = 1.03 \times 10^5 + 12 \times 10^3 + 486.67 \]
\[ P_{\text{inside}} = 103000 + 12000 + 486.67 = 115486.67 \text{ N/m}^2 \]

### Step 6: Final Result
5. The pressure inside the bubble is approximately:
\[ P_{\text{inside}} \approx 1.15487 \times 10^5 \text{ N/m}^2 \]