There is an air bubble of radius `1.0 mm ` in a liquid of surface tension `0.075"Nm"^(-1)` and density `10^(3) "kgm"^(-3)`. The bubble is at a depth of `10.0 cm` below the free surface of a liquid. By what amount is the pressure inside the bubble greater than the atmospheric pressure?

Answer

To find the amount by which the pressure inside the bubble is greater than the atmospheric pressure, we will follow these steps:

### Step 1: Identify the given values
- Radius of the bubble, \( R = 1.0 \, \text{mm} = 1.0 \times 10^{-3} \, \text{m} \)
- Surface tension of the liquid, \( T = 0.075 \, \text{N/m} \)
- Density of the liquid, \( \rho = 10^3 \, \text{kg/m}^3 \)
- Depth of the bubble, \( h = 10.0 \, \text{cm} = 0.1 \, \text{m} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### Step 2: Calculate the excess pressure due to surface tension
The excess pressure inside the bubble due to surface tension is given by the formula:
\[ \Delta P_{\text{surface}} = \frac{2T}{R} \]
Substituting the values:
\[ \Delta P_{\text{surface}} = \frac{2 \times 0.075 \, \text{N/m}}{1.0 \times 10^{-3} \, \text{m}} = \frac{0.15 \, \text{N/m}}{1.0 \times 10^{-3} \, \text{m}} = 150 \, \text{N/m}^2 \]

### Step 3: Calculate the hydrostatic pressure at the depth of the bubble
The hydrostatic pressure at a depth \( h \) is given by:
\[ P_{\text{hydrostatic}} = \rho g h \]
Substituting the values:
\[ P_{\text{hydrostatic}} = (10^3 \, \text{kg/m}^3) \times (9.8 \, \text{m/s}^2) \times (0.1 \, \text{m}) = 980 \, \text{N/m}^2 \]

### Step 4: Calculate the total pressure inside the bubble
The total pressure inside the bubble \( P_{\text{bubble}} \) is the sum of the atmospheric pressure \( P_0 \), the excess pressure due to surface tension, and the hydrostatic pressure:
\[ P_{\text{bubble}} - P_0 = \Delta P_{\text{surface}} + P_{\text{hydrostatic}} \]
Thus,
\[ P_{\text{bubble}} - P_0 = 150 \, \text{N/m}^2 + 980 \, \text{N/m}^2 = 1130 \, \text{N/m}^2 \]

### Step 5: Conclusion
The pressure inside the bubble is greater than the atmospheric pressure by:
\[ P_{\text{bubble}} - P_0 = 1130 \, \text{N/m}^2 \]

### Final Answer
The pressure inside the bubble is greater than the atmospheric pressure by **1130 N/m²**.

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