There is an air bubble of radius 1.0mm in a liquid of surface tension 0.075Nm−1 and density 103kgm−3. The bubble is at a depth of 10.0cm below the free surface of a liquid. By what amount is the pressure inside the bubble greater than the atmospheric pressure?

Explanation

To solve the problem of determining how much greater the pressure inside the air bubble is compared to the atmospheric pressure, we will follow these steps:

### Step 1: Understand the Problem
We have an air bubble submerged in a liquid at a certain depth. We need to find the pressure difference between the inside of the bubble and the atmospheric pressure.

### Step 2: Identify 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, \( \rho = 103 \, \text{kg/m}^3 \)
- Depth of the bubble, \( h = 10.0 \, \text{cm} = 0.1 \, \text{m} \)

### Step 3: Calculate the Excess Pressure
The excess pressure inside the bubble due to surface tension is given by the formula:
\[ \Delta P_{\text{excess}} = \frac{2T}{R} \]
Substituting the values:
\[ \Delta P_{\text{excess}} = \frac{2 \times 0.075 \, \text{N/m}}{1.0 \times 10^{-3} \, \text{m}} = \frac{0.15}{0.001} = 150 \, \text{N/m}^2 \]

### Step 4: Calculate the Hydrostatic Pressure
The hydrostatic pressure at the depth of the bubble is given by:
\[ P_{\text{hydrostatic}} = \rho g h \]
Where \( g \) (acceleration due to gravity) is approximately \( 9.8 \, \text{m/s}^2 \). Substituting the values:
\[ P_{\text{hydrostatic}} = 103 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 \times 0.1 \, \text{m} = 100.94 \, \text{N/m}^2 \approx 101 \, \text{N/m}^2 \]

### Step 5: Calculate the Total Pressure Inside the Bubble
The total pressure inside the bubble is the sum of the excess pressure and the hydrostatic pressure:
\[ P_{\text{bubble}} = \Delta P_{\text{excess}} + P_{\text{hydrostatic}} = 150 \, \text{N/m}^2 + 100.94 \, \text{N/m}^2 = 250.94 \, \text{N/m}^2 \]

### Step 6: Calculate the Pressure Greater than Atmospheric Pressure
The pressure inside the bubble greater than atmospheric pressure is:
\[ P_{\text{inside}} - P_{\text{atmospheric}} = P_{\text{bubble}} = 250.94 \, \text{N/m}^2 \]

### Final Answer
The pressure inside the bubble is approximately **250.94 N/m²** greater than the atmospheric pressure.

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