Calculate the total inside a spherical air bubble of radius 0.1 mm at a depth of 10 cm below the surface of a liquid of density 1.1 g/c.c and surface tension 50 dynes/cm. (Height of Hg barometer = 76 cm).

Explanation

To calculate the total pressure inside a spherical air bubble at a depth of 10 cm below the surface of a liquid with a density of 1.1 g/cm³ and surface tension of 50 dynes/cm, we will follow these steps:

### Step 1: Calculate the pressure due to the liquid column above the bubble
The pressure due to the liquid column can be calculated using the formula:
\[ P_{\text{liquid}} = \rho g h \]
Where:
- \( \rho \) = density of the liquid = 1.1 g/cm³ = 1100 kg/m³ (converting to SI units)
- \( g \) = acceleration due to gravity = 9.81 m/s²
- \( h \) = depth = 10 cm = 0.1 m (converting to SI units)

Substituting the values:
\[ P_{\text{liquid}} = 1100 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.1 \, \text{m} \]
\[ P_{\text{liquid}} = 1078.1 \, \text{Pa} \]

### Step 2: Calculate the atmospheric pressure
The atmospheric pressure can be given as:
\[ P_0 = 1.01 \times 10^5 \, \text{Pa} \]

### Step 3: Calculate the excess pressure inside the bubble
The excess pressure inside a spherical bubble can be calculated using the formula:
\[ P_{\text{excess}} = \frac{2s}{r} \]
Where:
- \( s \) = surface tension = 50 dynes/cm = 0.05 N/m (converting to SI units)
- \( r \) = radius of the bubble = 0.1 mm = 0.0001 m (converting to SI units)

Substituting the values:
\[ P_{\text{excess}} = \frac{2 \times 0.05 \, \text{N/m}}{0.0001 \, \text{m}} \]
\[ P_{\text{excess}} = 1000 \, \text{Pa} \]

### Step 4: Calculate the total pressure inside the bubble
The total pressure inside the bubble can be calculated by adding the atmospheric pressure, the pressure due to the liquid column, and the excess pressure:
\[ P_{\text{total}} = P_0 + P_{\text{liquid}} + P_{\text{excess}} \]

Substituting the values:
\[ P_{\text{total}} = 1.01 \times 10^5 \, \text{Pa} + 1078.1 \, \text{Pa} + 1000 \, \text{Pa} \]
\[ P_{\text{total}} = 1.0210781 \times 10^5 \, \text{Pa} \]

### Step 5: Convert the total pressure to dynes/cm²
To convert from pascals to dynes/cm²:
1 Pa = 10 dynes/cm²
\[ P_{\text{total}} = 1.0210781 \times 10^5 \, \text{Pa} \times 10 \, \text{dynes/cm}²/\text{Pa} \]
\[ P_{\text{total}} = 1.0210781 \times 10^6 \, \text{dynes/cm}² \]

### Final Answer
The total pressure inside the spherical air bubble is approximately:
\[ P_{\text{total}} \approx 1.021 \times 10^6 \, \text{dynes/cm}² \]

---