The pressure inside a soap bubble of radius 1 cm balances a 1.5 mm column of oil of density 800 `kg m^(-3)`. Find the surface tension of the soap solution.

Explanation

To solve the problem of finding the surface tension of the soap solution in a soap bubble, we will follow these steps:

### Step 1: Understand the given data
- Radius of the soap bubble, \( r = 1 \, \text{cm} = 1 \times 10^{-2} \, \text{m} \)
- Height of the oil column, \( h = 1.5 \, \text{mm} = 1.5 \times 10^{-3} \, \text{m} \)
- Density of the oil, \( \rho = 800 \, \text{kg/m}^3 \)

### Step 2: Write the formula for excess pressure in a soap bubble
The excess pressure \( \Delta P \) inside a soap bubble is given by:
\[ \Delta P = \frac{4T}{r} \]
where \( T \) is the surface tension and \( r \) is the radius of the bubble.

### Step 3: Write the formula for pressure due to the oil column
The pressure exerted by the oil column is given by:
\[ P = \rho g h \]
where \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

### Step 4: Set the excess pressure equal to the pressure from the oil column
Since the pressure inside the soap bubble balances the pressure from the oil column, we can equate the two expressions:
\[ \frac{4T}{r} = \rho g h \]

### Step 5: Rearrange the equation to solve for surface tension \( T \)
Rearranging the equation gives:
\[ T = \frac{\rho g h r}{4} \]

### Step 6: Substitute the known values into the equation
Now, substituting the values:
- \( \rho = 800 \, \text{kg/m}^3 \)
- \( g = 9.8 \, \text{m/s}^2 \)
- \( h = 1.5 \times 10^{-3} \, \text{m} \)
- \( r = 1 \times 10^{-2} \, \text{m} \)

So, we have:
\[ T = \frac{800 \times 9.8 \times 1.5 \times 10^{-3} \times 1 \times 10^{-2}}{4} \]

### Step 7: Calculate the surface tension
Calculating the values step by step:
1. Calculate \( 800 \times 9.8 = 7840 \)
2. Calculate \( 7840 \times 1.5 \times 10^{-3} = 11.76 \)
3. Calculate \( 11.76 \times 1 \times 10^{-2} = 0.1176 \)
4. Finally, divide by 4:
\[ T = \frac{0.1176}{4} = 0.0294 \, \text{N/m} \]

### Final Result
Thus, the surface tension \( T \) of the soap solution is:
\[ T = 29.4 \times 10^{-3} \, \text{N/m} \]