To solve the problem of finding the surface tension of the soap solution when the excess pressure inside a soap bubble is balanced by an oil column, we can follow these steps:
### Step 1: Understand the relationship between excess pressure and surface tension
The excess pressure \( P \) inside a soap bubble is given by the formula:
\[
P = \frac{4T}{r}
\]
where \( T \) is the surface tension and \( r \) is the radius of the bubble.
### Step 2: Calculate the pressure exerted by the oil column
The pressure \( P \) exerted by a column of liquid is given by:
\[
P = \rho g h
\]
where:
- \( \rho \) is the density of the oil,
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( h \) is the height of the oil column.
### Step 3: Convert the given values
1. The radius of the bubble \( r = 1 \, \text{cm} = 0.01 \, \text{m} \).
2. The density of the oil \( \rho = 0.8 \, \text{g/cm}^3 = 0.8 \times 10^{-3} \, \text{kg/cm}^3 = 800 \, \text{kg/m}^3 \).
3. The height of the oil column \( h = 2 \, \text{mm} = 0.002 \, \text{m} \).
### Step 4: Calculate the pressure from the oil column
Substituting the values into the pressure formula:
\[
P = \rho g h = (800 \, \text{kg/m}^3)(10 \, \text{m/s}^2)(0.002 \, \text{m}) = 16 \, \text{Pa}
\]
### Step 5: Set the two expressions for pressure equal to each other
Since the excess pressure inside the soap bubble is balanced by the pressure from the oil column, we can set the two expressions equal:
\[
\frac{4T}{r} = P
\]
Substituting \( P = 16 \, \text{Pa} \) and \( r = 0.01 \, \text{m} \):
\[
\frac{4T}{0.01} = 16
\]
### Step 6: Solve for surface tension \( T \)
Rearranging the equation gives:
\[
4T = 16 \times 0.01
\]
\[
4T = 0.16
\]
\[
T = \frac{0.16}{4} = 0.04 \, \text{N/m}
\]
### Step 7: Final result
Thus, the surface tension of the soap solution is:
\[
T = 0.04 \, \text{N/m} = 4 \times 10^{-2} \, \text{N/m}
\]
