When a capillary tube of radius `2.8xx10^(-4)m` is dipped vertically in alcohol , alcohol rises by `0.02m` above the outer level. Calculate the surface tension of alcohol, given density of alcohol is `790 kg-m^(-3)`.

Explanation

To calculate the surface tension of alcohol using the given parameters, we can follow these steps:

### Step 1: Identify the given values
- Radius of the capillary tube, \( r = 2.8 \times 10^{-4} \, \text{m} \)
- Height of the alcohol rise, \( h = 0.02 \, \text{m} \)
- Density of alcohol, \( \rho = 790 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 9.81 \, \text{m/s}^2 \)

### Step 2: Use the formula for capillary rise
The formula for the height of liquid rise in a capillary tube is given by:

\[ h = \frac{2 \sigma \cos \theta}{\rho g r} \]

Where:
- \( \sigma \) is the surface tension,
- \( \theta \) is the contact angle (we will assume \( \theta = 0 \) degrees, so \( \cos \theta = 1 \)),
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity,
- \( r \) is the radius of the capillary tube.

### Step 3: Rearrange the formula to solve for surface tension \( \sigma \)
Rearranging the formula to isolate \( \sigma \):

\[ \sigma = \frac{\rho g h r}{2} \]

### Step 4: Substitute the known values into the formula
Now we can substitute the known values into the rearranged formula:

\[ \sigma = \frac{790 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.02 \, \text{m} \times (2.8 \times 10^{-4} \, \text{m})}{2} \]

### Step 5: Calculate the surface tension
Now we perform the calculation step-by-step:

1. Calculate \( \rho g h \):
\[ \rho g h = 790 \times 9.81 \times 0.02 = 154.458 \, \text{N/m}^2 \]

2. Multiply by the radius \( r \):
\[ \rho g h r = 154.458 \times (2.8 \times 10^{-4}) = 0.043 \, \text{N/m} \]

3. Divide by 2 to find \( \sigma \):
\[ \sigma = \frac{0.043}{2} = 0.0215 \, \text{N/m} = 21.5 \times 10^{-3} \, \text{N/m} \]

### Final Result
Thus, the surface tension of alcohol is approximately:

\[ \sigma \approx 21.5 \times 10^{-3} \, \text{N/m} \]

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